MISQP

Introduction

Overview

Welcome to the TOMLAB /MISQP User's Guide. TOMLAB /MISQP includes the MISQP and MIQL solvers from Klaus Schittkowski and an interface to The MathWorks' MATLAB.

TOMLAB /MISQP solves mixed-integer nonlinear mathematical programming problems with equality and inequality constraints. It is assumed that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value.

The internal algorithm uses a modified sequential quadratic approximation method, stabilized by a trust region method including Yuan's second order corrections. The Hessian of the Lagrangian function is approximated by BFGS updates subject to the continous and integer variables.

TOMLAB /MIQL solves strictly convex mixed-integer quadratic mathematical programming problems with linear equality and inequality constraints. The mixed-integer problem is solved by a branch-and-cut algorithm.

Contents of this Manual

• #Introduction provides a basic overview of the TOMLAB /MISQP solver package.
• #Using the Matlab Interface provides an overview of the Matlab interface to MISQP.
• #Setting MISQP Options describes how to set MISQP solver options from Matlab.
• #MISQP Solver Reference gives detailed information about the interface routine misqpTL.
• #MIQL Solver Reference gives detailed information about the interface routine miqlTL.
• #QL Solver Reference gives detailed information about the interface routine qlTL.

More information

Please visit the following links for more information:

• http://tomopt.com/tomlab/products/misqp/

Prerequisites

In this manual we assume that the user is familiar with mixed-integer nonlinear programming, setting up problems in TOMLAB (in particular unconstrained and constrained mixed-integer nonlinear programming (minlp) problems) and the Matlab language in general.

Using the Matlab Interface

The MISQP solver is accessed via the tomRun driver routine, which calls the misqpTL interface routine. The solver itself is located in the MEX file misqp. The same applies for the other two solvers.

Observe that miqpAssign should be used when defining the problem for MIQL and qpAssign for QL.

Function	Description
misqpTL	The interface routine called by the TOMLAB driver routine tomRun. This routine then calls the MEX file misqp
miqlTL	The interface routine called by the TOMLAB driver routine tomRun. This routine then calls the MEX file miql
qlTL	The interface routine called by the TOMLAB driver routine tomRun. This routine then calls the MEX file ql

Setting MISQP Options

All MISQP control parameters are possible to set from Matlab.

Setting options using the Prob.MISQP structure

The parameters can be set as subfields in the Prob.MISQP structure. The following example shows how to set a limit on the maximum number of iterations.

Prob = minlpAssign(...)   %   Setup problem,  see help minlpAssign for more information
Prob.MISQP.maxit = 2000;  %   Setting maximum number of iterations

The maximum number of iterations can also be done through the TOMLAB parameter MaxIter:

Prob.optParam.MaxIter = 200;

In the cases where a solver specific parameter has a corresponding TOMLAB general parameter, the latter is used only if the user has not given the solver specific parameter.

A complete description of the available MISQP parameters can be found in #misqpTL.

Setting options using the Prob.MIQL structure

Options can be set for MIQL in the same way as for MISQP as described above in #Setting options using the Prob.MISQP structure. The only difference is in the name of the structure (MIQL instead of MISQP) and the assign-routine (miqpAssign instead of minlpAssign).

Setting options using the Prob.QL structure

Options can be set for QL in the same way as for MISQP as described above in #Setting options using the Prob.MISQP structure. The only difference is in the name of the structure (QL instead of MISQP) and the assign-routine (qpAssign instead of minlpAssign).

MISQP Solver References

Table: Solver routines in TOMLAB /MISQP.

Function	Description	Reference
ql	Quadratic programming using a primal-dual method with cholesky decomposition.	qlTL.m
miql	Mixed-integer quadratic programming using a branch-and-cut method with ql as subsolver.	miqlTL.m
misqp	Constrained, mixed-integer nonlinear minimization using a sequential quadratic approximation method. miql is used as MIQP subsolver.	misqpTL.m

A detailed description of the TOMLAB /MISQP solver interfaces is given below. Also see the M-file help for misqpTL.m, miqlTL.m and qlTL.m.

misqpTL

Purpose

Solves mixed-integer nonlinear optimization problems.

MISQP solves problems of the form

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

where $x,x_{L},x_{U}\in \mathbb {R} ^{n}$, $A\in \mathbb {R} ^{m_{1}\times n}$ and $b_{L},b_{U}\in \mathbb {R} ^{m_{1}}$ and $c(x),c_{L},c_{U}\in \mathbb {R} ^{m_{2}}$, Furthermore, $x_{i},\,i\in \mathbb {I}$ are restricted to integer values only.

Calling Syntax

Prob = minlpAssign( ... );
Result = tomRun('misqp',Prob,...);

Description of Inputs

Prob Problem description structure. The following fields are used:

Description of Outputs

Result Structure with result from optimization. The following fields are set:

Output	Description
Result	The structure with results (see ResultDef.m).
x_k	Solution vector.
x_0	Initial solution vector.
f_k	Function value at optimum.
g_k	Gradient of the objective function.
c_k	Nonlinear constraint residuals.
cJac	Nonlinear constraint gradients.
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;
cState	State of nonlinear constraints. Free == 0; Lower == 1; Upper == 2; Equality == 3;
ExitFlag	Exit status from misqp.m (similar to TOMLAB).
ExitText	Exit text from solver.
Inform	MISQP information parameter. 0 - The optimality conditions are satisfied. 1 - Termination after maxit iterations. 2 - Trust region radius lower than termination tolerance of the QP solver. 3 - Penalty parameter sigma tends to infinity. 4 - Termination at infeasible iterate. 5 - Termination with zero trust region for integer variables. 6 - Length of a working array is too short. 7 - There are false dimensions. 8 - Lower or upper bound of variable violates initial value. 9 - Linear constraints are inconsistent. 11 - QL could not solver the QP after 40*(n+m) iterations. 12 - The termination accuracy is insufficient for QL. 13 - QL terminated due to internal inconsistency, division by zero. 14 - Numerical instabilities in QL. 15 - More than mnfs steps without improvements of feasible function values. > 90 - QP sub problem error.
FuncEv	Number of function evaluations.
GradEv	Number of gradient evaluations.
ConstrEv	Number of constraint evaluations.
QP.B	Basis vector in TOMLAB QP standard.
Solver	Name of solver (misqp).
SolverAlgorithm	Description of the solver.

miqlTL

Purpose

Solves strictly convex mixed-integer quadratic programming problems.

MIQL solves problems of the form

${\begin{array}{ll}\min \limits _{x}&f(x)={\frac {1}{2}}x^{T}Fx+c^{T}x\\&\\s/t&{\begin{array}{lcccl}&&x&&,\\b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}\end{array}}$

where $c,x\in \mathbb {R} ^{n}$, $F\in \mathbb {R} ^{n\times n}$ and positive definite, $A\in \mathbb {R} ^{m_{1}\times n}$, and $b_{L},b_{U}\in \mathbb {R} ^{n+m_{1}}$.

The variables $x\in I$, the index subset of $1,...,n$, are restricted to be integers.

If F is empty, an LP or MILP problem is solved.

Calling Syntax

Prob = miqpAssign( ... );
Result = tomRun('miql',Prob,...);

Description of Inputs

Prob Problem description structure. The following fields are used:

Field	Description
x_L, x_U	Bounds on variables.
b_L, b_U	Bounds on linear constraints.
A	Linear constraint matrix.
QP.c	Linear coefficients in objective function.
QP.F	Quadratic matrix of size n x n.
PriLevOpt	Print Level in solver (0-4) 0 - No output. Default 1 - Root QP and final performance summary. 2 - Additional branch and bound iterations counter. 3 - Additional total output from all generated subproblems. 4 - Additional formal details.
MIP.IntVars	Vector of indices of the integer variables.
MIQL.PrintFile	Name of MIQL print file. Amount and type of printing determined by PriLevOpt. Default name miql.txt.
MIQL.eps	Desired final accuracy. The parameter value should not be smaller than the underlying machine precision.
MIQL.acc	The accuracy to identify integer values for integer variables. If acc is less than machine precision, e.g., acc = 0, then acc is set to the machine precision.
MIQL.ibr	Branching rule 1 - Maximal fractional branching (default). 2 - Minimal fractional branching.
MIQL.ins	Node selection strategy 1 - Best of all (large search trees) (default). 2 - Best of two (warmstarts, less memory for search tree.) 3 - Depth first (good warmstarts, smallest memory, many QPs.)
MIQL.maxnde	Maximal number of nodes, e.g., 100,000. Default: (2*n + 2*m + 6)^2 for cut == 1,3), (2*n + m + 4)^2 otherwise.
MIQL.wstart	Maximal number of successive warmstarts to avoid numerical instabilities. Default 100.
MIQL.impb	1 - Calculate improved bounds when best-of-all node selection strategy (ins = 1) is used. 0 - Default.
MIQL.dfdir	1 Select direction for depth-first node selection strategy (ins = 3) according to value of Lagrange function. 0 - Default.
MIQL.cholm	Cholesky decomposition mode. 0 - Calculate Cholesky decomposition once and reuse it. 1 - Calculate new Cholesky decomposition whenever warmstart is not active. Default.
MIQL.cut	Control the cutting process. 0 - No cuts. Default 1 - Use disjunctive cuts only. 2 - Complemented mixed integer rounding (CMIR) cuts only. 3 - Both disjunctive cuts and CMIR cuts.
MIQL.maxdc	Maximal number of rounds for disjunctive cuts. Default 1, if disjunctive cuts should be generated.
MIQL.maxcm	Maximal number of cuts for CMIR cuts. Default 1, if CMIR cuts should be generated.
MIQL.phm	Primal heuristic mode. 0 - No primal heuristics. Default 1 - Nearest integer. 2 - Feasibility pump.

Description of Outputs

Result Structure with result from optimization. The following fields are set:

Output	Description
Result	The structure with results (see ResultDef.m).
f_k	Function value at optimum.
x_k	Solution vector.
x_0	Initial solution vector.
g_k	Exact gradient computed at optimum.
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 (for bounds + dual solution vector).
ExitFlag	Exit status from miql.m (similar to TOMLAB).
Inform	MIQL information parameter. -4 - Branch and Bound root QP could not be solved. -3 - Relaxed QP without feasible solution. -2 - A feasible solution could not be computed by maxnde subproblems. -1 - A feasible solution does not exist. 0 - The optimal solution is found. 1 - A feasible solution is found, but tree search is terminated because of reaching maxnde nodes. 2 - Index in Prob.MIP.intVars out of bounds. 3 - Internal inconsistency of continuous QP. 4 - Length of a work array is too short. 5 - Sizes are incorrectly set. 6 - An integer option for Branch and Bound is incorrectly set. 7 - Independent Lagrangian multipliers could not be calculated, working arrays are too small, increase maxnde. 8 - Print file or print level is incorrectly set. 9 - Lower variable bound greater than upper variable bound. 11 - Subsolver QL could not solve QP after a maximal number of 40*(n+m) iterations. 12 - The termination accuracy is insufficient for QL to satisfy its convergence criteria. 13 - QL terminated due to an internal inconsistency, e.g., division by zero. 14 - Numerical instabilities prevent successful termination of QL. > 90 - Constraints are inconsistent and constraint number Inform - 100 is causing the conflict. The problem has no feasible solution.
rc	Reduced costs. If ninf=0, last m == -v_k.
Iter	Number of iterations.
FuncEv	Number of function evaluations. Set to Iter.
GradEv	Number of gradient evaluations. Set to Iter.
ConstrEv	Number of constraint evaluations. Set to 0.
QP.B	Basis vector in TOMLAB QP standard.
MinorIter	Number of minor iterations. NOT SET.
Solver	Name of solver (miql).
SolverAlgorithm	Description of the solver.
MIQL.cutges	Total number of cutting planes.
MIQL.nodes	Total number of branch and bound nodes.
MIQL.ctime	Time spent generating cutting planes.
MIQL.bbtime	Time spent for branch and bound process.
MIQL.cutgap	Gap reduced by cutting planes compared to original relaxed solution.
MIQL.rlxopt	Relaxed optimal value.
MIQL.gencut	Nonzero if cuts were generated.

qlTL

Solves strictly convex quadratic programming problems.

QL solves problems of the form

${\begin{array}{ll}\min \limits _{x}&f(x)={\frac {1}{2}}x^{T}Fx+c^{T}x\\&\\s/t&{\begin{array}{lcccl}&&x&&,\\b_{L}&\leq &Ax&\leq &b_{U}\\\end{array}}\end{array}}$

where $c,x\in \mathbb {R} ^{n}$, $F\in \mathbb {R} ^{n\times n}$ and positive definite, $A\in \mathbb {R} ^{m_{1}\times n}$, and $b_{L},b_{U}\in \mathbb {R} ^{n+m_{1}}$.

Calling Syntax

Prob = qpAssign( ... );
Result = tomRun('ql',Prob,...);

Description of Inputs

Prob Problem description structure. The following fields are used:

Field	Description
x_L, x_U	Bounds on variables.
b_L, b_U	Bounds on linear constraints.
A	Linear constraint matrix.
QP.c	Linear coefficients in objective function.
QP.F	Quadratic matrix of size n x n.
PriLevOpt	Print Level in solver (0 = off, 1 = only final error message).
QL.eps	Desired final accuracy. The parameter value should not be smaller than the underlying machine precision.
QL.PrintFile	Name of print file. Amount/print type determined by optPar(1). Default name ql.txt.

Description of Outputs

Result Structure with result from optimization. The following fields are set:

Output	Description
Result	The structure with results (see ResultDef.m).
f_k	Function value at optimum.
x_k	Solution vector.
x_0	Initial solution vector.
g_k	Exact gradient computed at optimum.
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 (for bounds + dual solution vector).
ExitFlag	Exit status from ql.m (similar to TOMLAB).
Inform	QL information parameter. 0 - Optimal solution with unique minimizer found 1 - Too many iterations 2 - Accuracy insufficient to attain convergence 3 - Internal inconsistency, division by zero 5 - An input parameter was invalide > 100 - Constraints are inconsistent and Inform=100+iCon where iCon denotes the index of the constraint causing the conflict. The problem has no feasible solution.
rc	Reduced costs. If ninf=0, last m == -v_k.
Iter	Number of iterations.
FuncEv	Number of function evaluations. Set to Iter.
GradEv	Number of gradient evaluations. Set to Iter.
ConstrEv	Number of constraint evaluations. Set to 0.
QP.B	Basis vector in TOMLAB QP standard.
MinorIter	Number of minor iterations. NOT SET.
Solver	Name of solver (ql).
SolverAlgorithm	Description of the solver.