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

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

Description of 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 subsolver, 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.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 in lpSimplex and DualSolve (in case they are used):
	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:
	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 TOMLAB Appendix B.
	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. 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