Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis

Bo Jiang, Tianyi Lin, Shiqian Ma, Shuzhong Zhang

Research output: Contribution to journalArticlepeer-review

67 Scopus citations


Nonconvex and nonsmooth optimization problems are frequently encountered in much of statistics, business, science and engineering, but they are not yet widely recognized as a technology in the sense of scalability. A reason for this relatively low degree of popularity is the lack of a well developed system of theory and algorithms to support the applications, as is the case for its convex counterpart. This paper aims to take one step in the direction of disciplined nonconvex and nonsmooth optimization. In particular, we consider in this paper some constrained nonconvex optimization models in block decision variables, with or without coupled affine constraints. In the absence of coupled constraints, we show a sublinear rate of convergence to an ϵ-stationary solution in the form of variational inequality for a generalized conditional gradient method, where the convergence rate is dependent on the Hölderian continuity of the gradient of the smooth part of the objective. For the model with coupled affine constraints, we introduce corresponding ϵ-stationarity conditions, and apply two proximal-type variants of the ADMM to solve such a model, assuming the proximal ADMM updates can be implemented for all the block variables except for the last block, for which either a gradient step or a majorization–minimization step is implemented. We show an iteration complexity bound of O(1 / ϵ2) to reach an ϵ-stationary solution for both algorithms. Moreover, we show that the same iteration complexity of a proximal BCD method follows immediately. Numerical results are provided to illustrate the efficacy of the proposed algorithms for tensor robust PCA and tensor sparse PCA problems.

Original languageEnglish (US)
Pages (from-to)115-157
Number of pages43
JournalComputational Optimization and Applications
Issue number1
StatePublished - Jan 15 2019
Externally publishedYes

Bibliographical note

Funding Information:
Bo Jiang: Research of this author was supported in part by NSFC Grants 11771269 and 11831002, and Program for Innovative Research Team of Shanghai University of Finance and Economics. Shiqian Ma: Research of this author was supported in part by a startup package in Department of Mathematics at UC Davis. Shuzhong Zhang: Research of this author was supported in part by the National Science Foundation (Grant CMMI-1462408), and in part by Shenzhen Fundamental Research Fund under Grant No. KQTD2015033114415450.

Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.


  • Alternating direction method of multipliers
  • Block coordinate descent method
  • Conditional gradient method
  • Iteration complexity
  • Structured nonconvex optimization
  • ϵ-Stationary


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