Numerische Mathematik

, Volume 136, Issue 2, pp 343–381 | Cite as

Nonconvex nonsmooth optimization via convex–nonconvex majorization–minimization

Article

Abstract

The class of majorization–minimization algorithms is based on the principle of successively minimizing upper bounds of the objective function. Each upper bound, or surrogate function, is locally tight at the current estimate, and each minimization step decreases the value of the objective function. We present a majorization–minimization approach based on a novel convex–nonconvex upper bounding strategy for the solution of a certain class of nonconvex nonsmooth optimization problems. We propose an efficient algorithm for minimizing the (convex) surrogate function based on the alternating direction method of multipliers. A preliminary convergence analysis for the proposed approach is provided. Numerical experiments show the effectiveness of the proposed method for the solution of nonconvex nonsmooth minimization problems.

Mathematics Subject Classification

65K05 Mathematical programming 65K10 Optimization and variational techniques 90C26 Nonconvex programming,  global optimization 49J52 Nonsmooth analysis 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • A. Lanza
    • 1
  • S. Morigi
    • 1
  • I. Selesnick
    • 2
  • F. Sgallari
    • 1
  1. 1.Department of MathematicsUniversity of BolognaBolognaItaly
  2. 2.Department of Electrical and Computer EngineeringNew York UniversityBrooklynUSA

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