Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization

Abstract.Based on the authors’ previous work which established theoretical foundations of two, conceptual, successive convex relaxation methods, i.e., the SSDP (Successive Semidefinite Programming) Relaxation Method and the SSILP (Successive Semi-Infinite Linear Programming) Relaxation Method, this paper proposes their implementable variants for general quadratic optimization problems. These problems have a linear objective function cTx to be maximized over a nonconvex compact feasible region F described by a finite number of quadratic inequalities. We introduce two new techniques, “discretization” and “localization,” into the SSDP and SSILP Relaxation Methods. The discretization technique makes it possible to approximate an infinite number of semi-infinite SDPs (or semi-infinite LPs) which appeared at each iteration of the original methods by a finite number of standard SDPs (or standard LPs) with a finite number of linear inequality constraints. We establish:¶•Given any open convex set U containing F, there is an implementable discretization of the SSDP (or SSILP) Relaxation Method which generates a compact convex set C such that F⊆C⊆U in a finite number of iterations.¶The localization technique is for the cases where we are only interested in upper bounds on the optimal objective value (for a fixed objective function vector c) but not in a global approximation of the convex hull of F. This technique allows us to generate a convex relaxation of F that is accurate only in certain directions in a neighborhood of the objective direction c. This cuts off redundant work to make the convex relaxation accurate in unnecessary directions. We establish:¶•Given any positive number ε, there is an implementable localization-discretization of the SSDP (or SSILP) Relaxation Method which generates an upper bound of the objective value within ε of its maximum in a finite number of iterations.

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

[2]  Alexander Schrijver,et al.  Cones of Matrices and Set-Functions and 0-1 Optimization , 1991, SIAM J. Optim..

[3]  M. Ramana.  An algorithmic analysis of multiquadratic and semidefinite programming problems , 1994 .

[4]  Y. Nesterov.  Semidefinite relaxation and nonconvex quadratic optimization , 1998 .

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

[6]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

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

[8]  D SheraliHanif,et al.  A hierarchy of relaxations and convex hull characterizations for mixed-integer zero-one programming problems , 1994 .

[9]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

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

[11]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[12]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations Between the Continuous and Convex Hull Representations for Zero-One Programming Problems , 1990, SIAM J. Discret. Math..

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

[14]  Egon Balas,et al.  A lift-and-project cutting plane algorithm for mixed 0–1 programs , 1993, Math. Program..

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

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

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

[18]  Michel X. Goemans,et al.  Semideenite Programming in Combinatorial Optimization , 1999 .

[19]  Mehran Mesbahi,et al.  A cone programming approach to the bilinear matrix inequality problem and its geometry , 1997, Math. Program..

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

[21]  Vladimir A. Yakubovich,et al.  Linear Matrix Inequalities in System and Control Theory (S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan) , 1995, SIAM Rev..

[22]  Satissed Now Consider.  Improved Approximation Algorithms for Maximum Cut and Satissability Problems Using Semideenite Programming , 1997 .

[23]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[24]  Hanif D. Sherali,et al.  A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique , 1992, J. Glob. Optim..

[25]  Hanif D. Sherali,et al.  A reformulation-convexification approach for solving nonconvex quadratic programming problems , 1995, J. Glob. Optim..

[26]  K. Goh,et al.  Control system synthesis via bilinear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[27]  Akiko Takeda,et al.  Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems , 2000 .

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

[29]  Hanif D. Sherali,et al.  A Hierarchy of Relaxations and Convex Hull Characterizations for Mixed-integer Zero-one Programming Problems , 1994, Discret. Appl. Math..