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.

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.

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.

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

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

where ${\displaystyle x,x_{L},x_{U}\in \mathbb {R} ^{n}}$, ${\displaystyle A\in \mathbb {R} ^{m_{1}\times n}}$ and ${\displaystyle b_{L},b_{U}\in \mathbb {R} ^{m_{1}}}$ and ${\displaystyle c(x),c_{L},c_{U}\in \mathbb {R} ^{m_{2}}}$, Furthermore, ${\displaystyle x_{i},\,i\in \mathbb {I} }$ 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:

Field Description
x_L, x_U Bounds on variables.
b_L, b_U Bounds on linear constraints.
A Linear constraint matrix.
PriLevOpt Print Level in solver
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

MISQP.PrintFile Name of MISQP print file. Amount and type of printing determined by PriLevOpt. Default name misqp_printfile.txt.
MISQP.maxit Maximum number of outer iterations, where one iteration corresponds to one formulation and solution of the quadratic programming subproblem, or, alternatively, one evaluation of gradients (default is to use optParam.MaxIter if set, otherwise set maxit to 1000).
MISQP.mnfs Maximum number of feasible steps without improvements, where the relative change of objective function values is measured by acc and feasibility is measured by sqrt(acc), must be greater than 1 (default maxit).
MISQP.maxnde Maximum number of branch-and-bound steps for solving MIQP. Default is (2*n+m+4)^2, where n is the number of variables and m is the number of constraints.
MISQP.nonmon Maximum number of successive iterations, which are to be considered for the non-monotone trust region algorithm (default 2).
MISQP.acc Desired final accuracy. The termination accuracy should not be smaller than the accuracy by which gradients are computed. If acc<=0, the machine precision is computed by MISQP and subsequently multiplied by 1.0e4 (default 1e-6).
MISQP.accqp The tolerance is needed for the QP solver to perform several tests, for example whether optimality conditions are satisfied or whether a number is considered as zero or not. If accqp is less or equal to zero, then the machine precision is computed by MISQP and subsequently multiplied by 1.0E4 (default 1e-9).
MISQP.rpen Factor for increasing a penalty parameter (sigma), must be greater than one (default 10).
MISQP.dltdec Factor for decreasing the internal descent parameter DELTA. dltdec must be less than than one (default 0.1).
MISQP.sigma Initial penalty parameter, greater than one (default 1000).
MISQP.delta Initial scaling parameter (default 0.05).
MISQP.trustc Initial trust region radius for continuous variables, must be greater than or equal to one (default 10.0).
MISQP.trusti Initial trust region radius for integer variables (default 10.0).
MISQP.scbnd Scaling bound in case of fscale=1, i.e., function values are scaled only if their absolute initial values are greater than the bound (default 1.0).
MISQP.mrs Maximum number of internal restarts without improving solution. Setting might lead to better results, but increases the number of function evaluations. The value must be less than 15 (default 2).
MISQP.fscale MISQP internally scales function values.

0 - No scaling.

1 - Subject to their absolute values, if greater than 1.0 or scbnd, respectively (default).

MISQP.scale MISQP internally scales continuous variables.

0 - No scaling.

1 - Scaling enabled (default).

MISQP.bmod MISQP modifies the Hessian approximation in order to get more accurate search directions. Calculation time is increased in case of integer variables.

0 - No modification

1 - (default 1).

Options for the MIQP subsolver MIQL:

MISQP.iprqp Print level of subsolver MIQL (default 0).
MISQP.ibr Branching rule for MIQP subproblem

1 - Maximal fractional branching (default).

2 - Minimal fractional branching.

MISQP.ins Node selection rule for MIQP subproblem.

1 - Best of all.

2 - Best of two.

3 - Depth first (default).

MISQP.wstart Maximal number of successive warmstarts. Default 100.
MISQP.impb Set nonzero to calculate improved bounds when Best-of-All node selection strategy is used (default 0).
MISQP.dfdir Set nonzero to select direction for Depth-First according to value of Lagrange function (default 0).
MISQP.cholm Cholesky decomposition mode.

0 - Calculate Cholesky decomposition once and reuse it.

1 - Calculate new Cholesky decomposition whenever warmstart is not active (default).

MISQP.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.

MISQP.maxdc Maximal number of rounds for disjunctive cuts. Default 5.
MISQP.maxcm Maximal number of cuts for CMIR cuts. Default 5.
MISQP.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).
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.
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.
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

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

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

The variables ${\displaystyle x\in I}$, the index subset of ${\displaystyle 1,...,n}$, 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.

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.
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

Purpose

Solves strictly convex quadratic programming problems.

QL solves problems of the form

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

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

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.