Name
Title | ||
|---|---|---|
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 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, 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. |
Year | DOI | Venue |
|---|---|---|
2000 | 10.1007/PL00011394 | Math. Program. |
Keywords | Field | DocType |
Key words: nonconvex quadratic optimization problem – semidefinite programming – linear matrix inequality – global optimization – SDP relaxation – semi-infinite LP relaxation – interior-point method | Discrete mathematics,Mathematical optimization,Relaxation (iterative method),Convex hull,Convex set,Feasible region,Proper convex function,Interior point method,Semidefinite programming,Linear matrix inequality,Mathematics | Journal |
Volume | Issue | ISSN |
89 | 1 | 0025-5610 |
Citations | PageRank | References |
9 | 1.27 | 11 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Masakazu Kojima | 1 | 1603 | 222.51 |
| Levent Tunçel | 2 | 429 | 72.12 |