Conic approximation to nonconvex quadratic programming with convex quadratic constraints

In this paper, a conic reformulation and approximation is proposed for solving a nonconvex quadratic programming problem subject to several convex quadratic constraints. The original problem is transformed into a linear conic programming problem, which can be approximated by a sequence of linear conic programming problems over the dual cone of the cone of nonnegative quadratic functions. Since the dual cone of the cone of nonnegative quadratic functions has a linear matrix inequality representation, each linear conic programming problem in the sequence can be solved efficiently using the semidefinite programming techniques. In order to speed up the convergence of the approximation sequence and relieve the computational effort in solving the linear conic programming problems, an adaptive scheme is adopted in the proposed algorithm. We prove that the lower bounds generated by the linear conic programming problems converge to the optimal value of the original problem. Several numerical examples are used to illustrate how the algorithm works and the computational results demonstrate the efficiency of the proposed algorithm.

[1]  A. J. Quist,et al.  Copositive realxation for genera quadratic programming , 1998 .

[2]  Shu-Cherng Fang,et al.  Adaptive computable approximation to cones of nonnegative quadratic functions , 2014 .

[3]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

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

[5]  Richard A. Tapia,et al.  A trust region strategy for nonlinear equality constrained op-timization , 1984 .

[6]  Samuel Burer,et al.  Second-Order-Cone Constraints for Extended Trust-Region Subproblems , 2013, SIAM J. Optim..

[7]  Panos M. Pardalos,et al.  Quadratic programming with one negative eigenvalue is NP-hard , 1991, J. Glob. Optim..

[8]  Nicholas I. M. Gould,et al.  Trust Region Methods , 2000, MOS-SIAM Series on Optimization.

[9]  Kurt M. Anstreicher,et al.  Institute for Mathematical Physics Semidefinite Programming versus the Reformulation–linearization Technique for Nonconvex Quadratically Constrained Quadratic Programming Semidefinite Programming versus the Reformulation-linearization Technique for Nonconvex Quadratically Constrained , 2022 .

[10]  Samuel Burer,et al.  A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations , 2008, Math. Program..

[11]  Franz Rendl,et al.  A recipe for semidefinite relaxation for (0,1)-quadratic programming , 1995, J. Glob. Optim..

[12]  Shuzhong Zhang,et al.  New Results on Quadratic Minimization , 2003, SIAM J. Optim..

[13]  Zhi-Quan Luo,et al.  Semidefinite Relaxation of Quadratic Optimization Problems , 2010, IEEE Signal Processing Magazine.

[14]  Warren P. Adams,et al.  A hierarchy of relaxation between the continuous and convex hull representations , 1990 .

[15]  Duan Li,et al.  Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations , 2011, J. Glob. Optim..

[16]  Jorge J. Moré,et al.  The NEOS Server , 1998 .

[17]  Nikolaos V. Sahinidis,et al.  BARON: A general purpose global optimization software package , 1996, J. Glob. Optim..

[18]  Franz Rendl,et al.  A semidefinite framework for trust region subproblems with applications to large scale minimization , 1997, Math. Program..

[19]  Shu-Cherng Fang,et al.  KKT Solution and Conic Relaxation for Solving Quadratically Constrained Quadratic Programming Problems , 2011, SIAM J. Optim..

[20]  Shuzhong Zhang,et al.  On Cones of Nonnegative Quadratic Functions , 2003, Math. Oper. Res..

[21]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[22]  Pierre Hansen,et al.  A branch and cut algorithm for nonconvex quadratically constrained quadratic programming , 1997, Math. Program..