A New Homotopy Proximal Variable-Metric Framework for Composite Convex Minimization

Quoc Tran-Dinh (), Ling Liang () and Kim-Chuan Toh ()
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Quoc Tran-Dinh: The University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599
Ling Liang: Department of Mathematics and Institute of Operations Research and Analytics, National University of Singapore, Singapore 119076
Kim-Chuan Toh: Department of Mathematics and Institute of Operations Research and Analytics, National University of Singapore, Singapore 119076

Mathematics of Operations Research, 2022, vol. 47, issue 1, 508-539

Abstract: This paper suggests two novel ideas to develop new proximal variable-metric methods for solving a class of composite convex optimization problems. The first idea is to utilize a new parameterization strategy of the optimality condition to design a class of homotopy proximal variable-metric algorithms that can achieve linear convergence and finite global iteration-complexity bounds. We identify at least three subclasses of convex problems in which our approach can apply to achieve linear convergence rates. The second idea is a new primal-dual-primal framework for implementing proximal Newton methods that has attractive computational features for a subclass of nonsmooth composite convex minimization problems. We specialize the proposed algorithm to solve a covariance estimation problem in order to demonstrate its computational advantages. Numerical experiments on the four concrete applications are given to illustrate the theoretical and computational advances of the new methods compared with other state-of-the-art algorithms.

Keywords: Primary: 90C25; 90C06; 90-08; homotopy method; proximal variable-metric algorithm; linear convergence rate; finite iteration complexity; primal-dual-primal framework; composite convex minimization (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:1:p:508-539

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