B-347 Towards the Implementation of Successive Convex Relaxation Method for Noncon- vex Quadratic Optimization Problems

Recently Kojima and Tun cel proposed new successive convex relaxation methods and their localized-discretized variants for general nonconvex quadratic programs. Although an upper bound of the objective function value within a prior precision can be found theoretically by solving a nite number of linear programs, several important implementation problems remain unsolved. In this paper we discuss these issues, present practically implementable algorithms and report numerical results.

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

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

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  Panos M. Pardalos,et al.  A Collection of Test Problems for Constrained Global Optimization Algorithms , 1990, Lecture Notes in Computer Science.

[5]  Ignacio E. Grossmann,et al.  Mixed-Integer Nonlinear Programming: A Survey of Algorithms and Applications , 1997 .

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

[7]  Gábor Pataki,et al.  On the generic properties of convex optimization problems in conic form , 2001, Math. Program..

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

[9]  Michael G. Safonov,et al.  Global optimization for the Biaffine Matrix Inequality problem , 1995, J. Glob. Optim..

[10]  Masakazu Kojima,et al.  Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization , 2000, Math. Program..

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

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

[13]  James E. Falk,et al.  Jointly Constrained Biconvex Programming , 1983, Math. Oper. Res..

[14]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

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

[16]  Susan W. Palocsay,et al.  Optimizing the sum of linear fractional functions , 1992 .

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

[18]  I. Grossmann,et al.  Reformulation of multiperiod MILP models for planning and scheduling of chemical processes , 1991 .

[19]  Hanif D. Sherali,et al.  A new reformulation-linearization technique for bilinear programming problems , 1992, J. Glob. Optim..

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

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

[22]  Efstratios N. Pistikopoulos,et al.  A Decomposition-Based Global Optimization Approach for Solving Bilevel Linear and Quadratic Programs , 1996 .