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Pure Mathematics 2025
用于求解多块可分凸优化问题的惯性临近严格收缩PRSM
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Abstract:
近年来，PRSM成为了处理具有线性约束的两部分可分凸优化问题的一个热门研究方向。本研究聚焦于目标函数由三个解耦变量函数之和构成的可分凸优化问题。单纯地运用PRSM可能无法保证其收敛性。因此，我们引入了一种带有惯性项和临近项的严格收缩PRSM。借助变分不等式、临近点算法以及基本不等式，我们对所提出的方法进行了全局收敛性的分析。此外，我们将这种新方法应用于鲁棒主成分分析(PCA)问题的求解，并提供了一些初步的数值结果，用以展示该方法的可行性和有效性。
In recent years, PRSM has become a popular research direction for dealing with two-block separable convex optimization problems with linear constraints. This study focuses on separable convex optimization problems where the objective function is composed of the sum of three decoupled variable functions. Simply applying the PRSM may not guarantee its convergence. Therefore, we introduce a strictly contractive PRSM with inertial and proximal terms. By means of variational inequalities, the proximal point algorithm, and fundamental inequalities, we have analyzed the global convergence of the proposed method. In addition, we applied this new method to the solution of the Robust Principal Component Analysis (PCA) problem and provided some preliminary numerical results to demonstrate the feasibility and effectiveness of the method.
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