ON SOLUTION SETS FOR CONVEX OPTIMIZATION PROBLEMS

On 1988, O. L. Mangasarian [8] proved that the gradient of the objective function of a convex optimization problem, which has a constrained set defined by a convex set, is constant on the solution set of the problem, and presented the characterization of the solution set by the gradient of the objective function when we know one solution of the problem. Since then, many researchers have studied the characterizations of the solution sets of several kinds of optimization problems. In particular, on 2004, V. Jeyakumar, G. M. Lee and N. Dinh [3] established that the Lagrangian function of a cone-constrained convex optimization problem, which has a cone-inequality constraint, is constant on its solution set, and then derive the characterizations of the solution set when we know one solution of the solution set, by using the well-known optimality conditions. On 2015, V. Jeyakumar, G. M. Lee and G. Li [5] investigated the characterizations of the solution set of a convex optimization problem with data uncertainty, which is called the robust convex optimization problem. On 2016, G. M. Lee and J. H. Lee [6] obtained the characterizations of the solution set for a convex optimization problem with convex integrable objective function and geometric constraint set, and G.M. Lee and J. C. Yao [7] gave the Lagrange multiplier based characterization on the solution set of a robust optimization problem which consists of pseudo-convex locally Lipschitz objective function and convex constraint function. In this paper, we review characterizations for solution sets for convex optimizations, which was given by O. L. Mangasarian [8], and then we review our main results on the characterizations of the solution sets of convex(psudoconvex)optimization problems given by G. M. Lee and J. H. Lee [6], and G. M. Lee and J. C. Yao [7].