AI 中文总结
研究非光滑优化中采样和逼近方案下集值积分稳定性，通过紧致参数集、紧致凸值等条件，得出超线性可积下积分映射图形收敛结论，为测度逼近提供原理，还讨论了假设尖锐性及应用。
AI 中文摘要
受非光滑优化中采样和逼近方案的启发，我们研究了在基础概率分布的弱扰动下参数化集值积分的稳定性。对于紧致参数集，我们表明紧致凸值且联合外半连续的被积函数会诱导集值积分映射，其在任何概率测度的弱收敛序列下沿超额距离图形收敛。该结果在超线性可积条件下成立，为集值期望的测度逼近提供了统一的稳定性原理。我们通过例子讨论了假设的尖锐性。特别强调一般需要联合外半连续，且超线性包络条件相对于经典独立同分布经验设置是紧的。由此，我们得到了随机广义方程解集的外稳定性。我们在几种情况下说明了稳定性结果，包括具有马尔可夫采样的随机非光滑优化、通过平滑函数进行平滑以及参数依赖的分布动力学。
英文摘要
Motivated by sampling and approximation schemes arising in nonsmooth optimization, we study the stability of parameterized set-valued integrals under weak perturbations of the underlying probability distribution. For a compact parameter set, we show that compact convex-valued and jointly outer semicontinuous integrands induce set-valued integral maps that converge graphically in excess distance along any weakly convergent sequence of probability measures. The result holds under a superlinear integrability condition and provides a unified stability principle for measure approximations of set-valued expectations. We discuss the sharpness of the assumptions through examples. In particular, we emphasize that joint outer semicontinuity is required in general and that the superlinear envelope condition is tight relative to the classical i.i.d. empirical setting. As a consequence, we obtain outer stability of solution sets for stochastic generalized equations. We illustrate the stability result in several settings, including stochastic nonsmooth optimization with Markovian sampling, smoothing by mollifiers, and parameter-dependent distributional dynamics.
