局部正则性下基于一阶增广拉格朗日方法的非凸复合函数约束

中文摘要 AI 辅助

我们研究了一类具有凸复合结构的非凸约束优化问题的原始对偶方法的非渐近收敛性。在此类问题中，目标函数和函数不等式约束均由与光滑非线性内映射复合的凸利普希茨外函数给出。非凸函数不等式系统中的约束违反以及乘子缺乏先验界使分析变得复杂。为解决这些问题，我们将对偶变量限制在一个辅助紧集上，并通过非光滑非凸 - 凹极小极大重新表述来分析一种平滑近线性增广拉格朗日方法。主要贡献是一种有限时间机制，可将截断极小极大问题的平稳性转化为原始约束问题的 KKT 证书。我们表明，对于足够大的惩罚参数，除了受控数量的迭代外，所有迭代都进入一个近可行区域。在此区域上，局部圆锥正则性条件统一界定相关的近线性乘子，从而使人工对偶截断在选定的迭代中无效。基于此机制，我们根据 KKT 残差为所提出的方法建立了明确的收敛速率。通过对偶正则化，全局对偶误差界与偏差平衡论证给出了\(O(K^{-1/3})\)的速率。在无正则化的情况下，在包括外函数分段线性在内的额外局部结构假设下，局部对偶误差界产生更尖锐的\(O(K^{-1/2})\)速率。

英文摘要

We study nonasymptotic convergence of primal-dual methods for a class of nonconvex constrained optimization problems with a convex-composite structure. In this class, both the objective and the functional inequality constraints are given by convex Lipschitz outer functions composed with smooth nonlinear inner mappings. The analysis is complicated by constraint violation in a nonconvex functional inequality system and by the lack of an a priori bound on the multipliers. To address these issues, we restrict the dual variable to an auxiliary compact set and analyze a smoothed prox-linear augmented Lagrangian method through a nonsmooth nonconvex-concave minimax reformulation. The main contribution is a finite-time mechanism for converting stationarity of the truncated minimax problem into a KKT certificate for the original constrained problem. We show that, for a sufficiently large penalty parameter, all but a controlled number of iterates enter a near-feasible region. On this region, a local conic regularity condition uniformly bounds the associated prox-linear multipliers and thereby makes the artificial dual truncation inactive at the selected iterates. Building on this mechanism, we establish explicit convergence rates for the proposed method in terms of the KKT residual. With dual regularization, a global dual error bound together with a bias-balancing argument gives an $O(K^{-1/3})$ rate. In the unregularized case, under additional local structural assumptions including piecewise linearity of the outer functions, a local dual error bound yields the sharper $O(K^{-1/2})$ rate.

发表机构

• H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA（赫尔曼·米勒工程与系统工程学院，佐治亚理工学院，美国亚特兰大，GA）
• Sauder School of Business, University of British Columbia, Vancouver, BC, Canada（萨德勒商学院，不列颠哥伦比亚大学，加拿大温哥华，BC）

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