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An approximate bundle method for solving nonsmooth equilibrium problems

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Abstract

We present an approximate bundle method for solving nonsmooth equilibrium problems. An inexact cutting-plane linearization of the objective function is established at each iteration, which is actually an approximation produced by an oracle that gives inaccurate values for the functions and subgradients. The errors in function and subgradient evaluations are bounded and they need not vanish in the limit. A descent criterion adapting the setting of inexact oracles is put forward to measure the current descent behavior. The sequence generated by the algorithm converges to the approximately critical points of the equilibrium problem under proper assumptions. As a special illustration, the proposed algorithm is utilized to solve generalized variational inequality problems. The numerical experiments show that the algorithm is effective in solving nonsmooth equilibrium problems.

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Acknowledgements

The authors thank two anonymous referees for a number of valuable and helpful suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Fan-Yun Meng, Li-Ping Pang & Jian Lv

  2. School of Computer, Qingdao Technological University, Qingdao, 266033, China

    Jin-He Wang

Authors
  1. Fan-Yun Meng
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  2. Li-Ping Pang
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  3. Jian Lv
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  4. Jin-He Wang
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Correspondence to Li-Ping Pang.

Additional information

The research is partially supported by the Natural Science Foundation of China, Grants 11171049, 11301347 and 31271077.

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Meng, FY., Pang, LP., Lv, J. et al. An approximate bundle method for solving nonsmooth equilibrium problems. J Glob Optim 68, 537–562 (2017). https://doi.org/10.1007/s10898-016-0490-9

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