AI 中文总结
针对带几何约束的优化问题，提出基于约束消解映射的不精确罚方法，其收敛性有理论保证，数值实验显示该方法解质量优且性能优于罚分解方法。
AI 中文摘要
带几何约束的优化问题应用广泛，涵盖机器学习、金融与控制等领域。约束消解方法是求解此类几何约束的有效算法工具，为此，我们提出适用于非凸几何约束的约束消解映射框架。基于该框架，我们开发了约束消解型不精确罚方法，用于求解带一般集合隶属约束及可能非凸几何约束的优化问题。我们证明了所提算法的收敛性，且每个可行聚点均为Mordukhovich平稳点。值得注意的是，我们仅依赖温和的渐近Mordukhovich正则性，该条件显著弱于现有约束消解方法文献中采用的约束规格。针对经典等式约束、互补约束、稀疏约束及低秩约束优化问题的数值实验表明，所提方法在解的质量上与带保护的增广拉格朗日方法相当，且显著优于罚分解方法。
英文摘要
Optimization problems with geometric constraints have a broad range of applications, including machine learning, finance, and control. A powerful algorithmic tool to resolve these geometric constraints are constraint dissolving methods. To this end, we propose a framework for constraint dissolving mappings for nonconvex geometric constraints. Leveraging these, we develop a constraint dissolving inexact penalty method to solve optimization problems with general set-membership constraints and possibly nonconvex geometric constraints. We establish the convergence of the proposed algorithm and prove that every feasible accumulation point is Mordukhovich stationary. Notably, we rely only on mild asymptotic Mordukhovich regularity, which is significantly weaker than the constraint qualifications adopted in the existing literature on constraint dissolving methods. Numerical experiments addressing classical equality-, complementarity-, sparsity-, and low-rank constrained optimization problems demonstrate that the proposed method is competitive with the safeguarded augmented Lagrangian method in terms of solution quality and significantly outperforms the penalty decomposition method.
