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

In this paper, we first give the definition of uniformly differentiable set and give the definitions of sets \(P(A,\eta , r)\) $P(A,\eta ,r)$ and \(P_{A,\delta }(f)\) $P_{A,\delta }(f)$ . Secondly, we prove that if the set A is bounded closed convex set, then A is uniformly differentiable if and only if for any \(\varepsilon , \eta , r>0\) $\epsilon ,\eta ,r>0$ , there exists \(\delta =\delta (\varepsilon ,\eta ,r )>0\) $\delta =\delta (\epsilon ,\eta ,r)>0$ such that \(\Vert x-y\Vert <\varepsilon \) $\Vert x-y\Vert <\epsilon$ whenever \(f\in P(A,\eta , r)\) $f\in P(A,\eta ,r)$ , \(y\in P_{A,\delta }(f)\) $y\in P_{A,\delta }(f)$ and \(x\in P_{A}(f)\) $x\in P_A(f)$ . 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).