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

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...

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Bibliographic Details
Published in:Journal of global optimization Vol. 82; no. 4; pp. 659 - 689
Main Authors: Bentobache, Mohand, Telli, Mohamed, Mokhtari, Abdelkader
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2022
Springer
Springer Nature B.V
Subjects:
Algorithms
Computer Science
Linear programming
Mathematics
Mathematics and Statistics
Mixed integer
Operations Research/Decision Theory
Optimization
Quadratic programming
Real Functions
90C20
Local minimizer
Nonconvex quadratic programming
Concave quadratic programming
Extreme point
90C26
90C59
Successive linear programming
Global minimizer
Numerical experiments
Approximate global minimizer
Simplex algorithm
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary: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.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-021-01108-w