SIAM Journal on Optimization, Volume 30, Issue 3, Page 2530-2558, January 2020.
This paper studies the class of two-stage stochastic programs with a linearly bi-parameterized recourse function defined by a convex quadratic program. A distinguishing feature of this new class of nonconvex stochastic programs is that the objective function in the second stage is linearly parameterized by the first-stage decision variable, in addition to the standard linear parameterization in the constraints. While a recent result has established that the resulting recourse function is of the difference-of-convex (dc) kind, the associated dc decomposition of the recourse function does not provide an easy way to compute a directional stationary solution of the two-stage stochastic program. Based on an implicit convex-concave property of the bi-parameterized recourse function, we introduce the concept of a generalized critical point of such a recourse function and provide a sufficient condition for such a point to be a directional stationary point of the stochastic program. We describe an iterative algorithm that combines regularization, convexification, and sampling and establish the subsequential convergence of the algorithm to a generalized critical point, with probability 1.

中文翻译：

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线性双参数二次求偿的两阶段随机规划

SIAM优化杂志，第30卷，第3期，第2530-2558页，2020年1月。
本文研究由凸二次程序定义的线性双参数追索函数的两阶段随机程序的类。这类新的非凸随机程序的一个显着特征是，除了约束中的标准线性参数化外，第二阶段的目标函数还由第一阶段决策变量线性化。尽管最近的一项结果表明，所得的追索函数是凸差（dc）类型的，但追索函数的相关dc分解并不能提供一种简便的方法来计算两阶段随机的有向静态解程序。基于双参数追索函数的隐式凹凸性质，我们介绍了这种追索函数的广义临界点的概念，并为该点成为随机程序的有向静止点提供了充分的条件。我们描述了一种将正则化，凸化和采样相结合的迭代算法，并建立了该算法对概率为1的广义临界点的后续收敛。