Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization

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• 作者：Yakui Huang ; Hongwei Liu
• 关键词：Smoothing projected Barzilai–Borwein algorithm ; Constrained non ; Lipschitz optimization ; Nonsmooth nonconvex optimization ; Smoothing approximation ; \(\ell _2\) ; \(\ell _p\) ; problem ; Image restoration ; Stochastic linear complementarity problem
• 刊名：Computational Optimization and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：65
• 期：3
• 页码：671-698
• 全文大小：1,171 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Optimization
  Operations Research and Mathematical Programming
  Operation Research and Decision Theory
  Statistics
  Convex and Discrete Geometry
  
• 出版者：Springer Netherlands
• ISSN：1573-2894
• 卷排序：65

文摘

We present a smoothing projected Barzilai–Borwein (SPBB) algorithm for solving a class of minimization problems on a closed convex set, where the objective function is nonsmooth nonconvex, perhaps even non-Lipschitz. At each iteration, the SPBB algorithm applies the projected gradient strategy that alternately uses the two Barzilai–Borwein stepsizes to the smooth approximation of the original problem. Nonmonotone scheme is adopted to ensure global convergence. Under mild conditions, we prove convergence of the SPBB algorithm to a scaled stationary point of the original problem. When the objective function is locally Lipschitz continuous, we consider a general constrained optimization problem and show that any accumulation point generated by the SPBB algorithm is a stationary point associated with the smoothing function used in the algorithm. Numerical experiments on \(\ell _2\)-\(\ell _p\) problems, image restoration problems, and stochastic linear complementarity problems show that the SPBB algorithm is promising.