近端点算法的半定实现及其对优化中渐进解耦的影响

中文摘要 AI 辅助

在凸优化中，用于寻找极大单调映射零点的近端点算法是驱动增广拉格朗日方法和各种分裂方案的引擎，其中仅正半定的近端项在预处理中可能是有帮助的。本文开发了该算法的一种半定扩展，具有新的特性，在比先前预见的更广泛的领域中保证收敛，通常甚至是线性收敛。此外，在变度量实现中允许子问题的非精确最小化，这可能开辟类似拟牛顿的方法。扩展的关键在于将正定近端点算法的一种高级形式应用于由给定映射导出的投影单调映射，并将其与完成步骤相结合。为使此方法有效，投影映射的极大性是至关重要的，但极大性并非自动成立。本文对保证极大性的情形进行了明确分析，并由此识别出极大性的便捷准则。该扩展算法的后果被应用于凸下投影、增广拉格朗日步骤以及基于Spingarn部分逆的括号化渐进解耦作为问题分解方案。

英文摘要

The proximal point algorithm for finding a zero of a maximal monotone mapping is the engine that drives augmented Lagrangian methods and various splitting schemes in convex optimization, where proximal terms that are merely positive semidefinite can be helpful in preconditioning. Here a semidefinite extension of that algorithm is developed with new features which guarantee convergence, typically even linear convergence, in broader territory than previously foreseen. Moreover, inexact minimization is allowed in the subproblems in a variable-metric implementation which might open up quasi-Newton-like approaches. The key to the extension is applying an advanced form of the positive definite proximal point algorithm to a projected monotone mapping derived from the given mapping, and combining that with a completion step. For this to work, the maximality of the projected mapping is essential, but maximality is not automatic. The circumstances that guarantee maximality are definitively analyzed and convenient criteria for maximality are thereby identified. Consequences of the extended algorithm are worked out for application to convex inf-projection, augmented Lagrangian steps, and bracketed progressive decoupling based on Spingarn's partial inverse as a scheme for problem decomposition.

发表机构

• School of Data Science, Fudan University（复旦大学数据科学学院）
• Department of Mathematics, University of Washington（华盛顿大学数学系）
• Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）

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