Moreau Envelope of Supremum Functions with Applications to Infinite and Stochastic Programming
研究了凸下半连续函数族上确界的Moreau包络，证明其等于各函数Moreau包络的上确界，从而用光滑函数逼近非光滑上确界函数，并应用于无限和随机规划中避免约束规格验证的对偶间隙消除与最优性条件。
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.