SIAM Journal on Optimization, Volume 31, Issue 3, Page 1635-1657, January 2021.
In this paper, we investigate the Moreau envelope of the supremum of a family of convex, proper, and lower semicontinuous functions. Under mild assumptions, we prove that the Moreau envelope of a supremum is the supremum of Moreau envelopes, which allows us to approximate possibly nonsmooth supremum functions by smooth functions that are also the suprema of functions. Consequently, we propose and study approximated optimization problems from infinite and stochastic programming for which we obtain zero-duality gap results and optimality conditions without the verification of constraint qualification conditions.

中文翻译：

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最高函数的莫罗包络及其在无限和随机规划中的应用

SIAM Journal on Optimization，第 31 卷，第 3 期，第 1635-1657 页，2021 年 1 月
。在本文中，我们研究了凸、适当和下半连续函数族的上确界的莫罗包络。在温和的假设下，我们证明了一个上确界的莫罗包络是莫罗包络的上确界，这使我们能够通过也是函数的上确界的光滑函数来近似可能不光滑的上确界函数。因此，我们提出并研究了来自无限和随机规划的近似优化问题，我们在没有验证约束条件条件的情况下获得了零对偶间隙结果和最优条件。