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Convergence of the proximal bundle algorithm for nonsmooth nonconvex optimization problems

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Abstract

A proximal bundle algorithm is proposed for solving unconstrained nonsmooth nonconvex optimization problems. At each iteration, using already generated information, the algorithm defines a convex model of the augmented objective function. Then by solving a quadratic subproblem a new candidate iterate is obtained and the algorithm is repeated. The novelty in our approach is that the objective function can be any arbitrary locally Lipschitz function without any additional assumptions. The global convergence, starting from any point, is also studied. At the end, some encouraging numerical results with a MATLAB implementation are reported.

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Acknowledgements

The first-named author was in part supported by a grant from IPM (No.1400900048). The authors would like to thank two anonymous referees for their comments that helped to improve the quality of the paper.

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Authors and Affiliations

  1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, 19395-5746, Iran

    N. Hoseini Monjezi

  2. Department of Applied Mathematics and Computer Science, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, Iran

    S. Nobakhtian

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  1. N. Hoseini Monjezi
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  2. S. Nobakhtian
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Correspondence to S. Nobakhtian.

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Hoseini Monjezi, N., Nobakhtian, S. Convergence of the proximal bundle algorithm for nonsmooth nonconvex optimization problems. Optim Lett 16, 1495–1511 (2022). https://doi.org/10.1007/s11590-021-01787-0

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