SIAM Journal on Optimization, Volume 31, Issue 1, Page 1017-1048, January 2021.
We present adaptive sequential SAA (sample average approximation) algorithms to solve large-scale two-stage stochastic linear programs. The iterative algorithm framework we propose is organized into outer and inner iterations as follows: during each outer iteration, a sample-path problem is implicitly generated using a sample of observations or “scenarios," and solved only imprecisely, to within a tolerance that is chosen adaptively, by balancing the estimated statistical error against solution error. The solutions from prior iterations serve as warm starts to aid efficient solution of the (piecewise linear convex) sample-path optimization problems generated on subsequent iterations. The generated scenarios can be independent and identically distributed, or dependent, as in Monte Carlo generation using Latin-hypercube sampling, antithetic variates, or randomized quasi-Monte Carlo. We first characterize the almost-sure convergence (and convergence in mean) of the optimality gap and the distance of the generated stochastic iterates to the true solution set. We then characterize the corresponding iteration complexity and work complexity rates as a function of the sample size schedule, demonstrating that the best achievable work complexity rate is Monte Carlo canonical and analogous to the generic $\mathcal{O}(\epsilon^{-2})$ optimal complexity for nonsmooth convex optimization. We report extensive numerical tests that indicate favorable performance, due primarily to the use of a sequential framework with an optimal sample size schedule, and the use of warm starts. The proposed algorithm can be stopped in finite time to return a solution endowed with a probabilistic guarantee on quality.

中文翻译：

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求解两阶段随机线性程序的自适应顺序样本平均逼近

SIAM优化杂志，第31卷，第1期，第1017-1048页，2021年1月。
我们提出了自适应顺序SAA（样本平均逼近）算法来解决大规模两阶段随机线性程序。我们提出的迭代算法框架分为以下外部迭代和内部迭代：在每次外部迭代期间，使用观察值或“场景”的样本隐式生成一个样本路径问题，并且只能在不超出公差的情况下不精确地求解该样本路径问题。通过平衡估计的统计误差与解误差来自适应选择先前迭代的解可以作为热启动，以帮助有效解决后续迭代中产生的（分段线性凸）样本路径优化问题。与使用拉丁超立方体采样的蒙特卡洛方法相同，分布相同或相关，对立变量，或随机准蒙特卡罗。我们首先描述最优间隙的几乎确定的收敛性（以及均值的收敛性），以及所生成的随机迭代与真实解集的距离。然后，我们将相应的迭代复杂度和工作复杂度比率表征为样本量表的函数，证明最佳可实现的工作复杂度是蒙特卡洛规范的，类似于通用$ \ mathcal {O}（\ epsilon ^ {-2 }）$非光滑凸优化的最佳复杂度。我们报告了广泛的数值测试，这些结果表明了良好的性能，这主要归因于使用具有最佳样本量计划的顺序框架以及热启动。