Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings

Isao Yamada, Masahiro Yukawa, Masao Yamagishi

Research output: Chapter in Book/Report/Conference proceedingChapter

56 Citations (Scopus)

Abstract

The first aim of this paper is to present a useful toolbox of quasinonexpansive mappings for convex optimization from the viewpoint of using their fixed point sets as constraints. Many convex optimization problems have been solved through elegant translations into fixed point problems. The underlying principle is to operate a certain quasi-nonexpansivemapping T iteratively and generate a convergent sequence to its fixed point. However, such a mapping often has infinitely many fixed points, meaning that a selection from the fixed point set Fix(T) should be of great importance. Nevertheless, most fixed point methods can only return an “unspecified” point from the fixed point set, which requires many iterations. Therefore, based on common sense, it seems unrealistic to wish for an “optimal” one from the fixed point set. Fortunately, considering the collection of quasi-nonexpansive mappings as a toolbox, we can accomplish this challenging mission simply by the hybrid steepest descent method, provided that the cost function is smooth and its derivative is Lipschitz continuous. A question arises: how can we deal with “nonsmooth” cost functions? The second aim is to propose a nontrivial integration of the ideas of the hybrid steepest descent method and the Moreau-Yosida regularization, yielding a useful approach to the challenging problem of nonsmooth convex optimization over Fix(T). The key is the use of smoothing of the original nonsmooth cost function by its Moreau-Yosida regularization whose derivative is always Lipschitz continuous. The field of application of hybrid steepest descent method can be extended to the minimization of the ideal smooth approximation over Fix(T). We present the mathematical ideas of the proposed approach together with its application to a combinatorial optimization problem: the minimal antenna-subset selection problem under a highly nonlinear capacity-constraint for efficient multiple input multiple output (MIMO) communication systems.

Original languageEnglish
Title of host publicationSpringer Optimization and Its Applications
PublisherSpringer International Publishing
Pages345-390
Number of pages46
Volume49
DOIs
Publication statusPublished - 2011
Externally publishedYes

Publication series

NameSpringer Optimization and Its Applications
Volume49
ISSN (Print)19316828
ISSN (Electronic)19316836

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Keywords

  • Hybrid steepest descent method
  • Moreau envelope
  • Nonsmooth convex optimization

ASJC Scopus subject areas

  • Control and Optimization

Cite this

Yamada, I., Yukawa, M., & Yamagishi, M. (2011). Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings. In Springer Optimization and Its Applications (Vol. 49, pp. 345-390). (Springer Optimization and Its Applications; Vol. 49). Springer International Publishing. https://doi.org/10.1007/978-1-4419-9569-8_17