We consider nonlinear multistage stochastic optimization problems in the spaces of integrable functions. We allow for nonlinear dynamics and general objective functionals, including dynamic risk measures. We study causal operators describing the dynamics of the system and derive the Clarke subdifferential for a penalty function involving such operators. Then we introduce the concept of subregular recourse in nonlinear multistage stochastic optimization and establish subregularity of the resulting systems in two formulations: with built-in nonanticipativity and with explicit nonanticipativity constraints. Finally, we derive optimality conditions for both formulations and study their relations.

中文翻译：

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非线性多级随机优化中的子正则资源

我们考虑可积函数空间中的非线性多级随机优化问题。我们允许非线性动力学和一般目标函数，包括动态风险度量。我们研究描述系统动力学的因果算子，并推导出涉及此类算子的惩罚函数的克拉克次微分。然后，我们在非线性多级随机优化中引入了次正则资源的概念，并在两种公式中建立了结果系统的次正则性：具有内置非预期性和显式非预期性约束。最后，我们推导出两个公式的最优条件并研究它们的关系。