New LP-based local and global algorithms for continuous and mixed-integer nonconvex quadratic programming

Abstract

In this work, we propose a new approach called "Successive Linear Programming Algorithm (SLPA)" for finding an approximate global minimizer of general nonconvex quadratic programs. This algorithm can be initialized by any extreme point of the convex polyhedron of the feasible domain. Furthermore, we generalize the simplex algorithm for finding a local minimizer of concave quadratic programs written in standard form. We prove a new necessary and sufficient condition for local optimality, then we describe the Revised Primal Simplex Algorithm (RPSA). Finally, we propose a hybrid local-global algorithm called "SLPLEX", which combines RPSA with SLPA for solving general concave quadratic programs. In order to compare the proposed algorithms to the branch-and-bound algorithm of CPLEX12.8 and the branch-and-cut algorithm of Quadproga, we develop an implementation with MATLAB and we present numerical experiments on 139 nonconvex quadratic test problems. KeywordsNonconvex quadratic programming • Concave quadratic programming • Successive linear programming • Simplex algorithm • Extreme point • Local minimizer • Global minimizer • Approximate global minimizer • Numerical experiments Mathematics Subject Classification 90C20 • 90C26 • 90C59 B Mohand Bentobache