SIAM Journal on Optimization, Volume 30, Issue 1, Page 823-852, January 2020.
This paper presents the stochastic decomposition (SD) algorithms for two classes of stochastic programming problems: (1) two-stage stochastic quadratic-linear programming (SQLP) in which a quadratic program defines the objective function in the first stage and a linear program defines the value function in the second stage and (2) two-stage stochastic quadratic-quadratic programming (SQQP) which has quadratic programming problems in both stages. Similar to their stochastic linear programming (SLP) predecessor, these iterative schemes in SD approximate the objective function using piecewise affine/quadratic minorants and then apply a stochastic proximal mapping to obtain the next iterate. In this paper we show that under some assumptions, the proximal mapping applied in SD obeys a contraction mapping property even though the approximations are based on sequential random samples. Following that, we demonstrate that under those assumptions, SD can provide a sequence of solutions converging to the optimal solution with a sublinear convergence rate in both SQLP and SQQP problems. Finally, we present an “in-sample” stopping rule to assess the optimality gap by constructing consistent bootstrap estimators.

中文翻译：

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两阶段随机二次规划的随机分解的渐近结果

SIAM优化杂志，第30卷，第1期，第823-852页，2020年1月。
本文针对两类随机规划问题提出了随机分解（SD）算法：（1）两阶段随机二次线性规划（SQLP），其中二次规划定义了第一阶段的目标函数，而线性规划则定义了第二阶段的值函数和（2）两阶段均具有二次规划问题的两阶段随机二次-二次规划（SQQP）。类似于其随机线性规划（SLP）的前身，SD中的这些迭代方案使用分段仿射/二次次要分量逼近目标函数，然后应用随机近端映射来获取下一个迭代。在本文中，我们表明，在某些假设下，即使近似值是基于顺序随机样本，SD中应用的近端映射也遵循收缩映射特性。随后，我们证明了在这些假设下，SD可以在SQLP和SQQP问题中以亚线性收敛速率提供收敛到最优解的一系列解。最后，我们提出了一个“样本中”停止规则，以通过构造一致的自举估计器来评估最佳差距。