Abstract

We propose new algorithms for (i) the local optimization of bound constrained quadratic programs, (ii) the solution of general definite quadratic programs, and (iii) finding either a point satisfying given linear equations and inequalities or a certificate of infeasibility. The algorithms are implemented in Matlab and tested against state-of-the-art quadratic programming software.

Highlights

  • B Waltraud HuyerWe only discuss the local solution of indefinite quadratic programs, needed repeatedly in branch and bound frameworks for finding good starting points for a global solver

  • We consider the quadratic programming problem with bound constraints min cT x + 1 2 x T Gx s.t. x ∈ x, (1)where c ∈ Rn, G is a symmetric n × n matrix, not necessarily semidefinite, x := [x, x] = {x ∈ Rn | xi ≤ xi ≤ xi, i = 1, . . . , n} (2) B Waltraud HuyerW

  • The numerical examples show that the MINQ8 performs well and is competitive with other quadratic programming algorithms implemented in Matlab

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Summary

B Waltraud Huyer

We only discuss the local solution of indefinite quadratic programs, needed repeatedly in branch and bound frameworks for finding good starting points for a global solver. For strictly convex bound constrained quadratic programs, many algorithms are available. MINQ5 [20] is a publicly available Matlab program for bound constrained quadratic programming and strictly convex general quadratic programming, based on rank 1 modifications. It finds a local minimizer of the problem (1). As MINQ5, our Matlab 8 implementation of MINQ8 solves both the general bound constrained quadratic program and the related problems discussed in Sect.

The MINQ8 algorithm
The reduced inactive set
Infinite bounds
Termination and convergence
Solving strictly convex quadratic programs
Numerical examples
Test Set 1: random sparse problems
Test Set 2: quadratic problems from CUTEr
Test Set 3: separable quadratic programs
Discussion summary
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