ON SOLUTION SET FOR CONVEX OPTIMIZATION PROBLEM WITH CONVEX INTEGRABLE OBJECTIVE FUNCTION AND GEOMETRIC CONSTRAINT SET

Abstract

In this paper, we consider a convex optimization prob- lem with a convex integrable objective function and a geometric constraint set. We characterize the solution set of the problem when we know its one solution. 1. Introduction and preliminaries Convex optimization problems often have multiple solutions. Re- cently, Mangasarian (10) established simple and complete characteriza- tions for the solution set of the problem when we knew one solution of the problem. Since then, many authors have studied such charac- terizations for solution sets of several classes of optimization problems (2, 3, 5, 6, 8, 9, 11, 12, 13, 15). In particular, Jeyakumar, Lee and Dinh (5) showed that the Lagrangian function of a cone-constrained convex optimization problem, which has a cone-inequality constraint, is con- stant on its solution set, and then derived the Lagrange multiplier based characterizations of the solution set when we know one solution of the solution set. Moreover, Jeyakumar, Lee and Li (7) developed the char- acterizations of the solution sets to convex optimization problems in the face of data uncertainty.