发表机构
University of the Bundeswehr Munich; Polytechnique Montréal（联邦国防军慕尼黑大学; 蒙特利尔理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
Envelopt 是一种全局收敛的迭代框架，用于处理带一般平滑约束的复合优化问题，通过 Moreau 包络和近端算子简化子问题，支持非凸目标，并提供 Julia 实现与多种数值实验验证。
AI 中文摘要
我们提出了 Envelopt，这是一个全局收敛的迭代框架，适用于一类广泛的结构化优化问题，其中平滑目标函数由非平滑凸正则化器与平滑映射的复合来增强，并且变量受一般平滑约束。所有平滑函数可以是非凸的。该方法类似于增广拉格朗日方法，其中对提升变量进行部分最小化会产生涉及非平滑正则化器的 Moreau 包络的平滑子问题，而原始约束被显式保留。仅需要正则化器的近端算子。子问题可以使用现成的平滑优化求解器来求解。我们陈述了全局收敛性质，证明了可行极限点是渐近平稳的，并开发了一种不可行性检测机制。当惩罚参数有界且不趋于零时，我们推导了最坏情况迭代复杂度界限。该框架包含了经典的增广拉格朗日方法，并适应重要的扩展，包括退化问题的稳定化公式、精确惩罚方法和锥约束。我们提供了 Julia 实现，this http URL，作为 JuliaSmoothOptimizers 生态系统的一部分。针对低秩矩阵补全、半定规划、互补约束优化和非凸正则化器的数值实验证明了 Envelopt 的有效性和多功能性。
英文摘要
We introduce Envelopt, a globally convergent iterative framework for a broad class of structured optimization problems where a smooth objective is augmented by a nonsmooth convex regularizer composed with a smooth mapping, and the variables are subject to general smooth constraints. All smooth functions may be nonconvex. The method is akin to an augmented-Lagrangian method in which partial minimization with respect to a lifting variable results in smooth subproblems involving the Moreau envelope of the nonsmooth regularizer, and the original constraints are retained explicitly. Only the proximal operator of the regularizer is required. Subproblems may be solved with off-the-shelf smooth optimization solvers. We state global convergence properties, establish that feasible limit points are asymptotically stationary, and develop an infeasibility detection mechanism. We derive worst-case iteration complexity bounds when the penalty parameter is and is not bounded away from zero. The framework subsumes the classical augmented Lagrangian method and accommodates important extensions, including stabilized formulations for degenerate problems, exact penalty methods, and conic constraints. We provide a Julia implementation, Envelopt.jl, as part of the JuliaSmoothOptimizers ecosystem. Numerical experiments with low-rank matrix completion, semidefinite programming, complementarity-constrained optimization, and nonconvex regularizers demonstrate the effectiveness and versatility of Envelopt.
Comments21 pages + references
