Convex Optimization Problems on Differentiable Sets

Abstract

Given a closed convex set in a Banach space , motivated by the continuity and Fréchet differentiability of introduced, respectively, in [D. Gale and V. Klee, Math. Scand., 7 (1959), pp. 379–391] and [X. Y. Zheng, SIAM J. Optim., 30 (2020), pp. 490–512], this paper considers the -differentiability, subdifferentiability, and Gâteaux differentiability of . Using the technique of variational analysis, it is proved that is -differentiable (resp., subdifferentiable or Gâteaux differentiable) if and only if for every continuous convex function with the corresponding constrained optimization problem is 2/ -order-well-posed solvable (resp., generalized well-posed solvable or weak well-posed solvable). It is also proved that if the conjugate function of a continuous convex function on is -differentiable on , then for every closed convex set in with the corresponding optimization problem is -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.