Adaptive computable approximation to cones of nonnegative quadratic functions

Cones of nonnegative quadratic functions are keys to the understanding of quadratic optimization problems, since any quadratically constrained quadratic programming problem can be reformulated as a linear conic programming problem over such a cone. This paper proposes an adaptive computable approximation scheme to cones of nonnegative quadratic functions and uses it for solving linear conic programming problems over such a cone. We study some basic properties of cones of nonnegative quadratic functions and present a class of simple cones with computable linear matrix inequalities representations. Building on these simple cones, we design a computable approximation scheme for handling a general cone of nonnegative quadratic functions. When the scheme is applied for solving linear conic programming problems over a cone of nonnegative quadratic functions, we incorporate an adaptive approach to enhance the performance of the proposed computable approximation scheme. The computational performance and theoretic convergence proof of the proposed adaptive computable approximation scheme are shown for solving box-constrained quadratic programming problems.

[1]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[2]  Hanif D. Sherali,et al.  Reformulation-Linearization Technique for Global Optimization , 2009, Encyclopedia of Optimization.

[3]  Wenxun Xing,et al.  CANONICAL DUALITY AND CONIC PROGRAMMING FOR SOLVING 0-1 QUADRATIC PROGRAMMING PROBLEMS , 2010 .

[4]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.

[5]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[6]  Duan Li,et al.  Convex relaxations for nonconvex quadratically constrained quadratic programming: matrix cone decomposition and polyhedral approximation , 2011, Math. Program..

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

[8]  Hanif D. Sherali,et al.  Reformulation-Linearization Methods for Global Optimization , 2007 .

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

[10]  R. Rockafellar.  Convex Analysis: (pms-28) , 1970 .

[11]  Mirjam Dür,et al.  An Adaptive Linear Approximation Algorithm for Copositive Programs , 2009, SIAM J. Optim..

[12]  Samuel Burer,et al.  Computable representations for convex hulls of low-dimensional quadratic forms , 2010, Math. Program..

[13]  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 .

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

[15]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[16]  Javier Peña,et al.  LMI Approximations for Cones of Positive Semidefinite Forms , 2006, SIAM J. Optim..

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

[18]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[19]  Samuel Burer,et al.  Separation and relaxation for cones of quadratic forms , 2013, Math. Program..

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

[21]  利久 亀井,et al.  California Institute of Technology , 1958, Nature.

[22]  Kurt M. Anstreicher,et al.  On convex relaxations for quadratically constrained quadratic programming , 2012, Math. Program..