SIAM Journal on Optimization, Volume 30, Issue 1, Page 490-512, January 2020.
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 rightarrow R bounded below on A the corresponding optimization problem inf_x in Af(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 rightarrow R bounded below on A the corresponding optimization problem inf_x in Af(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 Fréchet differentiable (resp., continuous) on dom(f^*), then for every closed convex set K in X with inf_x in Kf(x)>-infty 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.

中文翻译：

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可微或连续封闭凸集上凸优化问题的适定可解性

SIAM优化杂志，第30卷，第1期，第490-512页，2020年1月。
给定Banach空间X中的一个封闭凸集A，本文考虑了A的连续性和可微性。一个封闭凸集的连续性是由Gale和Klee [Math.M. [1]（Scand。，7（1959），pp。370--391），而封闭凸集的可微性又是从其支持功能方面引入的新概念。使用变分分析技术，我们证明，当且仅当对于以下连续的线性（或凸）函数f：X右箭头R限定在A上，相应的优化问题inf_x可适当解决时，A是可微的。在自反空间的情况下，我们证明A是连续的，当且仅当对于每个连续线性（或凸）函数f：X右箭头R的边界在A下方，Af（x）中的相应优化问题inf_x弱可解决。我们还证明，如果给定的连续凸函数f在X上的共轭函数f ^ *在dom（f ^ *）上是Fréchet微分的（分别是连续的），那么对于X中每个封闭凸集K，Kf中的inf_x （x）>-infty具有目标f和约束集K的相应优化问题可以很好地解决（分别为弱很好地解决）。在有限维空间的框架中，建立了一些更清晰的结果。灵活地解决具有目标f和约束集K的相应优化问题。在有限维空间的框架中，建立了一些更清晰的结果。灵活地解决具有目标f和约束集K的相应优化问题。在有限维空间的框架中，建立了一些更清晰的结果。