Penalty Dual Decomposition Method for Nonsmooth Nonconvex Optimization - Part I: Algorithms and Convergence Analysis

Qingjiang Shi, Mingyi Hong

Research output: Contribution to journalArticlepeer-review

Abstract

Many contemporary signal processing, machine learning and wireless communication applications can be formulated as nonconvex nonsmooth optimization problems. Often there is a lack of efficient algorithms for these problems, especially when the optimization variables are nonlinearly coupled in some nonconvex constraints. In this work, we propose an algorithm named penalty dual decomposition (PDD) for these difficult problems and discuss its various applications. The PDD is a double-loop iterative algorithm. Its inner iteration is used to inexactly solve a nonconvex nonsmooth augmented Lagrangian problem via block-coordinate-descent-type methods, while its outer iteration updates the dual variables and/or a penalty parameter. In Part I of this work, we describe the PDD algorithm and establish its convergence to KKT solutions. In Part II we evaluate the performance of PDD by customizing it to three applications arising from signal processing and wireless communications.

Original languageEnglish (US)
Article number9120361
Pages (from-to)4108-4122
Number of pages15
JournalIEEE Transactions on Signal Processing
Volume68
DOIs
StatePublished - 2020

Bibliographical note

Funding Information:
Manuscript received September 26, 2019; revised May 1, 2020 and June 8, 2020; accepted June 8, 2020. Date of publication June 18, 2020; date of current version July 24, 2020. The associate editor coordinating the review of this manuscript and approving it for publication was Nicolas Gillis. The work of Qingjiang Shi was supported in part by the National Key Research and Development Project under Grant 2017YFE0119300, and in part by the NSFC under Grants 61671411, 61731018, and U1709219. The work of Mingyi Hong was supported in part by the National Science Foundation under Grants CIF-1910385 and CMMI-172775 and in part by Army Research Office under Grant W911NF-19-1-0247. Part of this paper has been presented in IEEE International Conference on Acoustics, Speech and Signal Processing, New Orleans, LA, USA, 5–9 Mar. 2017 [1]. (Corresponding author: Mingyi Hong.) Qingjiang Shi is with the School of Software Engineering, Tongji University, Shanghai 201804, China, and also with the Shenzhen Research Institute of Big Data, Shenzhen 518172, China (e-mail: shiqj@tongji.edu.cn).

Keywords

  • BSUM
  • KKT
  • Penalty method
  • augmented Lagrangian
  • dual decomposition
  • nonconvex optimization

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