Two proximal splitting methods for multi-block separable programming with applications to stable principal component pursuit

Abstract

Recent years have witnessed the rapid development of the first order methods for multi-block separable programming involving large-scale date-set. The purpose of this paper is to introduce two new first order methods, the proximal splitting methods (PSMs), for the model under consideration. The first PSM fully utilizes the desired property of such problems and adopts fully Jacobian updating rule, which often results in easy subproblems in practice. The global convergence and the worst-case \(\mathcal {O}(1/t)\) convergence rate in an ergodic sense of the first PSM are proved under the condition that the involved functions are assumed to be strongly convex. Applying the hybrid Jacobian and Gauss–Seidel updating rule to the first PSM, we derive the second PSM, whose global convergence can be guaranteed only under the condition that the involved functions are convex. Furthermore, its worst-case \(\mathcal {O}(1/t)\) convergence rate in both the ergodic and non-ergodic senses is also established. Finally, numerical results on stable principal component pursuit are reported to testify the accuracy and speed of the second PSM, and some numerical comparisons are also reported.