Relaxing the optimality conditions of box QP

We present semidefinite relaxations of nonconvex, box-constrained quadratic programming, which incorporate the first- and second-order necessary optimality conditions, and establish theoretical relationships between the new relaxations and a basic semidefinite relaxation due to Shor. We compare these relaxations in the context of branch-and-bound to determine a global optimal solution, where it is shown empirically that the new relaxations are significantly stronger than Shor’s. An effective branching strategy is also developed.

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

[2]  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 .

[3]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[4]  Panos M. Pardalos,et al.  Quadratic Programming with Box Constraints , 1997 .

[5]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[6]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[7]  Abul Hasan Siddiqi,et al.  Trends in Industrial and Applied Mathematics , 2011 .

[8]  Yinyu Ye,et al.  Approximating quadratic programming with bound and quadratic constraints , 1999, Math. Program..

[9]  Panos M. Pardalos,et al.  GLOBAL OPTIMIZATION ALGORITHMS FOR LINEARLY CONSTRAINED INDEFINITE QUADRATIC PROBLEMS , 1991 .

[10]  Warren P. Adams,et al.  A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems , 1998 .

[11]  Aimo A. Törn,et al.  Global Optimization , 1999, Science.

[12]  Samuel Burer,et al.  Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound , 2009, Comput. Optim. Appl..

[13]  P. Toint,et al.  Numerical Methods for Large-Scale Non-Convex Quadratic Programming , 2002 .

[14]  George L. Nemhauser,et al.  A branch-and-cut algorithm for nonconvex quadratic programs with box constraints , 2005, Math. Program..

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

[16]  M. Er.  Quadratic optimization problems in robust beamforming , 1990 .

[17]  Y. Nesterov.  Global quadratic optimization via conic relaxation , 1998 .

[18]  George L. Nemhauser,et al.  A polyhedral study of nonconvex quadratic programs with box constraints , 2005, Math. Program..

[19]  Tibor Csendes,et al.  Developments in Global Optimization , 1997 .