CONVEX OPTIMIZATION PROBLEMS ON DIFFERENTIABLE SETS

Given a closed convex set A in a Banach space X, motivated by the continuity and Fre'\chet differentiability of A introduced, respectively, in [D. Gale and V. Klee, Math. Scand., 7 (1959), pp. 379--391] and [X. Y. Zheng, SIAM J. Optint., 30 (2020), pp. 490--512], this paper considers the CP-differentiability, subdifferentiability, and Ga<^>\teaux differentiability of A. Using the technique of variational analysis, it is proved that A is CP-differentiable (resp., subdifferentiable or Ga<^>\teaux differentiable) if and only if for every continuous convex function f : X-o J with infxEA f (x) > infxEX f (x) the corresponding constrained optimization problem PA(f) is 2/p-order -well-posed solvable (resp., generalized well-posed solvable or weak well-posed solvable). It is also proved that if the conjugate function f* of a continuous convex function f on X is CP-differentiable on dom(f *), then for every closed convex set A in X with infxEA f (x) > ooo the corresponding opti-mization problem PA(f) is 2/p-order-well-posed solvable. As a byproduct, every constrained convex optimization problem with a strongly convex quadratic objective function is proved to be globally second-order-well-posed solvable. Our main results are new even in the case of finite dimensional spaces.