Well-Posed Uniform Solvability of Convex Optimization Problems on a Uniform Differentiable Closed Convex Set

Abstract

In this paper, we first give the definition of uniformly differentiable set and give the definitions of sets \(P(A,\eta , r)\) and \(P_{A,\delta }(f)\). Secondly, we prove that if the set A is bounded closed convex set, then A is uniformly differentiable if and only if for any \(\varepsilon , \eta , r>0\), there exists \(\delta =\delta (\varepsilon ,\eta ,r )>0\) such that \(\Vert x-y\Vert <\varepsilon \) whenever \(f\in P(A,\eta , r)\), \(y\in P_{A,\delta }(f)\) and \(x\in P_{A}(f)\). Moreover, we also prove that if A is a bounded closed convex set in a finite-dimensional space X, then A is differentiable if and only if A is uniformly differentiable. Finally, we give some examples of uniformly differentiable set. Therefore, we extend some conclusions (SIAM J. Optim. Vol. 30, No. 1, pp. 490–512).