通过逐步采样优化大规模等式约束问题，提升求解效率。
Progressively Sampled Equality-Constrained Optimization
- 逐次增加样本规模，分阶段求解优化问题。
- 理论证明比一次性使用全部样本更优，样本复杂度更低。
- 适合大规模期望型优化，实测有效。
本文提出、分析并测试了一种求解连续非线性等式约束优化问题的算法，其中目标函数和约束函数由大量有限项的期望或平均值定义。算法核心思想是求解一系列相关问题，每个问题基于逐渐增长的目标与约束函数项的有限采样集。在合理假设下（包括问题函数及其一阶、二阶导数性质），若初始样本量足够大，则通过渐进采样求解序列问题，相比直接使用全部样本求解单个问题，能获得更优的最坏情况样本复杂度上界。数值实验在一组测试问题上验证了该方法在实际应用中的有效性。
原文摘要 · Abstract (English)
An algorithm is proposed, analyzed, and tested for solving continuous nonlinear-equality-constrained optimization problems where the objective and constraint functions are defined by expectations or averages over large, finite numbers of terms. The main idea of the algorithm is to solve a sequence of related problems, each involving finite samples of objective- and constraint-function terms, over which the sample sets grow progressively. Under assumptions about the problem functions and their first- and second-order derivatives that are reasonable in real-world settings of interest, it is shown that -- with sufficiently large initial sample sizes -- solving a sequence of problems defined through progressive sampling yields a better worst-case sample complexity bound compared to solving a single problem with the full sets of samples. The results of numerical experiments with a set of test problems demonstrate that the proposed approach can be effective in practice.
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