## Discretization and Localization in Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems

### Masakazu Kojima and Levent Tuncel

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 c^T x 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, an implementable discretization of the SSDP (or SSILP) Relaxation Method generates a compact convex set C such that F \subseteq C \subseteq 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 $\epsilon$, an implementable localization -discretization of the SSDP (or SSILP) Relaxation Method generates an upper bound of the objective values within $\epsilon$ of their maximum in a finite number of iterations.

Research Report B-341, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Oh-Okayama, Meguoro, Tokyo 152, July, 1998. Also issued as COOR 98-34, Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.

Contact: kojima@is.titech.ac.jp