WELL-POSED SOLVABILITY OF CONVEX OPTIMIZATION PROBLEMS ON A DIFFERENTIABLE OR CONTINUOUS CLOSED CONVEX SET

Given a closed convex set A in a Banach space X, this paper considers the continuity and differentiability of A. The continuity of a closed convex set was introduced and studied by Gale and Klee [Math. Scand., 7 (1959), pp. 370-391] in terms of its support functional, and the differentiability of a closed convex set is a new notion introduced again in terms of its support functional. Using the technique of variational analysis, we prove that A is differentiable if and only if for every continuous linear (or convex) function f : X -> R bounded below on A the corresponding optimization problem inf(x is an element of A) f(x) is well-posed solvable. In the reflexive space case, we prove that A is continuous if and only if for every continuous linear (or convex) function f : X -> R bounded below on A the corresponding optimization problem inf(x is an element of A) (x) is weakly well-posed solvable. We also prove that if the conjugate function f* of a given continuous convex function f on X is Frechet differentiable (resp., continuous) on dom(f*), then for every closed convex set K in X with inf(x is an element of K) f(x) > -infinity the corresponding optimization problem with objective f and constraint set K is well-posed (resp., weakly well-posed) solvable. In the framework of finite-dimensional spaces, several sharper results are established.