TOMLAB Solver Reference

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This page is part of the TOMLAB Manual. See TOMLAB Manual.

Detailed descriptions of the TOMLAB solvers, driver routines and some utilities are given in the following sections. Also see the M-file help for each solver. All solvers except for the TOMLAB Base Module are described in separate manuals.

For a description of solvers called using the MEX-file interface, see the M-file help, e.g. for the MINOS solver minosTL.m. For more details, see the User's Guide for the particular solver.

clsSolve

Solves dense and sparse nonlinear least squares optimization problems with linear inequality and equality con- straints and simple bounds on the variables.

• clsSolve

conSolve

Solve general constrained nonlinear optimization problems.

• conSolve

cutPlane

Solve mixed integer linear programming problems (MIP).

• cutPlane

DualSolve

Solve linear programming problems when a dual feasible solution is available.

• dualSolve

expSolve

Solve exponential fitting problems for given number of terms p.

• expSolve

glbDirect

Solve box-bounded global optimization problems.

• glbDirect

glbSolve

Solve box-bounded global optimization problems.

• glbSolve

glcCluster

Solve general constrained mixed-integer global optimization problems using a hybrid algorithm.

• glcCluster

glcDirect

Solve global mixed-integer nonlinear programming problems.

• glcDirect

glcSolve

Solve general constrained mixed-integer global optimization problems.

• glcSolve

infLinSolve

Finds a linearly constrained minimax solution of a function of several variables with the use of any suitable TOMLAB solver. The decision variables may be binary or integer.

• infLinSolve

infSolve

Find a constrained minimax solution with the use of any suitable TOMLAB solver.

• infSolve

Calling Syntax

Result=infSolve(Prob,PriLev)

Description of Inputs

Prob	Problem description structure. Should be created in the cls format. infSolve uses two special fields in Prob:
	SolverInf	Name of solver used to solve the transformed problem.
		Valid choices are conSolve, nlpSolve, sTrustr and clsSolve.
		If TOMLAB /SOL is installed: minos, snopt, npopt.
	InfType	1 - constrained formulation (default).
		2 - LS penalty approach (experimental).
	The remaining fields of Prob should be defined as required by the selected subsolver.
	PriLev	Print level in infSolve.
		= 0 Silent except for error messages.
		> 0 Print summary information about problem transformation.
		Calls PrintResult with specified PriLev.
		= 2Standard output from PrintResult (default).

Description of Outputs

Result	Structure with results from optimization. Output depends on the solver used.
	The fields x_k, r_k, J_k, c_k, cJac, x_0, xState, cState, v_k are transformed back to match the original problem.
	g_k is calculated as Failed to parse (unknown function "\VAR"): {\displaystyle \VAR{J\_k} $\$^{T}\$$Failed to parse (syntax error): {\displaystyle }} $\$\cdot \$$ Failed to parse (unknown function "\VAR"): {\displaystyle \VAR{r\_k}} .
	The output in Result.Prob is the result after infSolve transformed the problem, i.e. the altered Prob structure

Description

The minimax problem is solved in infSolve by rewriting the problem as a general constrained optimization problem. One additional variable Failed to parse (unknown function "\MATHSET"): {\displaystyle $z\in \MATHSET{R}$} , stored as $\$x_{n+1}\$$ is added and the problem is rewritten as:

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \multicolumn{6}{l}{\min\limits_x z}\\\\\mbox{subject to} & x_L & \leq & (x_1,x_2,\ldots,x_n)^T & \leq & x_U \\& -\infty & \leq & z & \leq & \infty \\& b_L & \leq & A x & \leq & b_U \\& c_L & \leq & c(x) & \leq & c_U \\& -\infty & \leq & r(x) - z e & \leq & 0 \\\end{array} }

where Failed to parse (unknown function "\Rdim"): {\displaystyle $e \in \Rdim{N},\; e(i)=1 \ \forall i$} .

To handle cases where an element $\$r_{i}(x)\$$ in $\$r(x)\$$ appears in absolute value: $\$\min \max |r_{i}(x)|\$$, expand the problem with extra residuals with the opposite sign: $\$[r_{i}(x);-r_{i}(x)]\$$

Examples

minimaxDemo.m.

See Also

clsAssign.

linRatSolve

Purpose

Finds a linearly constrained solution of a function of the ratio of two linear functions with the use of any suitable TOMLAB solver. Binary and integer variables are not supported.

linRatSolve solves problems of the type:

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{ \Large (c1 x) \over (c2 x) } \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\& b_L & \leq & Ax & \leq & b_U \\ \end{array} }

where Failed to parse (unknown function "\Rdim"): {\displaystyle $c1,c2,x,x_L,x_U \in \Rdim{n}$} , Failed to parse (unknown function "\Rdim"): {\displaystyle $b_L,b_U \in \Rdim{m_1}$} and Failed to parse (unknown function "\Rdim"): {\displaystyle $A \in \Rdim{m_1 \times n}$} .

Calling Syntax

Result=linRatSolve(Prob,PriLev)

Description of Inputs

Prob	Structure Prob. Prob must be defined. Best is to use Prob = lpAssign(.....), if using the TQ format. Prob.QP.c1/c2 matrices should then be set (Prob.QP.c ignored).
PriLev	Print level in linRatSolve.
	= 0Silent except for error messages.
	> 0 Print summary information about problem transformation.
	Calls PrintResult with specified PriLev.
	= 2 Standard output from PrintResult (default).
Extra fields used in Prob:
SolverRat	Name of the TOMLAB solver. Valid names are: cplex, minos, snopt, xa and more. See SolverList('lp');
QP.c1	The numerator in the objective.
QP.c2	The denominator in the objective.
z1_L	Lower bound for z1 (default 1e-5). See description below

Description of Outputs

Result	Structure with results from optimization. Output depends on the solver used.
	The fields x_k, f_k, x_0, xState, bState, v_k are transformed back to match the original problem.
	The output in Result.Prob is the result after linRatSolve transformed the problem, i.e. the altered Prob structure

Description

The linear ratio problem is solved by linRatSolve by rewriting the problem as a linear constrained optimization problem. n+1 variables z1 and z2(2:n+1) are needed, stored as x(1:n+1). The n original variables are removed so one more variable exists in the final problem.

${\begin{array}{ccc}\\z1&=&1/(c2x)\\z2&=&xz1\\z1(c1x)&=&(c1z1x)=c1z2\\\end{array}}$

The problem then becomes:

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \multicolumn{6}{l}{\min\limits_x c1 z2}\\\\\mbox{subject to} & z1_L & \leq & z1 & \leq & \infty \\& 1 & \leq & c2 z2 & \leq & 1 \\& 0 & \leq & A z2 - z1 beq & \leq & 0 \\& -\infty & \leq & A z2 - z1 b_U & \leq & 0 \\& -\infty & \leq & - A z2 + z1 b_L & \leq & 0 \\\\& 0 & \leq & A1 z2 - z1 xeq & \leq & 0 \\& -\infty & \leq & A1 z2 - z1 x_U & \leq & 0 \\& -\infty & \leq & - A1 z2 + z1 x_L & \leq & 0 \\ \end{array} }

where Failed to parse (unknown function "\Rdim"): {\displaystyle $A1 \in \Rdim{N},\; A1=speye(N)$} .

OBSERVE the denominator c2x must always be positive. It is normally a good a idea to run the problem with both signs (multiply each side by -1).

See Also

lpAssign.

lpSimplex

Purpose

Solve general linear programming problems.

lpSimplex solves problems of the form

${\begin{array}{cccccl}\min \limits _{x}&f(x)&=&c^{T}x&&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}$

where Failed to parse (unknown function "\MATHSET"): {\displaystyle $x,x_{L},x_{U}\in \MATHSET{R} ^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $c \in \MATHSET{R}^{n}$} , Failed to parse (unknown function "\MATHSET"): {\displaystyle $A\in \MATHSET{R}^{m\times n}$} and Failed to parse (unknown function "\MATHSET"): {\displaystyle $b_{L},b_{U} \in \MATHSET{R}^{m}$} .

Calling Syntax

Result = lpSimplex(Prob) or

Result = tomRun('lpSimplex', Prob, 1);

Description of Inputs

Prob	Problem description structure. The following fields are used:
	QP.c	Constant vector.
	A	Constraint matrix for linear constraints.
	b_L	Lower bounds on the linear constraints.
	b_U	Upper bounds on the linear constraints.
	x_L	Lower bounds on the variables.
	x_U	Upper bounds on the variables.
	x_0	Starting point.
	Solver.Alg	Variable selection rule to be used:
		0: Minimum reduced cost.
		1: Bland's rule (default).
		2: Minimum reduced cost. Dantzig's rule.
	QP.B	Active set B 0 at start:
		B(i) = 1: Include variable x(i) is in basic set.
		B(i) = 0: Variable x(i) is set on its lower bound.
		B(i) = -1: Variable x(i) is set on its upper bound.
	optParam	Structure with special fields for optimization parameters, see Table 141.
		Fields used are: MaxIter, PriLev, wait, eps_f, eps_Rank, xTol and bTol.

Description of Outputs

Result	Structure with result from optimization. The following fields are changed:
	x_k	Optimal point.
	f_k	Function value at optimum.
	g_k	Gradient value at optimum, c.
	v_k	Lagrange multipliers.
	x_0	Starting point.
	f_0	Function value at start.
	xState	State of each variable, described in Table 150.
	ExitFlag	0: Optimal solution found.
		1: Maximal number of iterations reached.
		2: Unbounded feasible region.
		5: Too many active variables in given initial point.
		6: No feasible point found with Phase 1.
		10: Errors in input parameters.
		11: Illegal initial x as input.
	Inform	If ExitF lag > 0, I nf orm = ExitF lag.
	QP.B	Optimal active set. See input variable QP.B.
	Solver	Solver used.
	SolverAlgorithm	Solver algorithm used.
	Iter	Number of iterations.
	FuncEv	Number of function evaluations. Equal to Iter. ConstrEv Number of constraint evaluations. Equal to Iter.
	Prob	Problem structure used.

Description

The routine lpSimplex implements an active set strategy (Simplex method) for Linear Programming using an additional set of slack variables for the linear constraints. If the given starting point is not feasible then a Phase I objective is used until a feasible point is found.

M-files Used

ResultDef.m

See Also

qpSolve

L1Solve

Purpose

Find a constrained L1 solution of a function of several variables with the use of any suitable nonlinear TOMLAB solver.

L1Solve solves problems of the type:

Failed to parse (unknown function "\multicolumn"): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{\sum_i |r_i(x)|} \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\{} & b_L & \leq & Ax & \leq & b_U \\{} & c_L & \leq & c(x) & \leq & c_U \\ \end{array} }

where Failed to parse (unknown function "\Rdim"): {\displaystyle $x,x_L,x_U \in \Rdim{n}$} , Failed to parse (unknown function "\Rdim"): {\displaystyle $r(x) \in \Rdim{N}$} , Failed to parse (unknown function "\Rdim"): {\displaystyle $c(x),c_L,c_U\in \Rdim{m_1}$} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $b_L,b_U \in \Rdim{m_2}$} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $A\in \Rdim{m_2 \times n}$} .

Calling Syntax

Result = L1Solve(Prob,PriLev)

Description of Inputs

Prob	Problem description structure. Prob should be created in the cls constrained nonlinear format.
	L1Solve uses one special field in Prob:
	SolverL1	Name of the TOMLAB solver used to solve the augmented general nonlinear problem generated by L1Solve.
	Any other fields are passed along to the solver specified by Prob.SolverL1. In particular:
	A	Linear constraint matrix.
	b_L	Lower bounds on variables.
	b_U	Upper bounds on variables.
	c_L	Lower bounds for nonlinear constraints.
	c_U	Upper bounds for nonlinear constraints..
	x_L	Lower bounds on variables.
	x_U	Upper bounds on variables.
	x_0	Starting point.
	ConsPattern	Nonzero patterns of constraint and residual Jacobians.
	JacPattern	Prob.LS.y must have the correct residual length if JacPattern is empty but ConsPattern is not.
		L1Solve will create the new patterns for the sub-solver using the information supplied in these two fields.
	PriLev	Print level in L1Solve.
	= 0	silent except for error messages.
	> 0	print summary information about problem transformation.
		Calls PrintResult with specified PriLev.
	= 2	standard output from PrintResult.

Description of Outputs

Result

Structure with results from optimization. Fields changed depends on which solver was used for the extended problem.

The fields x_k, r_k, J_k, c_k, cJac, x_0, xState, cState, v_k, are transformed back to the format of the original L1 problem. g k is calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {J\_k} Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $^T$} Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle }} · r k. The returned problem structure Result.Prob is the result after L1Solve transformed the problem, i.e. the altered Prob structure. Description L1Solve solves the L1 problem by reformulating it as the general constrained optimization problem Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & \multicolumn{5}{l}{\sum_i (y_i+z_i) } \\ \mbox{subject to} & x_L & \leq & x & \leq & x_U \\ {} & 0 & \leq & y & \leq & \infty \\ {} & 0 & \leq & z & \leq & \infty \\ {} & b_L & \leq & Ax & \leq & b_U \\ {} & c_L & \leq & c(x) & \leq & c_U \\ {} & 0 & \leq & r(x)+y-z & \leq & 0 \\ \end{array} } A problem with N residuals is extended with 2N nonnegative variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $y,z \in \Rdim{N}$} along with N equality constraints Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle $r_i(x) + y_i - z_i = 0$} . See Also infSolve MILPSOLVE Purpose Solve mixed integer linear programming problems (MILP). MILPSOLVE solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & = & c^{T}x & \\s/t & x_{L} & \leq & x & \leq & x_{U} \\& b_{L} & \leq & Ax & \leq & b_{U} \\& & & \multicolumn{3}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c, x, x_L, x_U \in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in\MATHSET{R}^{m\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_L, b_U \in MATHSET{R}^{m}} . The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in I} , the index subset of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,...,n} are restricted to be integers. Calling Syntax Result = tomRun('MILPSOLVE',Prob, 1); or Prob = ProbCheck(Prob, 'MILPSOLVE'); Result = milpsolveTL(Prob); PrintResult(Result,1); Description of Inputs Prob Problem description structure. The following fields are used: x_L, x_U Lower and upper bounds on variables. (Must be dense). b_L, b_U Lower and upper bounds on linear constraints. (Must be dense). A Linear constraint matrix. (Sparse or dense). QP.c Linear objective function coefficients, size nx1. BIG Definition of infinity. Default is 1e30. LargeScale Defines if milpsolveTL will convert the A matrix to a sparse matrix or not. Largescale ! = 0 - sparse LargeScale = 0 - dense Default is to use the A matrix just as it is defined. PriLevOpt Specifies the printlevel that will be used by MILPSOLVE. 0 (NONE) No outputs 1 (NEUTRAL) Only some specific debug messages in debug print rou- tines are reported. 2 (CRITICAL) Only critical messages are reported. Hard errors like instability, out of memory. 3 (SEVERE) Only severe messages are reported. Errors. 4 (IMPORTANT) Only important messages are reported. Warnings and Errors. 5 (NORMAL) Normal messages are reported. 6 (DETAILED) Detailed messages are reported. Like model size, con- tinuing B&B improvements. 7 (FULL) All messages are reported. Useful for debugging purposes and small models. Default print level is 0, no outputs. PriLevOpt < 0 is interpreted as 0, and larger than 7 is interpreted as 7. MaxCPU Maximal CPU Time (in seconds) to be used by MILPSOLVE, stops with best point found. Fields used in Prob.MILPSOLVE (Structure with MILPSOLVE specific parameters) ANTI_DEGEN Binary vector. If empty, no anti-degeneracy handling is applied. If the length (i) of the vector is less than 8 elements,only the i first modes are considered. Also if i is longer than 8 elements, the elements after element8 are ignored. ANTI_DEGEN specifies if special handling must be done to reduce de- generacy/cycling while solving. Setting this flag can avoid cycling, but can also increase numerical instability. ANTIDEGEN FIXEDVARS  ! = 0 Check if there are equality slacks in the basis and try to drive them out in order to reduce chance of degeneracy in Phase 1. ANTIDEGEN COLUMNCHECK  != 0 ANTIDEGEN STALLING  != 0 ANTIDEGEN NUMFAILURE != 0 ANTIDEGEN LOSTFEAS != 0 ANTIDEGEN INFEASIBLE  != 0 ANTIDEGEN DYNAMIC != 0 ANTIDEGEN DURINGBB != 0 basis If empty or erroneous, default basis is used. Default start base is the all slack basis (the default simplex starting basis). Prob.MILPSOLVE.basis stores the basic variables. If an element is less then zero then it means on lower bound, else on upper bound. Element 0 of the array is unused. The default initial basis is bascolumn\[x\] = -x. By MILPSOLVE convention, a basic variable is always on its lower bound, meaning that basic variables is always represented with a minus sign. When a restart is done, the basis vector must be assigned a correct starting basis. BASIS_CRASH The set basiscrash function specifies which basis crash mode MILP- SOLVE will used. When no base crash is done (the default), the initial basis from which MILPSOLVE starts to solve the model is the basis containing all slack or artificial variables that is automatically associates with each constraint. When base crash is enabled, a heuristic "crash procedure" is executed before the first simplex iteration to quickly choose a basis matrix that has fewer artificial variables. This procedure tends to reduce the num- ber of iterations to optimality since a number of iterations are skipped. MILPSOLVE starts iterating from this basis until optimality. BASIS_CRASH ! = 2 - No basis crash BASIS_CRASH = 2 - Most feasible basis Default is no basis crash. BB_DEPTH_LIMIT Sets the maximum branch-and-bound depth. This value makes sense only if there are integer, semi-continuous or SOS variables in the model so that the branch-and-bound algorithm is used to solve the model. The branch-and-bound algorithm will not go deeper than this level. When BB DEPTH LIMIT i set to 0 then there is no limit to the depth. The default value is -50. A positive value means that the depth is absolute. A negative value means a relative B&B depth. The "order" of a MIP problem is defined to be 2 times the number of binary variables plus the number of SC and SOS variables. A relative value of -x results in a maximum depth of x times the order of the MIP problem. BB_FLOOR_FIRST Specifies which branch to take first in branch-and-bound algorithm. Default value is 1. BB_FLOOR_FIRST = 0 (BRANCH CEILING) Take ceiling branch first BB_FLOOR_FIRST = 1 (BRANCH FLOOR) Take floor branch first BB_FLOOR_FIRST = 2 (BRANCH AUTOMATIC) MILPSOLVE decides which branch being taken first BB_RULE Specifies the branch-and-bound rule. Default value is 0. BB_RULE = 0 (NODE FIRSTSELECT) Select lowest indexed non- integer column BB_RULE = 1 (NODE GAPSELECT) Selection based on distance from the current bounds BB_RULE = 2 (NODE RANGESELECT) Selection based on the largest current bound BB_RULE = 3 (NODE FRACTIONSELECT) Selection based on largest fractional value BB_RULE = 4 (NODE PSEUDOCOSTSELECT4) Simple, unweighted pseudo-cost of a variable BB_RULE = 5 (NODE PSEUDONONINTSELECT) This is an ex- tended pseudo-costing strategy based on minimizing the number of in- teger infeasibilities. BB RULE = 6 (NODE PSEUDORATIOSELECT) This is an extended pseudo-costing strategy based on maximizing the normal pseudo-cost divided by the number of infeasibilities. Effectively, it is similar to (the reciprocal of ) a cost/benefit ratio. BB_RULE = 7 (NODE USERSELECT) BB_RULE_ADD Additional values for the BB RULE. BB RULE is a vector. If the length i of the vector is less than 10 elements, only the i first modes are considered. Also if i is longer than 10 elements, the elements after element 10 is ignored. BB_RULE ADD(1) ! = 0 (NODE WEIGHTREVERSEMODE) BB_RULE ADD(2)  ! = 0 (NODE BRANCHREVERSEMODE) In case when get bb floorfirst is BRANCH AUTOMATIC, select the opposite direction (lower/upper branch) that BRANCH AUTOMATIC had cho- sen. BB_RULE_ADD(3) ! = 0 (NODE GREEDYMODE) BB_RULE_ADD(4) ! = 0 (NODE PSEUDOCOSTMODE) BB_RULE_ADD(5) ! = 0 (NODE DEPTHFIRSTMODE) Select the node that has already been selected before the number of times BB_RULE_ADD(6) ! = 0 (NODE RANDOMIZEMODE) BB_RULE_ADD(7) ! = 0 (NODE DYNAMICMODE) When NODE DEPTHFIRSTMODE is selected, switch off this mode when a first solution is found. BB_RULE_ADD(8) ! = 0 (NODE RESTARTMODE) BB_RULE_ADD(9) ! = 0 (NODE BREADTHFIRSTMODE) Select the node that has been selected before the fewest number of times or not at all BB RULE ADD(10) ! = 0 (NODE AUTOORDER) BFP Defines which Basis Factorization Package that will be used by MILP- SOLVE. BFP = 0 : LUSOL BFP = 1 : built in etaPHI from MILPSOLVE v3.2 BFP = 2 : Additional etaPHI BFP = 3 : GLPK Default BFP is LUSOL. BREAK_AT_FIRST Specifies if the branch-and-bound algorithm stops at the first found so- lution (BREAK_AT_FIRST != 0) or not (BREAK_AT_FIRST = 0). De- fault is not to stop at the first found solution. BREAK_AT_VALUE' Specifies if the branch-and-bound algorithm stops when the object value is better than a given value. The default value is (-) infinity. EPAGAP Specifies the absolute MIP gap tolerance for the branch and bound algo- rithm. This tolerance is the difference between the best-found solution yet and the current solution. If the difference is smaller than this toler- ance then the solution (and all the sub-solutions) is rejected. The default value is 1e-9. EPGAP Specifies the relative MIP gap tolerance for the branch and bound algo- rithm. The default value is 1e-9. EPSB Specifies the value that is used as a tolerance for the Right Hand Side (RHS) to determine whether a value should be considered as 0. The default epsb value is 1.0e-10 EPSD Specifies the value that is used as a tolerance for reduced costs to deter- mine whether a value should be considered as 0. The default epsd value is 1e-9. If EPSD is empty, EPSD is read from Prob.optParam.eps f. EPSEL Specifies the value that is used as a tolerance for rounding values to zero. The default epsel value is 1e-12. EPSINT Specifies the tolerance that is used to determine whether a floating- point number is in fact an integer. The default value for epsint is 1e-7. Changing this tolerance value can result in faster solving times, but the solution is less integer. EPSPERTURB Specifies the value that is used as perturbation scalar for degenerative problems. The default epsperturb value is 1e-5. EPSPIVOT Specifies the value that is used as a tolerance pivot element to determine whether a value should be considered as 0. The default epspivot value is 2e-7 IMPROVEMENT_LEVEL Specifies the iterative improvement level. IMPROVEMENT LEVEL = 0 (IMPROVE NONE) improve none IMPROVEMENT LEVEL = 1 (IMPROVE FTRAN) improve FTRAN IMPROVEMENT LEVEL = 2 (IMPROVE BTRAN) improve BTRAN IMPROVEMENT LEVEL = 3 (IMPROVE SOLVE) improve FTRAN + BTRAN. IMPROVEMENT LEVEL = 4 (IMPROVE INVERSE) triggers automatic inverse accuracy control in the dual simplex, and when an error gap is exceeded the basis is reinverted Choice 1,2,3 should not be used with MILPSOLVE 5.1.1.3, because of problems with the solver. Default is 0. LoadFile File that contains the model. If LoadFile is a nonempty string which corresponds to actual file, then the model is read from this file rather than from the Prob struct. LoadMode 1 - LP - MILPSOLVE LP format 2 - MPS - MPS format 3 - FMPS - Free MPS format A default value for this field does not exist. Both LoadFile and Load- Mode must be set if a problem will be loaded. If there is something wrong with LoadMode or LoadFile, an error mes- sage will be printed and MILPSOLVE will be terminated. Leave Load- Mode and LoadFile empty if the problem not will be loaded from file. LogFile Name of file to print MILPSOLVE log on. MAXIMIZE If MAXIMIZE ! = 0, MILPSOLVE is set to maximize the objective function, default is to minimize. MAX_PIVOT Sets the maximum number of pivots between a re-inversion of the matrix. Default is 42. NEG_RANGE Specifies the negative value below which variables are split into a negative and a positive part. This value must always be zero or negative. If a positive value is specified, then 0 is taken. The default value is -1e6. PRESOLVE Vector containing possible presolve options. If the length i of the vector is less than 7 elements, only the i first modes are considered. Also if i is longer than 7 elements, the elements after element 7 is ignored. PRESOLVE(1) ! = 0 (PRESOLVE ROWS) Presolve rows PRESOLVE(2) ! = 0 (PRESOLVE COLS) Presolve columns PRESOLVE(3) ! = 0 (PRESOLVE LINDEP) Eliminate linearly depen- dent rows PRESOLVE(4) ! = 0 (PRESOLVE SOS) Convert constraints to SOSes (only SOS1 handled) PRESOLVE(5) ! = 0 (PRESOLVE REDUCEMIP) If the phase 1 solu- tion process finds that a constraint is redundant then this constraint is deleted. PRESOLVE(6) ! = 0 (PRESOLVE DUALS) Calculate duals PRESOLVE(7) ! = 0 (PRESOLVE SENSDUALS) Calculate sensitivity if there are integer variables Default is not to do any presolve. PRICING_RULE The pricing rule can be one of the following rules. PRICING_RULE = 0 Select first (PRICER_FIRSTINDEX) PRICING_RULE = 1 Select according to Dantzig (PRICER_DANTZIG) PRICING_RULE = 2 Devex pricing from Paula Harris (PRICER_DEVEX) PRICING_RULE = 3 Steepest Edge (PRICER STEEPESTEDGE) PRICING_MODE Additional pricing settings, any combination of the modes below. This is a binary vector. If the length i of the vector is less than 7 elements, only the i first modes are considered. Also if i is longer than 7 elements, the elements after element 7 is ignored. PRICE_PRIMALFALLBACK ! = 0 In case of Steepest Edge, fall back to DEVEX in primal. PRICE_MULTIPLE ! = 0 Preliminary implementation of the multiple pricing scheme. This means that attractive candidate entering columns from one iteration may be used in the subsequent iteration, avoiding full updating of reduced costs. In the current implementation, MILPSOLVE only reuses the 2nd best entering column alternative. PRICE_PARTIAL ! = 0 Enable partial pricing PRICE_ADAPTIVE ! = 0 Temporarily use First Index if cycling is detected PRICE_RANDOMIZE  ! = 0 Adds a small randomization effect to the selected pricer PRICE_LOOPLEFT  ! = 0 Scan entering/leaving columns left rather than right PRICE_LOOPALTERNATE ! = 0 Scan entering/leaving columns alternatingly left/right Default basis is PRICER DEVEX combined with PRICE ADAPTIVE. sa Struct containing information of the sensitivity analysis (SA) MILPSOLVE will perform. sa.obj =! 0 Perform sensitivity analysis on the objective function sa.obj = 0 Do not perform sensitivity analysis on the objective function sa.rhs =! 0 Perform sensitivity analysis on the right hand sides. sa.rhs = 0 Do not perform sensitivity analysis on the right hand sides. SaveFileAfter Name of a file to save the MILPSOLVE object after presolve.The name must be of type string (char), Example: Prob.MILPSOLVE.SaveFileAfter = 'save2' If the type is not char SaveFileBefore is set to save2.\[file extension\]. SaveFileBefore Name of a file to save the MILPSOLVE object before presolve. The name must be of type string (char), Example: Prob.MILPSOLVE.SaveFileBefore = 'save1'. If the type is not char SaveFileBefore is set to save1.\[file extension\]. SaveMode 1 - LP - MILPSOLVE LP format 2 - MPS - MPS format 3 - FMPS - Free MPS format If empty, the default format LP is used. SCALE_LIMIT Sets the relative scaling convergence criterion to the absolute value of SCALE_LIMIT for the active scaling mode. The integer part of SCALE_LIMIT specifies the maximum number of iterations. Default is 5. SCALING_ALG Specifies which scaling algorithm will be used by MILPSOLVE. 0 No scaling algorithm 1 (SCALE EXTREME) Scale to convergence using largest absolute value 2 (SCALE RANGE) Scale based on the simple numerical range 3 (SCALE MEAN) Numerical range-based scaling 4 (SCALE GEOMETRIC) Geometric scaling 7 (SCALE CURTISREID) Curtis-reid scaling Default is 0, no scaling algorithm. SCALING_ADD Vector containing possible additional scaling parameters. If the length (i) of the vector is less than 7 elements, only the i first modes are considered. Also if i is longer than 7 elements, the elements after element 7 is ignored. SCALING ADD ! = 0 (SCALE_QUADRATIC) SCALING ADD ! = 0 (SCALE_LOGARITHMIC) Scale to convergence using logarithmic mean of all values SCALING_ADD  ! = 0 (SCALE_USERWEIGHT) User can specify scalars SCALING_ADD ! = 0 (SCALE_POWER2) also do Power scaling SCALING_ADD  ! = 0 (SCALE_EQUILIBRATE) Make sure that no scaled number is above 1 SCALING_ADD ! = 0 (SCALE_INTEGERS) Also scaling integer vari- ables SCALING_ADD ! = 0 (SCALE_DYNUPDATE) Dynamic update Default is 0, no additional mode. Settings SCALE_DYNUPDATE is a way to make sure that scaling factors are recomputed. In that case, the scaling factors are recomputed also when a restart is done. SIMPLEX_TYPE Sets the desired combination of primal and dual simplex algorithms. 5 (SIMPLEX_PRIMAL_PRIMAL) Phase1 Primal, Phase2 Primal 6 (SIMPLEX_DUAL_PRIMAL) Phase1 Dual, Phase2 Primal 9 (SIMPLEX_PRIMAL_DUAL) Phase1 Primal, Phase2 Dual 10 (SIMPLEX_DUAL_DUAL) Phase1 Dual, Phase2 Dual Default is SIMPLEX DUAL PRIMAL (6). SOLUTION_LIMIT Sets the solution number that will be returned. This value is only consid- ered if there are integer, semi-continuous or SOS variables in the model so that the branch-and-bound algorithm is used. If there are more solutions with the same objective value, then this number specifies which solution must be returned. Default is 1. sos List of structs containing data about Special Ordered Sets (SOS). See below for further description. Fields used in Prob.MIP (Structure with MIP specific parameters) IntVars Defines which variables are integers, of general type I or binary type B Variable indices should be in the range \[1,...,n\]. IntVars is a logical vector ==> x(f ind(I ntV ars > 0)) are integers IntVars is a vector of indices ==> x(I ntV ars) are integers (if \[\], then no integers of type I or B are defined) variables with x L=0 and x U=1, is are set to binary. It is possible to combine integer and semi-continuous type to obtain the semi-integer type. fIP This parameter specifies the initial "at least better than" guess for objective function. This is only used in the branch-and-bound algorithm when integer variables exist in the model. All solutions with a worse objective value than this value are immediately rejected. The default is infinity. SC A vector with indices for variables of type semi-continuous (SC), a logical vector or a scalar (see MIP.IntVars). A semi-continuous variable i takes either the value 0 or some value in the range \[x L(i), x U(i)\]. It is possible to combine integer and semi-continuous type to obtain the semi-integer type. sos1 List of structures defining the Special Ordered Sets of Order One (SOS1). For SOS1 set k, sos1(k).var is a vector of indices for variables of type SOS1 in set k, sos1(k).row is the priority of SOS k in the set of SOS1 and sos1(k).weight is a vector of the same length as sos1(k).var and it describes the order MILPSOLVE will weight the variables in SOS1 set k. a low number of a row and a weight means high priority. sos2 List of n structures defining the Special Ordered Sets (SOS) of Order Two (SOS2). (see MIP.sos1) Description of Outputs Result Structure with result from optimization. The following fields are changed: x_k Optimal solution (or some other solution if optimum could not been found) f_k Optimal objective value. v_k \[rc; duals\]. If Reduced cost and dual variables are not available, then v_k is empty. ExitFlag TOMLAB information parameter. 0 = Optimal solution found. 1 = Suboptimal solution or user abort. 2 = Unbounded solution. 3 = Numerical failure. 4 = Infeasible model. 10 = Out of memory. 11 = Branch and bound stopped. ExitText Status text from MILPSOLVE. Inform MILPSOLVE information parameter. -2 = Out of memory. 0 = Optimal solution found. 1 = Suboptimal solution. 2 = Infeasible model. 3 = Unbounded solution. 4 = Degenerate solution. 5 = Numerical failure. 6 = User abort. 7 = Timeout. 10 = Branch and bound failed. 11 = Branch and bound stopped. 12 = Feasible branch and bound solution. 13 = No feasible branch and bound solution. Other = Unknown status. Iter The total number of nodes processed in the branch-and-bound algorithm. Is only applicable if the model contains integer variables. In the case of an LP model Result.Iter contains the number of iterations. This is however not documented. MinorIter The total number of Branch-and-bound iterations. When the problem is LP, MinorIter equals Result.Iter MILPSOLVE.basis Optimal basis, on the format described above under Prob.MILPSOLVE.basis. MILPSOLVE.MaxLevel The deepest Branch-and-bound level of the last solution. Is only applicable if the model contains integer variables. MILPSOLVE.sa.objStatus 1 successful 0 SA not requested -1 Error: error from MILPSOLVE -3 no SA available MILPSOLVE.sa.ObjLower An array that will contain the values of the lower limits on the objective function. MILPSOLVE.sa.ObjUpper An array that will contain the values of the upper limits on the objective function. MILPSOLVE.sa.RhsStatus see MILPSOLVE.sa.objStatus. MILPSOLVE.sa.RhsLower An array that will contain the values of the lower limits on the RHS. MILPSOLVE.sa.RhsUpper An array that will contain the values of the upper limits on the RHS. xState State of each variable 0 - free variable, 1 - variable on lower bound, 2 - variable on upper bound, 3 - variable is fixed, lower bound = upper bound. bState State of each linear constraint 0 - Inactive constraint, 1 - Linear constraint on lower bound, 2 - Linear constraint on upper bound, 3 - Linear equality constraint. minlpSolve Purpose Branch & Bound algorithm for Mixed-Integer Nonlinear Programming (MINLP) with convex or nonconvex sub problems using NLP relaxation (Formulated as minlp-IP). The parameter Convex (see below) determines if to assume the NLP subproblems are convex or not. minlpSolve depends on a suitable NLP solver. minlpSolve solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) \\s/t & x_{L} & \leq & x & \leq & x_{U} \\& b_{L} & \leq & Ax & \leq & b_{U} \\& c_{L} & \leq & c(x) & \leq & c_{U} \\& & & \multicolumn{3}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \MATHSET{R}^{m_{2}\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} . The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in I} , the index subset of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,...,n} are restricted to be integers. Calling Syntax Result = tomRun('minlpSolve',Prob,...) Description of Inputs Prob Problem description structure. The following fields are used: x_L Lower bounds on x. x_U Upper bounds on x. A The linear constraint matrix. b_L Lower bounds on linear constraints. b_U Upper bounds on linear constraints. c_L Lower bounds on nonlinear constraints. c_U Upper bounds on nonlinear constraints. x_0 Starting point. Convex If Convex==1, assume NLP problems are convex, and only one local NLP solver call is used at each node. If Convex==0 (Default), multiMin is used to do many calls to a local solver to determine the global minima at each node. The global minimum with most components integer valued is chosen. MaxCPU Maximal CPU Time (in seconds) to be used by minlpSolve, stops with best point found PriLev Print level in minlpSolve (default 1). Also see optParam.IterPrint PriLevOpt Print level in sub solvers (SNOPT and other NLP solvers): =0 No output; >0 Convergence results >1 Output every iteration, >2 Output each step in the NLP alg For other NLP solvers, see the documentation for the solver WarmStart If true, >0, minlpSolve reads the output from the last run from Prob.minlpSolve, if it exists. If it doesn't exist, minlpSolve attempts to open and read warm start data from mat-file minlpSolveSave.mat. minlpSolve uses the warm start information to continue from the last run. The mat-file minlp- SolveSave.mat is saved every Prob.MIP.SaveFreq iteration. SolverNLP Name of the solver used for NLP subproblems. If empty, the default solver is found calling GetSolver('con',1); If TOMLAB /SOL installed, SNOPT is the default solver. If SolverNLP is a SOL solver (SNOPT, MINOS or NPSOL), the SOL.optPar and SOL.PrintFile is used: See help minosTL.m, npsolTL.m or snoptTL.m for how to set these parameters RandState If Convex == 0, RandState is sent to multiMin to initialize the random gen- erator. RandState is used as follows: If > 0, rand('state',RandState) is set to initialize the pseudo-random generator if < 0, rand('state',sum(100\*clock)) is set to give a new set of random values each run if RandState == 0, rand('state',) is not called. Default RandState = -1 MIP Structure in Prob, Prob.MIP. Defines integer optimization parameters. Fields used: IntVars If empty, all variables are assumed non-integer. If islogical(IntVars) (=all el- ements are 0/1), then 1 = integer variable, 0 = continuous variable. If any element >1, IntVars is the indices for integer variables. VarWeight Weight for each variable in the variable selection phase. A lower value gives higher priority. Setting Prob.MIP.VarWeight might improve convergence. DualGap minlpSolve stops if the duality gap is less than DualGap. DualGap = 1, stop at first integer solution e.g. DualGap = 0.01, stop if solution < 1% from optimal solution. fIP An upper bound on the IP value wanted. Makes it possible to cut branches and avoid node computations. Used even if xIP not given. xIP The x-values giving the fIP value, if a solution (xIP,fIP) is known. NodeSel Node selection method in branch and bound = 0 Depth First. Priority on nodes with more integer components = 1 Breadth First. Priority on nodes with more integer components = 2 Depth First. When integer solution found, use NodeSel = 1 (default) = 3 Pure LIFO (Last in, first out) Depth First = 4 Pure FIFO (First in, first out) Breadth First = 5 Pure LIFO Depth First. When integer solution found, use NodeSel 4 VarSel Variable selection method in branch and bound: = 1 Use variable with most fractional value = 2 Use gradient and distance to nearest integer value KNAPSACK If = 1, use a knapsack heuristic. Default 0. ROUNDH If = 1, use a rounding heuristic. Default 0. SaveFreq Warm start info saved on minlpSolveSave.mat every SaveFreq iteration (default -1, i.e. no warm start info is saved) optParam Structure in Prob. Fields used in Prob.optParam, also in sub solvers: MaxIter Maximal number of iterations, default 10000 IterPrint Print short information each iteration (PriLev > 0 ==> IterPrint = 1). Iter- ation number: Depth in tree (symbol L\[\] - empty list, symbol Gap - Dual Gap convergence), fNLP (Optimal f(x) current node), fIPMin (Best integer feasi- ble f(x) found), LowBnd (Lower bound on optimal integer feasible f(x)), Dual Gap in absolut value and percent, The length of the node list L, -L-, The Inform and ExitFlag the solver returned at the current node, FuEv (Number of function evaluations used by solver at current node), date/time stamp. bTol Linear constraint violation convergence tolerance. cTol Constraint violation convergence tolerance. Description of Outputs Result Structure with result from optimization. The following fields are changed: Iter Number of iterations. ExitFlag 0: Global optimal solution found, or integer solution with duality gap less than user tolerance. 1: Maximal number of iterations reached. 2: Empty feasible set, no integer solution found. 4: No feasible point found running NLP relaxation. 5: Illegal x 0 found in NLP relaxation. 99: Maximal CPU Time used (cputime > Prob.MaxCPU). Inform Code telling type of convergence, returned from subsolver. ExitText Text string giving ExitFlag and Inform information. DualGap Relative duality gap, max(0,fIPMin-fLB)/-fIPMin-, if fIPMin =0; max(0,fIPMin-fLB) if fIPMin == 0. If fIPMin =0: Scale with 100, 100\*Dual- Gap, to get the percentage duality gap. For absolute value duality gap: scale with fIPMin, fIPMin \* DualGap x_k Solution. v_k Lagrange multipliers. Bounds, Linear and Nonlinear Constraints, n + mLin + mNonLin. f_k Function value at optimum. g_k Gradient vector at optimum. x_0 Starting point x 0. f_0 Function value at start. c_k Constraint values at optimum. cJac Constraint derivative values at optimum. xState State of each variable, described in Table 150. bState State of each constraint, described in Table 151. cState State of each general constraint, described in Table 152. Solver Solver used ('mipSolve'). SolverAlgorithm Text description of solver algorithm used. Prob Problem structure used. minlpSolve A structure with warm start information. Use with WarmDefGLOBAL, see example below. Description To make a restart (warm start), just set the warm start flag, and call minlpSolve once again: Prob.WarmStart = 1; Result = tomRun('minlpSolve', Prob, 2); minlpSolve will read warm start information from the minlpSolveSave.mat file. Another warm start (with same MaxFunc) is made by just calling tomRun again: Result = tomRun('minlpSolve', Prob, 2); To make a restart from the warm start information in the Result structure, make a call to WarmDefGLOBAL before calling minlpSolve. WarmDefGLOBAL moves information from the Result structure to the Prob structure and sets the warm start flag, Prob.WarmStart = 1; Prob = WarmDefGLOBAL('minlpSolve', Prob, Result); where Result is the result structure returned by the previous run. A warm start (with same MaxIter) is done by just calling tomRun again: Result = tomRun('minlpSolve', Prob, 2); To make another warm start with new MaxIter 100, say, redefine MaxIter as: Prob.optParam.MaxIter = 100; Then repeat the two lines: Prob = WarmDefGLOBAL('minlpSolve', Prob, Result); Result = tomRun('minlpSolve', Prob, 2); mipSolve Purpose Solve mixed integer linear programming problems (MIP). mipSolve solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & = & c^{T}x & \\s/t & x_{L} & \leq & x & \leq & x_{U} \\& b_{L} & \leq & Ax & \leq & b_{U} \\ & & & \multicolumn{3}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c, x, x_L, x_U \in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \MATHSET{R}^{m\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_L, b_U \in\MATHSET{R}^{m}} .The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in I} , the index subset of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,...,n} are restricted to beintegers. Starting with TOMLAB version 4.0, mipSolve accepts upper and lower bounds on the linear constraints like most other TOMLAB solvers. Thus it is no longer necessary to use slack variables to handle inequality constraints. Calling Syntax Result = tomRun('mipSolve',Prob,...) Description of Inputs Prob Problem description structure. The following fields are used: c The vector c in Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^{T}x} . A Constraint matrix for linear constraints. b_L Lower bounds on the linear constraints. If empty, Prob.b U is used. b_U Upper bounds on the linear constraints. x_L Lower bounds on the variables. x_U Upper bounds on the variables. x_0 Starting point. MaxCPU Maximal CPU Time (in seconds) to be used by mipSolve, stops with best point found. QP.B Active set B 0 at start: B(i) = 1: Include variable x(i) is in basic set. B(i) = 0: Variable x(i) is set on its lower bound. B(i) = -1: Variable x(i) is set on its upper bound. SolverLP Name of solver used for initial LP subproblem. Default solver is used if empty, see GetSolver.m and tomSolve.m. SolverDLP Name of solver used for the dual LP subproblems. Default solver is used if empty, see GetSolver.m and tomSolve.m. PriLevOpt Print level in lpSimplex and DualSolve: 0: No output; > 0: Convergence result; > 1: Output every iteration; > 2: Output each step in simplex algorithm. PriLev Print level in mipSolve. SOL.optPar Parameters for the SOL solvers, if they are used as subsolvers. SOL.PrintFile Name of print file for SOL solvers, if they are used as subsolvers. MIP Structure with fields for integer optimization The following fields are used: IntVars The set of integer variables. If empty, all variables are assumed non-integer (LP problem) VarWeight Weight for each variable in the variable selection phase. A lower value gives higher priority. Setting Prob.MIP.VarWeight = Prob.c improves convergence for knapsack problems. DualGap mipSolve stops if the duality gap is less than DualGap. To stop at the first found integer solution, set DualGap =1. For example, DualGap = 0.01 makes the algorithm stop if the solution is < 1% from the optimal solution. fIP An upper bound on the IP value wanted. Makes it possible to cut branches and avoid node computations. xIP The x-value giving the fIP value. KNAPSACK If solving a knapsack problem, set to true (1) to use a knapsack heuristic. optParam Structure with special fields for optimization parameters, see Table 141.Fields used are: IterPrint, MaxIter, PriLev, wait, eps f and eps Rank. Solver Structure with fields for algorithm choices: Alg Node selection method: 0: Depth first 1: Breadth first 2: Depth first. When integer solution found, switch to Breadth. method Rule to select new variables in DualSolve/lpSimplex: 0: Minimum reduced cost, sort variables increasing. (Default) 1: Bland's rule (default). 2: Minimum reduced cost. Dantzig's rule. Description of Outputs Result Structure with result from optimization. The following fields are changed: x_k Optimal point. f_k Function value at optimum. g_k Gradient value at optimum, c. v_k Lagrange multipliers, \[Constraints + lower + upper bounds\]. x_0 Starting point. f_0 Function value at start. xState State of each variable, described in Table 150, page 239. Inform If ExitF lag > 0, I nf orm = ExitF lag. QP.B Optimal active set. See input variable QP.B. QP.y Dual parameters y (also part of Result.v_k. p_dx Search steps in x. alphaV Step lengths for each search step. ExitFlag 0: OK. 1: Maximal number of iterations reached. 2: Empty feasible set, no integer solution found. 3: Rank problems. Can not find any solution point. 4: No feasible point found running LP relaxation. 5: Illegalx_0 found in LP relaxation. 99: Maximal CPU Time used (cputime > Prob.MaxCPU). Iter Number of iterations. Solver Solver used ('mipSolve'). SolverAlgorithm Text description of solver algorithm used. Prob Problem structure used. Description The routine mipSolve is an implementation of a branch and bound algorithm from Nemhauser and Wolsey \[58, chap. 8.2\]. mipSolve normally uses the linear programming routines lpSimplex and DualSolve to solve relaxed subproblems. mipSolve calls the general interface routines SolveLP and SolveDLP. By changing the setting of the structure fields Prob.Solver.SolverLP and Prob.Solver.SolverDLP, different sub-solvers are possible to use, see the help for the interface routines. Algorithm See \[58, chap. 8.2\] and the code in mipSolve.m. Examples See exip39, exknap, expkorv. M-files Used lpSimplex.m, DualSolve.m, GetSolver.m, tomSolve.m See Also cutplane, balas, SolveLP, SolveDLP multiMin Purpose multiMin solves general constrained mixed-integer global optimization problems. It tries to find all local minima by a multi-start method using a suitable nonlinear programming subsolver. multiMin solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & & & & \\ s/t & x_{L} & \leq & x & \leq & x_{U} \\ & b_{L} & \leq & Ax & \leq & b_{U} \\ & c_{L} & \leq & c(x) & \leq & c_{U} \\ & & & \multicolumn{3}{c}{x_i \in \MATHSET{N} \; \forall i \in I} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \MATHSET{R}^{m_{2}\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} . The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in I} , the index subset of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,...,n} are restricted to be integers. The integer components of every x_0 is rounded to the nearest integer value inside simple bounds, and these components are fixed during the nonlinear local search. If generating random points and there are linear constraints, multiMin checks feasibility with respect to the linear constraints, and for each initial point tries 100 times to get linear feasibility before accepting the initial point. Calling Syntax Result = multiMin(Prob, xInit) Result = tomRun('multiMin', Prob, PriLev) (driver call) Description of Inputs 3 = tomRun (PrintResult) output from every optimization, print level 1. Prob Problem description structure. The following fields are used: xInit Either, 1x1 number - The number of random initial points, default 10\*Prob.N dxm matrix - Every column is an initial point (of length d=Prob.N). May also be set as Prob.xInit. fCut If initial f(x_0) > fCut, no local optimization is done. WarmStart If true, > 0, multiMin assumes the field multiMin defined with the output from a previous run on the same problem. See the Output fields of Result.multiMin. Use WarmDefGLOBAL to set the correct fields in the Prob structure. Nec- essary fields are fOpt and xOpt. If any of the other fields are missing, the corresponding variables are initialized to 0. These other fields are: localTry, Iter, FuncEv, GradEv, HessEv, ConstrEv Inform (is set to zeros(length(fOpt,1) if not defined). In WarmDefGLOBAL the Result structure for the optimal run will be fed back to multiMin as Prob.multiMin.ResOpt In case this field is not defined, and no better point is found during the runs, a special call to the localSolver is used to generate a proper Result structure. RandState If WarmStart and isscalar(xInit), RandState is used as follows: If > 0, rand('state',RandState) is set to initialize the pseudo-random generator if < 0, rand('state',sum(100\*clock)) is set to give a new set of random values each run if RandState == 0, rand('state',) is not called Default RandState = -1 xEqTol Tolerance to test if new point x_k already defined as optimum: norm(xk - xOpt(:, i)) <= xEqTol * max(1, norm(xk)) If test fulfilled x_k is assumed to be too close to xOpt(:,i) Default xEqTol = 1E-5 x_L Lower bounds for each element in x. If generating random points, -inf elements of x L are set to -10000. x_U Upper bounds for each element in x. If generating random points, inf elements of x U are set to 10000. A Constraint matrix for linear constraints. b_L Lower bounds on the linear constraints. b_U Upper bounds on the linear constraints. c_L Lower bounds on the general constraints. c_U Upper bounds on the general constraints. PriLevOpt 0 = silent. 1 = Display one row for each unique local minima found. The minima are sorted, lowest value first (possibly the global minima) The following 4 information values are displayed: 1. Order \# 2. Function value f(x) at local minima 3. Point x at local minima. Only up to 10 values are displayed 4. Inform value returned from local Solver (normally 0) 2 = One row of output from each multiMin local optimization trial The following 6 information values are displayed: 1. Step \# 2. Text Old (previously found local minima), FAIL (solver failed to verify local minima) or blank (solver success, new local minima found) 3. Inform value from local solver 4. f(x_0) - function value f(x_0) for initial x_0 5. f(x) - best f(x) value found in this local search 6. x - point for which best f(x) value was found in this local search. Only up to 10 values are displayed. 4 = tomRun (PrintResult) output from every optimization, print level 2. For constrained problems output of sum(-constr-) and information if optimal point was accepted w.r.t. feasibility. 5 = tomRun (PrintResult) output from every optimization, print level 3. GO Structure in Prob, Prob.GO. Fields used: localSolver The local solver used to run all local optimizations. Default is the license dependent output of GetSolver('con',1,0). optParam Defines optimization parameters. Fields used: fGoal Goal for function value f(x), if empty not used. If fGoal is reached, no further local optimizations are done. eps_f Relative accuracy for function value, fTol == eps_f. Stop if abs(f-fGoal) <= abs(fGoal) \* fTol , if fGoal = 0. Stop if abs(f-fGoal) <= fTol , if fGoal ==0. bTol The local solver used to run all local optimizations. Default is the license dependent output of GetSolver('con',1,0). MIP.IntVars Structure in Prob, Prob.MIP. If empty, all variables are assumed non-integer (LP problem). If length(I ntV ars) > 1 ==> length(I ntV ars) == length(c) should hold. Then I ntV ars(i) == 1 ==> x(i) integer. I ntV ars(i) == 0 ==> x(i) real. If length(I ntV ars) < n, IntVars is assumed to be a set of indices. It is advised to number the integer values as the first variables, before the continuous. The tree search will then be done more efficiently. varargin Other parameters directly sent to low level routines. Description of Outputs Result Structure with result from optimization. The following fields are changed: The following fields in Result are changed by multiMin before return: ExitFlag = 0 normal output, of if fGoal set and found. = 1 A solution reaching the user defined fGoal was not found The Solver, SolverAlgorithm and ExitText fields are also reset. A special field in Result is also returned, Result.multiMin: xOpt Prob.N x k matrix with k distinct local optima, the test being norm(xk -xOpt(:, i)) <= xEqT ol * max(1, norm(xk )) that if fulfilled assumes x k to be to closeto xOpt(:,i) fOpt The k function values in the local optima xOpt(:,i),i=1,...,k. Inform The Inform value returned by the local solver when finding each of the local optima xOpt(:,i); i=1,...,k. The Inform value can be used to judge the validity of the local minimum reported. localTry Total number of local searches. Iter Total number of iterations. FuncEv Total number of function evaluations. GradEv Total number of gradient evaluations. HessEv Total number of Hessian evaluations. ConstrEv Total number of constraint function evaluations. ExitText Text string giving ExitFlag and Inform information. multiMINLP Purpose multiMINLP solves general constrained mixed-integer global nonlinear optimization problems. It is aimed for problems where the number of integer combinations nMax is huge and relaxation of the integer constraint is possible. If no integer variables, multiMINLP calls multiMin. If nMax <= min(Prob.optParam.MaxFunc,5000), glcDirect is used. Otherwise, multiMINLP first finds a set M of local minima calling multiMin with no integer restriction on any variable. The best local minimum is selected. glcDirect is called to find the best integer feasible solution fIP in a small area (< +- 2 units) around the best local minimum found. The other local minima are pruned, if fOpt(i) > fIP, no integer feasible solution could be found close to this local minimum i. The area close to the remaining candidate local minima are searched one by one by calling glcDirect to find any fIPi < fIP. multiMINLP solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) \\ s/t & x_{L} & \leq & x & \leq & x_{U} \\ & b_{L} & \leq & Ax & \leq & b_{U} \\ & c_{L} & \leq & c(x) & \leq & c_{U} \\ & & & \multicolumn{3}{l}{x_{j} \in \MATHSET{N} \ \ \forall j \in $I$} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \MATHSET{R}^{m_{2}\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} . The variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in I} , the index subset of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,...,n} are restricted to be integers. Calling Syntax Result = tomRun('multiMINLP',Prob,...) Description of Inputs Prob Problem description structure. The following fields are used: Prob is a structure, defined as to solve a standard MINLP problem. The Prob structure is fed to the localSolver. See e.g. minlpAssign. See multiMin and glcDirect for input to the subsolvers e.g. Prob.xInit is used in multiMin (and fCut, RandState, xEQTol). x_L Lower bounds for each element in x. If generating random points, -inf elements of x L are set to min(-L,xMin,x U-L) xMin is the minimum of the finite x L values. x_U Upper bounds for each element in x. If generating random points, inf elements of x U are set to max(L,xMax,x L+L) xMax is the maximum of the finite x U values. L is 100 for nonlinear least squares, otherwise 1000. b_L Lower bounds on linear constraints. b_U Upper bounds on linear constraints. A The linear constraint matrix. c_L Lower bounds on nonlinear constraints. c_U Upper bounds on nonlinear constraints. PriLev Print Level: 0 = Silent 1 = Display 2 summary rows 2 = Display some extra summary rows 5 = Print level 1 in tomRun call 6 = Print level 2 in tomRun call 7 = Print level 3 in tomRun call xInit Used in multiMin. See help for multiMin. GO.localSolver The local solver used to run all local optimizations. Default is the license dependent output of GetSolver('con',1,0). optParam Structure in Prob, Prob.optParam. Defines optimization parameters. Fields used: MaxFunc Max number of function evaluations in each subproblem fGoal Goal for function value f(x), if empty not used. If fGoal is reached, no further local optimizations are done eps_f Relative accuracy for function value, fTol == eps_f. Stop if abs(f-fGoal) <= abs(fGoal) \* fTol , if fGoal = 0. Stop if abs(f-fGoal) <= fTol , if fGoal ==0. Default 1e-8. bTol Linear constraint feasibility tolerance. Default 1e-6 cTol Nonlinear constraint feasibility tolerance. Default 1e-6 MIP Structure in Prob, Prob.MIP. Defines integer optimization parameters. Fields used: IntVars If empty, all variables are assumed non-integer. If islogical(IntVars) (=all el- ements are 0/1), then 1 = integer variable, 0 = continuous variable. If any element >1, IntVars is the indices for integer variables. nMax Number of integer combinations possible, if empty multiMINLP computes nMax. Rfac Reduction factor for real variables in MINLP subproblem close to local multiMINLP minimum. Bounds set to x_L = max(x_L,x-Rfac\*(x_U-x_L)) and x_U= min(x_U,x+Rfac\*(x_U-x_L)). Default 0.25. Description of Outputs Result Structure with result from optimization. The following fields are changed: Result structure from the last good optimization step giving the best f(x) value, the possible global MINLP minimum. The following fields in Result are changed by multiMINLP before return: ExitFlag = 0 normal output, of if fGoal set and found. = 1 A solution reaching the user defined fGoal was not found. = 2 Unbounded problem. =4 Infeasible problem. The Solver, SolverAlgorithm and ExitText fields are also reset. A special field in Result is also returned, Result.multiMINLP: xOpt Prob.N x k matrix with k distinct local optima, the test being norm(x k- xOpt(:,i)) <= xEqTol\*max(1,norm(x k)) that if fulfilled assumes x_k to be to close to xOpt(:,i). fOpt The k function values in the local optima xOpt(:,i),i=1,...,k. Inform The Inform value returned by the local solver when finding each of the local optima xOpt(:,i); i=1,...,k. The Inform value can be used to judge the validity of the local minimum reported. localTry Total number of local searches. Iter Total number of iterations. FuncEv Total number of function evaluations. GradEv Total number of gradient evaluations. HessEv Total number of Hessian evaluations. ConstrEv Total number of constraint function evaluations. ExitText Text string giving Inform information. nlpSolve Purpose Solve general constrained nonlinear optimization problems. nlpSolve solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & & & & \\ s/t & x_{L} & \leq & x & \leq & x_{U} \\ & b_{L} & \leq & Ax & \leq & b_{U} \\ & c_{L} & \leq & c(x) & \leq & c_{U} \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x),c_{L},c_{U}\in \MATHSET{R} ^{m_{1}}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \MATHSET{R}^{m_{2}\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} . Calling Syntax Result = nlpSolve(Prob, varargin) Result = tomRun('nlpSolve', Prob); Description of Inputs Prob Problem description structure. The following fields are used: A Constraint matrix for linear constraints. b_L Lower bounds on the linear constraints. b_U Upper bounds on the linear constraints. c_L Lower bounds on the general constraints. c_U Upper bounds on the general constraints. x_L Lower bounds on the variables. x_U Upper bounds on the variables. x_0 Starting point. PriLevOpt Print level: 0 Silent, 1 Final result, 2 Each iteration, short, 3 Each iteration, more info, 4 Matrix update information. FUNCS.f Name of m-file computing the objective function f (x). FUNCS.g Name of m-file computing the gradient vector g(x). FUNCS.H Name of m-file computing the Hessian matrix H (x). FUNCS.c Name of m-file computing the vector of constraint functions c(x). FUNCS.dc Name of m-file computing the matrix of constraint normals ?c(x)/dx. FUNCS.d2c Name of m-file computing the second derivatives of the constraints, weighted by an input Lagrange vector NumDiff How to obtain derivatives (gradient, Hessian). ConsDiff How to obtain the constraint derivative matrix. SolverQP Name of the solver used for QP subproblems. If empty, the default solver is used. See GetSolver.m and tomSolve.m. SolverFP Name of the solver used for FP subproblems. If empty, the default solver is used. See GetSolver.m and tomSolve.m. optParam Structure with special fields for optimization parameters, see Table 141. Fields used are: eps_g, eps_x, MaxIter, wait, size_x, method, IterPrint, xTol, bTol, cTol, and QN InitMatrix. varargin Other parameters directly sent to low level routines. Description of Outputs Result Structure with result from optimization. The following fields are changed: x_k Optimal point. f_k Function value at optimum. g_k Gradient value at optimum. c_k Value of constraints at optimum. H_k Hessian value at optimum. v_k Lagrange multipliers. x_0 Starting point. f_0 Function value at start. cJac Constraint Jacobian at optimum. xState State of each variable, described in Table 150. bState State of each linear constraint, described in Table 151. cState State of each general constraint. Inform Type of convergence. ExitFlag Flag giving exit status. ExitText 0: Convergence. Small step. Constraints fulfilled. 1: Infeasible problem? 2: Maximal number of iterations reached. 3: No progress in either function value or constraint reduction. Inform 1: Iteration points are close. 2: Small search direction 3: Function value below given estimate. Restart with lower fLow if minimum not reached. 4: Projected gradient small. 10: Karush-Kuhn-Tucker conditions fulfilled. Iter Number of iterations. Solver Solver used. SolverAlgorithm Solver algorithm used. Prob Problem structure used. Description The routine nlpSolve implements the Filter SQP by Roger Fletcher and Sven Leyffer presented in the paper \[21\]. M-files Used tomSolve.m, ProbCheck.m, iniSolve.m, endSolve.m See Also conSolve, sTrustr pdcoTL Purpose pdcoTL solves linearly constrained convex nonlinear optimization problems of the kind Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L & \leq & x & \leq & x_U \\ {} & b_L & \leq & Ax & \leq & b_U \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is a convex nonlinear function, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_L,x_U \in \RR^n} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \RR^{m \times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_L, b_U \in \RR^m} . Calling Syntax Result=tomRun('pdco',Prob,...); Description of Inputs Prob Problem description structure. The following fields are used: x_0 Initial x vector, used if non-empty. A The linear constraint matrix. b_L,b_U Lower and upper bounds for the linear constraints. PriLevOpt Print level in pdsco solver. If > 0: prints summary information. SOL Structure with SOL special parameters: pdco Options structure with fields as defined by pdcoSet. d1 Primal regularization vector. Must be a positive vector (length n) or scalar, in which case D1 = diag(d1) is used. Default: 10-4 . d2 Dual regularization vector. Must be a positive vector (length m) or a scalar value, in which case D2 = diag(d2) is used. Default: 10-4 . y0 Initial dual parameters for linear constraints (default 0) z0 Initial dual parameters for simple bounds (default 1/N ) xsize,zsize are used to scale (x, y, z). Good estimates should improve the performance of the barrier method. xsize Estimate of the biggest x at the solution. (default 1/N ) zsize Estimate of the biggest z at the solution. (default 1/N ) optParam Structure with optimization parameters. The following fields are used: MaxIter Maximum number of iterations. (Prob.SOL.pdco.MaxIter). MinorIter Maximum number of iterations in LSQR (Prob.SOL.pdco.LSQRMaxIter). eps_x Accuracy for satisfying x1 . * z1 = 0, x2 .z1 = 0, where z = z1 - z2 and z1 , z2 > 0.(Prob.SOL.pdco.OptTol) bTol Accuracy for satisfying Ax + D2 r = b, AT y + z = ?f (x) and x - x1 = bL , x +x2 = bU , where x1 , x2 > 0 (Prob.SOL.pdco.FeaTol) wait 0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (Prob.SOL.pdco.wait) Description of Outputs Result Structure with result from optimization. The following fields are set by pdcoTL x_k Solution vector f_k Function value at optimum g_k Gradient of the function at the solution H_k Hessian of the function at the solution, diagonal only x_0 Initial solution vector f_0 Function value at start, x = x_0 xState State of variables. Free == 0; On lower == 1; On upper == 2; Fixed == 3; bState State of linear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3; v_k Lagrangian multipliers (orignal bounds + constraints ) y_k Lagrangian multipliers (for bounds + dual solution vector) The full \[z; y\] vec- tor as returned from pdco, including slacks and extra linear constraints after rewriting constraints: -inf < b L < A * x < b U < inf ; non-inf lower AND upper bounds ExitFlag Tomlab Exit status from pdco MEX Inform pdcoinformation parameter: 0 = Solution found; 0 Solution found 1 Too many iterations 2 Linesearch failed too often Iter Number of iterations FuncEv Number of function evaluations GradEv Number of gradient evaluations HessEv Number of Hessian evaluations Solver Name of the solver ('pdco') SolverAlgorithm Description of the solver Description pdco implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL). The problem (19) is first reformulated into SOL PDCO form: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L & \leq & x & \leq & x_U \\ {} & & & Ax & = & b \\ {} & \multicolumn{5}{l}{r \mathrm{\ unconstrained}} \\ \end{array} } The problem actually solved by pdco is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lllcll} \min\limits_{x,r} & \multicolumn{5}{l}{\phi(x) + \frac{1}{2}\|D_1 x\|^2 + \frac{1}{2}\|r\|^2 } \\ \\ \mathrm{s/t} & x_L & \leq & x & \leq & x_U \\ {} & Ax & + & D_{2}r & = & b \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_2} are positive-definite diagonal matrices defined from Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_1} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_2} given in Prob.SOL.d1 and Prob.SOL.d2. In particular, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_2} indicates the accuracy required for satisfying each row of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Ax=b} . See pdco for a detailed discussion of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_2} . Note that in pdco, the objective Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is denoted Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x)} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle bl == x\_L} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle bu == x\_U} . Examples Problem 14 and 15 in tomlab/testprob/con prob.m.m are good examples of the use of pdcoTL. M-files Used pdcoSet.m, pdco.m, Tlsqrmat.m See Also pdscoTL.m pdscoTL Purpose pdscoTL solves linearly constrained convex nonlinear optimization problems of the kind Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & f(x) \\ \mathrm{s/t} & x_L & \leq & x & \leq & x_U \\ {} & b_L & \leq & Ax & \leq & b_U \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is a convex separable nonlinear function, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_L,x_U \in \RR^n} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \RR^{m \times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_L, b_U \in\RR^m} . Calling Syntax Result=tomRun('pdsco',Prob,...); Description of Inputs Prob Problem description structure. The following fields are used: x_0 Initial x vector, used if non-empty. A The linear constraints coefficient matrix. b_L,b_U Lower and upper bounds for the linear constraints. HessPattern Non-zero pattern for the objective function. Only the diagonal is needed. Default if empty is the unit matrix. PriLevOpt Print level in pdsco solver. If > 0: prints summary information. SOL Structure with SOL special parameters: pdco Options structure with fields as defined by pdscoSet. gamma Primal regularization parameter. delta Dual regularization parameter. y0 Initial dual parameters for linear constraints (default 0) z0 Initial dual parameters for simple bounds (default 1/N ) xsize,zsize are used to scale (x, y, z). Good estimates should improve the performance of the barrier method. xsize Estimate of the biggest x at the solution. (default 1/N ) zsize Estimate of the biggest z at the solution. (default 1/N ) optParam Structure with optimization parameters. The following fields are used: MaxIter Maximum number of iterations. (Prob.SOL.pdco.MaxIter). MinorIter Maximum number of iterations in LSQR (Prob.SOL.pdco.LSQRMaxIter). eps_x Accuracy for satisfying x. * z = 0 bTol Accuracy for satisfying Ax + r = b, AT y + z = ?f (x) and x - x1 = bL , x + x2 =bU, where x1 , x2 > 0. (Prob.SOL.pdco.FeaTol) wait 0 - solve the problem with default internal parameters; 1 - pause: allows interactive resetting of parameters. (Prob.SOL.pdco.wait) Description of Outputs Result Structure with result from optimization. The following fields are set by pdscoTL: x_k Solution vector f_k Function value at optimum g_k Gradient of the function at the solution H_k Hessian of the function at the solution, diagonal only x_0 Initial solution vector f_0 Function value at start, x = x_0 xState State of variables. Free == 0; On lower == 1; On upper == 2; Fixed == 3; bState State of linear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3; v_k Lagrangian multipliers (orignal bounds + constraints ) y k Lagrangian multipliers (for bounds + dual solution vector) The full \[z; y\] vec- tor as returned from pdsco, including slacks and extra linear constraints after rewriting constraints: -inf < b L < A * x < b U < inf ; non-inf lower AND upper bounds ExitFlag Tomlab Exit status from pdsco MEX Inform pdsco information parameter: 0 = Solution found; 0 Solution found 1 Too many iterations 2 Linesearch failed too often Iter Number of iterations FuncEv Number of function evaluations GradEv Number of gradient evaluations HessEv Number of Hessian evaluations Solver Name of the solver ('pdsco') SolverAlgorithm Description of the solver Description pdsco implements an primal-dual barrier method developed at Stanford Systems Optimization Laboratory (SOL). The problem (20) is first reformulated into SOL PDSCO form: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{clll} \min\limits_x & f(x) \\ \mathrm{s/t} & x & \geq & x_U \\ {} & Ax & = & b. \\ \end{array} } The problem actually solved by pdsco is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lllcll} \min\limits_{x,r} & \multicolumn{5}{l}{f(x) + \frac{1}{2}\|\gamma x\|^2 + \frac{1}{2}\|r / \delta \|^2 } \\ \\ \mathrm{s/t} & & & x & \geq & 0 \\ {} & Ax & + & r & = & b \\ {} & \multicolumn{5}{l}{r \mathrm{\ unconstrained}} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma} is the primal regularization parameter, typically small but 0 is allowed. Furthermore, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} is the dual regularization parameter, typically small or 1; must be strictly greater than zero. With positive Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma,\delta} the primal-dual solution Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y,z)} is bounded and unique. See pdsco for a detailed discussion of </math>\gamma</math> and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} . Note that in pdsco, the objective Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is denoted Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi(x)} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle bl == x\_L} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle bu == x\_U} . Examples Problem 14 and 15 in tomlab/testprob/con prob.m are good examples of the use of pdscoTL. M-files Used pdscoSet.m, pdsco.m, Tlsqrmat.m See Also pdcoTL.m qpSolve Purpose Solve general quadratic programming problems. qpSolve solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccl} \min\limits_{x} & f(x) & = & \frac{1}{2}(x)^{T}Fx + c^{T}x & & \\s/t & x_{L} & \leq & x & \leq & x_{U} \\& b_{L} & \leq & Ax & \leq & b_{U} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R} ^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F\in \MATHSET{R}^{n\times n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c \in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \MATHSET{R}^{m\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m}} . Calling Syntax Result = qpSolve(Prob) or Result = tomRun('qpSolve', Prob, 1); Description of Inputs Prob Problem description structure. The following fields are used: QP.F Constant matrix, the Hessian. QP.c Constant vector. A Constraint matrix for linear constraints. b_L Lower bounds on the linear constraints. b_U Upper bounds on the linear constraints. x_L Lower bounds on the variables. x_U Upper bounds on the variables. x_0 Starting point. optParam Structure with special fields for optimization parameters, see Table 141.Fields used are: eps f, eps Rank, MaxIter, wait, bTol and PriLev. Description of Outputs Result Structure with result from optimization. The following fields are changed: x_k Optimal point. f_k Function value at optimum. g_k Gradient value at optimum. H_k Hessian value at optimum. v_k Lagrange multipliers. x_0 Starting point. f_0 Function value at start. xState State of each variable, described in Table 150 . Iter Number of iterations. ExitFlag 0: OK, see Inform for type of convergence. 2: Can not find feasible starting point x_0. 3: Rank problems. Can not find any solution point. 4: Unbounded solution. 10: Errors in input parameters. Inform If ExitF lag > 0, I nf orm = ExitF lag, otherwise I nf orm show type of convergence: 0: Unconstrained solution. 1: ? = 0. 2: ? = 0. No second order Lagrange mult. estimate available. 3: ? and 2nd order Lagr. mult. positive, problem is not negative definite. 4: Negative definite problem. 2nd order Lagr. mult. positive, but releasing variables leads to the same working set. Solver Solver used. SolverAlgorithm Solver algorithm used. Prob Problem structure used. Description Implements an active set strategy for Quadratic Programming. For negative definite problems it computes eigen- values and is using directions of negative curvature to proceed. To find an initial feasible point the Phase 1 LP problem is solved calling lpSimplex. The routine is the standard QP solver used by nlpSolve, sTrustr and conSolve. M-files Used ResultDef.m, lpSimplex.m, tomSolve.m, iniSolve.m, endSolve.m See Also qpBiggs, qpe, qplm, nlpSolve, sTrustr and conSolve slsSolve Purpose Find a Sparse Least Squares (sls) solution to a constrained least squares problem with the use of any suitable TOMLAB NLP solver. slsSolve solves problems of the type: Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_x & \frac{1}{2}r(x)^T r(x)\\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\{} & b_L & \leq & Ax & \leq & b_U \\{} & c_L & \leq & c(x) & \leq & c_U \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_L,x_U \in \Rdim{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r(x) \in \Rdim{m}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A \in\Rdim{m_1,n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_L,b_U \in \Rdim{m_1}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x),c_L,c_U \in\Rdim{m_2}} . The use of slsSolve is mainly for large, sparse problems, where the structure in the Jacobians of the residuals and the nonlinear constraints are utilized by a sparse NLP solver, e.g. SNOPT. Calling Syntax Result=slsSolve(Prob,PriLev) Description of Inputs Prob Problem description structure. Should be created in the cls format, preferably by callingProb=clsAssign(...) if using the TQ format. slsSolve uses two special fields in Prob: SolverL2 Text string with name of the NLP solver used for solving the reformulated problem. Valid choices are conSolve, nlpSolve, sTrustr, clsSolve. Suitable SOL solvers, if available: minos, snopt, npopt. L2Type Set to 1 for standard constrained formulation. Currently this is the only allowed choice. All other fields should be set as expected by the nonlinear solver selected. In particular: A Linear constraint matrix. b_L Lower bounds on the linear constraints. b_U Upper bounds on the linear constraints. c_L Upper bounds on the nonlinear constraints. c_U Lower bounds on the nonlinear constraints. x_L Lower bounds on the variables. x_U Upper bounds on the variables. x_0 Starting point. ConsPattern The nonzero pattern of the constraint Jacobian. JacPattern The nonzero pattern of the residual Jacobian. Note that Prob.LS.y must be of correct length if JacPattern is empty (but ConsPattern is not). slsSolve will create the new Prob.ConsPattern to be used by the nonlinear solver using the information in the supplied ConsPattern and JacPattern. PriLev Print level in slsSolve. Default value is 2. 0 Silent except for error messages. > 1 Print summary information about problem transformation. slsSolve calls Print- Result(Result,PriLev). 2 Standard output in PrintResult. Description of Outputs Result Structure with results from optimization. The contents of Result depend on which nonlinear solver was used to solved slsSolve transforms the following fields of Result back to the format of the original problem: x_k Optimal point. r_k Residual at optimum. J_k Jacobian of residuals at optimum. c_k Nonlinear constraint vector at optimum. v_k Lagrange multipliers. g_k The gradient vector is calculated as J kT · r k. cJac Jacobian of nonlinear constraints at optimum. x_0 Starting point. xState State of variables at optimum. cState State of constraints at optimum. Result.Prob The problem structure defining the reformulated problem. Description The constrained least squares problem is solved in slsSolve by rewriting the problem as a general constrained optimization problem. A set of m (the number of residuals) extra variables Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=(z_1,z_2,\ldots,z_m)} are added at the end of the vector of unknowns. The reformulated problem Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min \limits_x & \multicolumn{5}{l}{\frac{1}{2} z^T z} \\ \mbox{subject to} & x_L & \leq & (x_1,x_2,\ldots,x_n) & \leq & x_U \\ {} & b_L & \leq & Ax & \leq & b_U \\ {} & c_L & \leq & c(x) & \leq & c_U \\ {} & 0 & \leq & r(x) - z & \leq & 0 \\ \end{array} } is then solved by the solver given by Prob.SolverL2. Examples slsDemo.m M-files Used iniSolve.m, GetSolver.m sTrustr Purpose Solve optimization problems constrained by a convex feasible region. sTrustr solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & & & & \\ s/t & x_{L} & \leq & x & \leq & x_{U} \\ & b_{L} & \leq & Ax & \leq & b_{U} \\ & c_{L} & \leq & c(x) & \leq & c_{U} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R}^{n}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c(x),c_{L},c_{U}\in \MATHSET{R}^{m_{1}}} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A\in \MATHSET{R}^{m_{2}\times n}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_{L},b_{U}\in \MATHSET{R}^{m_{2}}} . Calling Syntax Result = sTrustr(Prob, varargin) Description of Inputs Prob Problem description structure. The following fields are used: A Constraint matrix for linear constraints. b_L Lower bounds on the linear constraints. b_U Upper bounds on the linear constraints. c_L Lower bounds on the general constraints. c_U Upper bounds on the general constraints. x_L Lower bounds on the variables. x_U Upper bounds on the variables. x_0 Starting point. FUNCS.f Name of m-file computing the objective function f (x). FUNCS.g Name of m-file computing the gradient vector g(x). FUNCS.H Name of m-file computing the Hessian matrix H (x). FUNCS.c Name of m-file computing the vector of constraint functions c(x). FUNCS.dc Name of m-file computing the matrix of constraint normals ?c(x)/dx. optParam Structure with special fields for optimization parameters, see Table 141. Fields used are: eps f, eps g, eps c, eps x, eps Rank, MaxIter, wait, size x, size f, xTol, LowIts, PriLev, method and QN InitMatrix. PartSep Structure with special fields for partially separable functions, see Table 142. varargin Other parameters directly sent to low level routines. Description of Outputs Result Structure with result from optimization. The following fields are changed: x_k Optimal point. f_k Function value at optimum. g_k Gradient value at optimum. c_k Value of constraints at optimum. H_k Hessian value at optimum. v_k Lagrange multipliers. x_0 Starting point. f_0 Function value at start. cJac Constraint Jacobian at optimum. xState State of each variable, described in Table 150. Iter Number of iterations. ExitFlag Flag giving exit status. Inform Binary code telling type of convergence: 1: Iteration points are close. 2: Projected gradient small. 3: Iteration points are close and projected gradient small. 4: Relative function value reduction low for LowIts iterations. 5: Iteration points are close and relative function value reduction low for LowIts iterations. 6: Projected gradient small and relative function value reduction low for LowIts iterations. 7: Iteration points are close, projected gradient small and relative function value reduction low for LowIts iterations. 8: Too small trust region. 9: Trust region small. Iteration points close. 10: Trust region and projected gradient small. 11: Trust region and projected gradient small, iterations close. 12: Trust region small, Relative f(x) reduction low. 13: Trust region small, Relative f(x) reduction low. Iteration points are close. 14: Trust region small, Relative f(x) reduction low. Projected gradient small. 15: Trust region small, Relative f(x) reduction low. Iteration points close, Projected gradient small. 101: Maximum number of iterations reached. 102: Function value below given estimate. 103: Convergence to saddle point (eigenvalues computed). Solver Solver used. SolverAlgorithm Solver algorithm used. Prob Problem structure used. Description The routine sTrustr is a solver for general constrained optimization, which uses a structural trust region algorithm combined with an initial trust region radius algorithm (itrr). The feasible region defined by the constraints must be convex. The code is based on the algorithms in \[13\] and \[67\]. BFGS or DFP is used for the Quasi-Newton update, if the analytical Hessian is not used. sTrustr calls internal routine itrr. M-files Used qpSolve.m, tomSolve.m, iniSolve.m, endSolve.m See Also conSolve, nlpSolve, clsSolve Tfmin Purpose Minimize function of one variable. Find miniumum x in [x_L, x_U] for function Func within tolerance xTol. Solves using Brents minimization algorithm. Reference: "Computer Methods for Mathematical Computations", Forsythe, Malcolm, and Moler, Prentice-Hall, 1976. Calling Syntax [x, nFunc] = Tfmin(Func, x_L, x_U, xTol, Prob) Description of Inputs Variable Description Func Function of x to be minimized. Func must be defined as: f = Func(x) if no 5th argument Prob is given or f = Func(x, Prob) if 5th argument Prob is given. x_L Lower bound on x. x_U Upper bound on x. xTol Tolerance on accuracy of minimum. Prob Structure (or any Matlab variable) sent to Func. If many parameters are to be sent to Func set them in Prob as a structure. Example for parameters a and b: Prob.user.a = a; Prob.user.b = b; [x, nFunc] = Tfmin('myFunc',0,1,1E-5,Prob); In myFunc: function f = myFunc(x, Prob) a = Prob.user.a; b = Prob.user.b; f = "matlab expression dependent of x, a and b"; Description of Outputs Variable Description x Solution. nFunc Number of calls to Func. Tfzero Purpose Tfzero, TOMLAB fzero routine. Tfzero, TOMLAB fzero routine.\\ \\Find a zero for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} in the interval Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [x_L, x_U]} . Tfzero searches for a zero of a function Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} between the given scalar values Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_L} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_U} until the width of the interval (xLow, xUpp) has collapsed to within a tolerance specified by the stopping criterion, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle abs(xLow-xUpp) <= 2.*(RelErr*abs(xLow)+AbsErr)} . The method used is an efficient combination of bisection and the secant rule and is due to T. J. Dekker. Calling Syntax [xLow, xUpp, ExitFlag] = Tfzero(x L, x U, Prob, x 0, RelErr, AbsErr) Description of Inputs Variable Description x_L Lower limit on the zero x to f(x). x_U Upper limit on the zero x to f(x). Prob Structure, sent to Matlab routine ZeroFunc. The function name should be set in Prob.FUNCS.f0. Only the function will be used, not the gradient. x_0 An initial guess on the zero to f(x). If empty, x 0 is set as the middle point in [x_L, x_U]. RelErr Relative error tolerance, default 1E-7. AbsErr Absolute error tolerance, default 1E-14. Description of Outputs Variable Description xLow Lower limit on the zero x to f(x). xUpp Upper limit on the zero x to f(x). ExitFlag Status flag 1,2,3,4,5. 1: xLow is within the requested tolerance of a zero. The interval (xLow, xUpp) collapsed to the requested tolerance, the function changes sign in (xLow, xUpp), and f(x) decreased in magnitude as (xLow, xUpp) collapsed. 2: f(xLow) = 0. However, the interval (xLow, xUpp) may not have collapsed to the requested tolerance. 3: xLow may be near a singular point of f(x). The interval (xLow, xUpp) collapsed to the requested tolerance and the function changes sign in (xLow, xUpp), but f(x) increased in magnitude as (xLow, xUpp) collapsed, i.e. abs(f (xLow)) > max(abs(f (xLow - I N )), abs(f (xU pp - I N ))). 4: No change in sign of f(x) was found although the interval (xLow, xUpp) collapsed to the requested tolerance. The user must examine this case and decide whether xLow is near a local minimum of f(x), or xLow is near a zero of even multiplicity, or neither of these. 5: Too many (> 500) function evaluations used. ucSolve Purpose Solve unconstrained nonlinear optimization problems with simple bounds on the variables. ucSolve solves problems of the form Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{cccccc} \min\limits_{x} & f(x) & & & & \\ s/t & x_{L} & \leq & x & \leq & x_{U} \\ \end{array} } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,x_{L},x_{U}\in \MATHSET{R} ^{n}} . Calling Syntax Result = ucSolve(Prob, varargin) Description of Inputs Prob Problem description structure. The following fields are used: x_L Lower bounds on the variables. x_U Upper bounds on the variables. x_0 Starting point. FUNCS.f Name of m-file computing the objective function f (x). FUNCS.g Name of m-file computing the gradient vector g(x). FUNCS.H Name of m-file computing the Hessian matrix H (x). f_Low Lower bound on function value. Solver.Alg Solver algorithm to be run: 0: Gives default, either Newton or BFGS. 1: Newton with subspace minimization, using SVD. 2: Safeguarded BFGS with inverse Hessian updates (standard). 3: Safeguarded BFGS with Hessian updates. 4: Safeguarded DFP with inverse Hessian updates. 5: Safeguarded DFP with Hessian updates. 6: Fletcher-Reeves CG. 7: Polak-Ribiere CG. 8: Fletcher conjugate descent CG-method. Solver.Method Method used to solve equation system: 0: SVD (default). 1: LU-decomposition. 2: LU-decomposition with pivoting. 3: Matlab built in QR. 4: Matlab inversion. 5: Explicit inverse. Solver.Method Restart or not for C-G method: 0: Use restart in CG-method each n:th step. 1: Use restart in CG-method each n:th step. LineParam Structure with line search parameters, see routine LineSearch and Table 140. optParam Structure with special fields for optimization parameters, see Table 141. Fields used are: eps absf, eps f, eps g, eps x, eps Rank, MaxIter, wait, size x, xTol, size f, LineSearch, LineAlg, xTol, IterPrint and QN InitMatrix. PriLevOpt Print level. varargin Other parameters directly sent to low level routines. Description of Outputs Result Structure with result from optimization. The following fields are changed: x_k Optimal point. f_k Function value at optimum. g_k Gradient value at optimum. H_k Hessian value at optimum. B_k Quasi-Newton approximation of the Hessian at optimum. v_k Lagrange multipliers. x_0 Starting point. f_0 Function value at start. xState State of each variable, described in Table 150. Iter Number of iterations. ExitFlag 0 if convergence to local min. Otherwise errors. Inform Binary code telling type of convergence: 1: Iteration points are close. 2: Projected gradient small. 4: Relative function value reduction low for LowIts iterations. 101: Maximum number of iterations reached. 102: Function value below given estimate. 104: Convergence to a saddle point. Solver Solver used. SolverAlgorithm Solver algorithm used. Prob Problem structure used. Description The solver ucSolve includes several of the most popular search step methods for unconstrained optimization. The search step methods included in ucSolve are: the Newton method, the quasi-Newton BFGS and DFP methods, the Fletcher-Reeves and Polak-Ribiere conjugate-gradient method, and the Fletcher conjugate descent method. The quasi-Newton methods may either update the inverse Hessian (standard) or the Hessian itself. The Newton method and the quasi-Newton methods updating the Hessian are using a subspace minimization technique to handle rank problems, see Lindstr¨om \[53\]. The quasi-Newton algorithms also use safe guarding techniques to avoid rank problem in the updated matrix. The line search algorithm in the routine LineSearch is a modified version of an algorithm by Fletcher \[20\]. Bound constraints are treated as described in Gill, Murray and Wright \[28\]. The accuracy in the line search is critical for the performance of quasi-Newton BFGS and DFP methods and for the CG methods. If the accuracy parameter Prob.LineParam.sigma is set to the default value 0.9, ucSolve changes it automatically according to: Prob.Solver.Alg Prob.LineParam.sigma 4,5 (DFP) 0.2 6,7,8 (CG) 0.01 M-files Used ResultDef.m, LineSearch.m, iniSolve.m, tomSolve.m, endSolve.m See Also clsSolve c Additional solvers Documentation for the following solvers is only available at http://tomopt.com and in the m-file help. goalSolve - For sparse multi-objective goal attainment problems, with linear and nonlinear constraints. Tlsqr - Solves large, sparse linear least squares problem, as well as unsymmetric linear systems. lsei - For linearly constrained least squares problem with both equality and inequality constraints. Tnnls - Also for linearly constrained least squares problem with both equality and inequality constraints. qld - For convex quadratic programming problem.