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arXiv 2605.03400math.OC
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基于二次近似的随机逼近方法求解弱凸随机规划

A Quadratic-Approximation-Based Stochastic Approximation Method for Weakly Convex Stochastic Programming

Dalian Maritime University（大连海事大学）；Peking University（北京大学）；Dalian University of Technology（大连理工大学）；Northeastern University（东北大学）

Yule Zhang, Benqi Liu, Xiantao Xiao, Liwei Zhang

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40 pages, 8 figures, 1 table
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AI总结：

提出PMQSopt算法，结合乘子近端法与二次近似求解弱凸随机优化问题，证明在期望值下梯度范数、约束违背和互补松弛违背的收敛率为O(T^{-1/4})，高概率下可达O(T^{-1/8})和O(T^{-1/4})。

摘要：

我们提出了一种新颖的随机逼近算法，称为PMQSopt，用于求解涉及期望值函数的弱凸随机优化问题。该算法通过将乘子近端法与原始随机问题的二次近似相结合而构建。我们分析了PMQSopt在所需随机梯度评估总数方面的样本复杂度。算法的收敛性由与ε-KKT条件相关的三个指标表征：拉格朗日函数Moreau包络梯度的平均平方范数、平均约束违背和平均互补松弛违背。对于每个指标，我们建立了在T次迭代后期望收敛率为O(T^{-1/4})。此外，我们证明，以至少1-1/T^{2/3}的概率，拉格朗日函数的梯度满足O(T^{-1/8})界；以至少1-2/T^{2/3}的概率，约束违背达到O(T^{-1/4})界；以至少1-3/T^{2/3}的概率，互补松弛违背达到O(T^{-1/4})界。所有结果均在两个温和条件下建立：(i) 所有问题函数的弱凸性，(ii) 存在严格可行点。所提出的PMQSopt算法是一种易于实现的序列强凸规划方法。数值实验说明了其实际性能。

英文摘要：

We propose a novel stochastic approximation algorithm, termed PMQSopt, for solving weakly convex stochastic optimization problems involving expectation-valued functions. The algorithm is constructed by integrating the proximal method of multipliers with quadratic approximations of the original stochastic problem. We analyze the sample complexity of PMQSopt in terms of the total number of stochastic gradient evaluations required. The convergence of the algorithm is characterized by three metrics associated with the $ε$-KKT conditions: the average squared norm of the gradient of the Moreau envelope of the Lagrangian, the average constraint violation, and the average complementarity violation. For each of these metrics, we establish an expected convergence rate of $\mathcal{O}(T^{-1/4})$ after $T$ iterations. Furthermore, we show that with probability at least $1-1/T^{2/3}$, the gradient of the Lagrangian satisfies an $\mathcal{O}(T^{-1/8})$ bound; with probability at least $1-2/T^{2/3}$, the constraint violation achieves an $\mathcal{O}(T^{-1/4})$ bound; and with probability at least $1-3/T^{2/3}$, the complementarity violation attains an $\mathcal{O}(T^{-1/4})$ bound. All results are established under two mild conditions: (i) weak convexity of all problem functions, and (ii) the existence of a strictly feasible point. The proposed PMQSopt algorithm is a sequentially strongly convex programming method that is readily implementable. Numerical experiments illustrate its practical performance.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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