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A proximal bundle method for a class of nonconvex nonsmooth composite optimization problems

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Abstract

In this paper, a proximal bundle method is proposed for a class of nonconvex nonsmooth composite optimization problems. The composite problem considered here is the sum of two functions: one is convex and the other is nonconvex. Local convexification strategy is adopted for the nonconvex function and the corresponding convexification parameter varies along iterations. Then the sum of the convex function and the extended function is dynamically constructed to approximate the primal problem. To choose a suitable cutting plane model for the approximation function, here we consider the sum of two cutting planes, which are designed respectively for the convex function and the extended function. By choosing appropriate descent condition, our method can keep track of the relationship between primal problem and approximate models. Under mild conditions, the convergence is proved and the accumulation point of iterations is a stationary point of the primal problem. Two polynomial problems and twelve DC (difference of convex) problems are referred in numerical experiments. The preliminary numerical results show that the proposed method is effective for solving these testing problems.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Key Laboratory for Computational Mathematics and Data Intelligence of Liaoning Province, Dalian, 116024, China

    Liping Pang

  2. School of Science, Department of Mathematics Science, Zhejiang Sci-Tech University, Hangzhou, 310018, China

    Xiaoliang Wang

  3. School of Information and Control Engineering, Qingdao University of Technology, Qingdao, 266520, China

    Fanyun Meng

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  1. Liping Pang
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  2. Xiaoliang Wang
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  3. Fanyun Meng
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Correspondence to Xiaoliang Wang.

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This work is supported by Science Foundation of Zhejiang Sci-Tech Uni-versity (ZSTU) under Grant No. 21062347-Y and also supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LQ23A010020.

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Pang, L., Wang, X. & Meng, F. A proximal bundle method for a class of nonconvex nonsmooth composite optimization problems. J Glob Optim 86, 589–620 (2023). https://doi.org/10.1007/s10898-023-01279-8

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