MipSolve

Purpose

Solve mixed-integer linear programming problems (MIP).

mipSolve solves problems of the form

${\begin{array}{cccccc}\min \limits _{x}&f(x)&=&c^{T}x&\\s/t&x_{L}&\leq &x&\leq &x_{U}\\&b_{L}&\leq &Ax&\leq &b_{U}\\&&&{x_{j}\in \mathbb {N} \ \ \forall j\in I}\\\end{array}}$

where $c,x,x_{L},x_{U}\in \mathbb {R} ^{n}$ , $A\in \mathbb {R} ^{m\times n}$ and $b_{L},b_{U}\in \mathbb {R} ^{m}$.The variables $x\in I$ , the index subset of $1,...,n$ 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

Recommended is to first set IterPrint, to get information each iteration

Prob.optParam.IterPrint = 1;

Driver call, including printing with level 2:

Result = tomRun('mipSolve',Prob,2)

Direct solver call:

Result = mipSolve(Prob);
PrintResult(Result);

Warm Start

To make a restart (warm start), just set the warm start flag, and call mipSolve once again:

Prob.WarmStart = 1;
Result = tomRun('mipSolve', Prob, ...);

mipSolve will read warm start information from the mipSolveSave.mat file.

Another warm start (with same MaxFunc) is made by just calling tomRun again.

Result = tomRun('mipSolve', Prob, ...);

To make a restart from the warm start information in the Result structure, make a call to WarmDefGLOBAL before calling mipSolve. WarmDefGLOBAL moves information from the Result structure to the Prob structure and sets the warm start flag, Prob.WarmStart = 1;

Prob = WarmDefGLOBAL('mipSolve', 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('mipSolve', Prob, ...);

To make another warm start with new MaxIter 100, say, redefine MaxIter as:

Prob.optParam.MaxIter = 100;

Then repeat the two lines:

Prob = WarmDefGLOBAL('mipSolve', Prob, Result);
Result = tomRun('mipSolve', Prob, ...);

Inputs

Prob	Problem description structure. The following fields are used:
	c	The vector c in cTx.
	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.
	PriLevOpt	Print level in sub-solver, see the documentation for the solver.
	PriLev	Print level in mipSolve (default 1). Also see optParam.IterPrint.
	WarmStart	If true, >0, mipSolve reads the output from the last run from Prob.mipSolve, if it exists. If it doesn't exist, mipSolve attempts to open and read warm start data from mat-file mipSolveSave.mat. mipSolve uses the warm start information to continue from the last run. The mat-file mipSolveSave.mat is saved every Prob.MIP.SaveFreq iteration
	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 the solver used for the initial LP subproblem. If empty, the best available solver with a license is picked from a list. Note - If using MINOS, use the special LP interface LP-MINOS
	SolverDLP	Name of the solver used for dual LP subproblems. If empty, SolverLP is used. If SolverLP/SolverDLP 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.
	MIP	Structure in Prob, Prob.MIP. Defines integer optimization parameters:
	MIP.IntVars	If empty, all variables are assumed non-integer. If islogical(IntVars) (i.e. all elements are 0/1), then 1 = integer variable, 0 = continuous variable. If any element >1, IntVars is the indices for integer variables
	MIP.VarWeight	Weight for each variable in the variable selection phase. A lower value gives higher priority. Setting Prob.MIP.VarWeight = c; for knapsack problems improve convergence.
	MIP.DualGap	mipSolve stops if the duality gap is less than DualGap. E.g. DualGap = 1, implies a stop at the first integer solution. E.g. DualGap = 0.01, stop if the solution less than one percent from the optimal solution.
	MIP.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.
	MIP.xIP	The x-value giving the fIP value, if a solution (xIP, fIP) is known.
	MIP.NodeSel	Node selection method: 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.
	MIP.VarSel	Variable selection method in branch and bound. 1: Use variable with most fractional value 2: Use gradient and distance to nearest integer value
	MIP.KNAPSACK	If solving a knapsack problem, set to true (1) to use a knapsack heuristic.
	MIP.ROUNDH	If = 1, use a rounding heuristic. Default 1.
	MIP.SaveFreq	Warm start info saved on mipSolveSave.mat every SaveFreq iteration. (default -1, i.e. no warm start info is saved)
	Solver	Options for the LP subproblem:
	Solver.method	Rule to select new variables in DualSolve/lpSimplex: 0: Minimum reduced cost, sort variables increasing (default). 1: Bland's Anti-Cycling Rule 2: Minimum reduced cost. Dantzig's rule.
	optParam	Structure in Prob. Fields used in Prob.optParam, also in sub-solvers:
	optParam.MaxIter	Maximal number of iterations. max(10*dim(x),100) is default.
	optParam.IterPrint	Print short information each iteration (PriLev > 0 ==> IterPrint = 1). One line each iteration with: Iteration number Depth in tree (symbol L[] - empty list, symbol Gap - Dual Gap convergence; date/time stamp, fLP (Optimal f(x) current node), fIPMin (Best integer feasible 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
	optParam.bTol	Linear constraint violation convergence tolerance
		Fields in Prob.optParam that are used by lpSimplex and DualSolve (assuming they are set as sub-solvers):
	optParam.eps_f	Tolerance used for function value tests
	optParam.eps_Rank	Rank tolerance

Outputs

Result	Structure with result from optimization. The following fields are changed:
	Iter	Number of iterations.
	ExitFlag	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. 3: Rank problems (not used). 4: No feasible point found running LP relaxation. 5: Illegal x_0 found in LP relaxation. 99: Maximal CPU Time used (cputime > Prob.MaxCPU).
	ExitText	Text string giving ExitFlag and Inform information.
	Inform	If ExitFlag > 0, Inform = ExitFlag, except if duality gap less than given tolerance (Inform = 6)
	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*DualGap, to get the percentage duality gap. For absolute value duality gap: scale with fIPMin, fIPMin * DualGap
	x_k	Solution.
	v_k	Lagrange parameters (n+m) vector, [Bounds(n); Linear constraints (m)].
	QP.B	B Optimal set. B(i)==0 implies inclusion of variable x(i) in basic set. sum(B==1)==length(b) holds. See QP.B as input.
	QP.y	Dual parameters y (also part of Result.v_k).
	f_k	Function value cTx_k, fIPMin (Best integer feasible solution).
	g_k	Gradient value at optimum, c.
	Solver	Solver used (mipSolve).
	SolverAlgorithm	Description of method used.
	x_0	Starting point.
	xState	State of each variable, described in TOMLAB Appendix B.
	mipSolve	A structure with warm start information. Use with WarmDefGLOBAL, see example.
	Prob	Problem structure used.

Description

The routine mipSolve is an implementation of a branch and bound algorithm. 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 mipSolve.m.

Examples

See exip39, exknap, expkorv.

M-files Used

lpSimplex.m, DualSolve.m, GetSolver.m, tomSolve.m

See Also

• cutPlane
• balas
• solveLP
• solveDLP