A Quadratically Constrained Quadratic Optimization Model for Completely Positive Cone Programming

We propose a class of quadratic optimization problems which can be reformulated by completely positive cone programming with the same optimal values. The objective function can be any quadratic form. The constraints of each problem are described in terms of quadratic forms with no linear terms, and all constraints are homogeneous equalities, except one inhomogeneous equality where a quadratic form is set to be a positive constant. For the equality constraints, “a hierarchy of copositvity” condition is assumed. This model is a generalization of the standard quadratic optimization problem of minimizing a quadratic form over the standard simplex, and covers many of the existing quadratic optimization problems studied for exact copositive cone and completely positive cone programming relaxations. In particular, it generalizes the recent results on quadratic optimization problems by Burer and the set-semidefinite representation by Eichfelder and Povh [Optim. Lett., 7 (2013), pp. 1373--1386].

[1]  N. Shor.  Dual quadratic estimates in polynomial and Boolean programming , 1991 .

[2]  Samuel Burer,et al.  Representing quadratically constrained quadratic programs as generalized copositive programs , 2012, Oper. Res. Lett..

[3]  Franz Rendl,et al.  Copositive and semidefinite relaxations of the quadratic assignment problem , 2009, Discret. Optim..

[4]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[5]  Samuel Burer,et al.  On the copositive representation of binary and continuous nonconvex quadratic programs , 2009, Math. Program..

[6]  Peter J. C. Dickinson.  Geometry of the copositive and completely positive cones , 2011 .

[7]  Gabriele Eichfelder,et al.  On the set-semidefinite representation of nonconvex quadratic programs over arbitrary feasible sets , 2013, Optim. Lett..

[8]  Franz Rendl,et al.  Semidefinite programming and integer programming , 2002 .

[9]  J. Gallier.  Quadratic Optimization Problems , 2020, Linear Algebra and Optimization with Applications to Machine Learning.

[10]  K. P. Hadeler,et al.  On copositive matrices , 1983 .

[11]  Stanislav Busygin,et al.  Copositive Programming , 2009, Encyclopedia of Optimization.

[12]  F. Rendl,et al.  Copositive and Semidenite Relaxations of the Quadratic Assignment Problem y (appeared in Discrete Optimization 6 (2009) 231{241) , 2009 .

[13]  Etienne de Klerk,et al.  On Copositive Programming and Standard Quadratic Optimization Problems , 2000, J. Glob. Optim..

[14]  Masakazu Kojima,et al.  Semidefinite Programming Relaxation for Nonconvex Quadratic Programs , 1997, J. Glob. Optim..

[15]  Etienne de Klerk,et al.  Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming , 2002, J. Glob. Optim..

[16]  Masakazu Kojima,et al.  Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets , 1999, SIAM J. Optim..

[17]  Franz Rendl,et al.  A Copositive Programming Approach to Graph Partitioning , 2007, SIAM J. Optim..

[18]  P. H. Diananda.  On non-negative forms in real variables some or all of which are non-negative , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[19]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[20]  Etienne de Klerk,et al.  Approximation of the Stability Number of a Graph via Copositive Programming , 2002, SIAM J. Optim..

[21]  Mirjam Dür,et al.  Copositive Programming – a Survey , 2010 .

[22]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[23]  M. Kojima,et al.  A primal-dual interior point algorithm for linear programming , 1988 .