A hybrid penalty method for a class of optimization problems with multiple rank constraints

TIANXIANG LIU, IVAN MARKOVSKY, TING KEI PONG, AKIKO TAKEDA

Research output: Journal article publicationJournal articleAcademic researchpeer-review

Abstract

In this paper, we consider the problem of minimizing a smooth objective over multiple rank constraints on Hankel structured matrices. These kinds of problems arise in system identification, system theory, and signal processing, where the rank constraints are typically "hard constraints.""To solve these problems, we propose a hybrid penalty method that combines a penalty method with a postprocessing scheme. Specifically, we solve the penalty subproblems until the penalty parameter reaches a given threshold, and then switch to a local alternating "pseudoprojection""method to further reduce constraint violation. Pseudoprojection is a generalization of the concept of projection. We show that a pseudoprojection onto a single low-rank Hankel structured matrix constraint can be computed efficiently by existing software such as SLRA [I. Markovsky and K. Usevich, J. Comput. Appl. Math., 256 (2014), pp. 278-292], under mild assumptions. We also demonstrate how the penalty subproblems in the hybrid penalty method can be solved by pseudoprojection-based optimization methods, and then present some convergence results for our hybrid penalty method. Finally, the efficiency of our method is illustrated by numerical examples.

Original languageEnglish
Pages (from-to)1260-1283
Number of pages24
JournalSIAM Journal on Matrix Analysis and Applications
Volume41
Issue number3
DOIs
Publication statusE-pub ahead of print - Sep 2020

Keywords

  • Hankel structure
  • Hybrid penalty method
  • Pseudoprojection
  • System identification

ASJC Scopus subject areas

  • Analysis

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