Computational Optimization and Applications, 26, 143–154, 2003 c  2003 Kluwer Academic Publishers. Manufactured in The Netherlands. Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations SUNYOUNG KIM ∗ skim@ewha.ac.kr; skim@is.titech.ac.jp Department of Mathematics, Ewha Women’s University, 11-1 Dahyun-dong, Sudaemoon-gu, Seoul 120-750, Korea MASAKAZU KOJIMA kojima@is.titech.ac.jp Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152-8552, Japan Received July 2002; Revised February 2003 Abstract. We show that SDP (semidefinite programming) and SOCP (second order cone programming) relax- ations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a gener- alization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization prob- lems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation. Keywords: nonconvex quadratic optimization problem, semidefinite programming relaxation, second order cone programming relaxation, sparsity 1. Introduction We are concerned with solving quadratic optimization problems (QOPs) with quadratic con- straints by semidefinite programming (SDP) relaxation and second order cone programming (SOCP) relaxation. QOPs have been a subject of extensive study for their theoretical and practical importance in optimization. The focus of this paper, in particular, is on nonconvex QOPs which involve indefinite coefficient matrices in the objective function and constraints. QOPs arise in a broad range of fields such as combinatorial optimization, numerical partial differential equations from engineering, control and finance, and general nonlinear programming problems. Nonconvex QOPs include indefinite symmetric matrices in the ob- jective and constraints, as opposed to convex QOPs whose coefficient matrices are positive semidefinite. Local optimizers of convex QOPs serve as global optimizers, hence, an ap- proximate global optimizer can be found using many publically available codes [1, 8, 9]. For ∗ A considerable amount of this work was conducted while this author was visiting Tokyo Institute of Technology, Department of Mathematical and Computing Sciences. Research supported by Kosef R004-000-2001-00200.