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A primal nonsmooth reformulation for bilevel optimization problems

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Abstract

The solution of bilevel optimization problems with possibly nondifferentiable upper objective functions and with smooth and convex lower-level problems is discussed. A new approximate one-level reformulation for the original problem is introduced. An algorithm based on this reformulation is developed that is proven to converge to a solution of the bilevel problem. Each iteration of the algorithm depends on the solution of a nonsmooth optimization problem and its implementation leverages recent advances on nonsmooth optimization algorithms, which are fundamental to obtain a practical method. Experimental work is performed in order to demonstrate some characteristics of the algorithm in practice.

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Acknowledgements

This work was partially funded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grants 310893/2019-4 and 305010/2020-4 and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grants 2018/24293-0, 2016/22989-2 and 2013/07375-0. The authors are thankful to the anonymous referees, who provided insightful comments that improved the presentation of this work.

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  1. University of São Paulo, São Carlos, SP, Brazil

    Elias S. Helou

  2. University of Campinas, Campinas, SP, Brazil

    Sandra A. Santos & Lucas E. A. Simões

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  1. Elias S. Helou
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  2. Sandra A. Santos
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  3. Lucas E. A. Simões
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Helou, E.S., Santos, S.A. & Simões, L.E.A. A primal nonsmooth reformulation for bilevel optimization problems. Math. Program. 198, 1381–1409 (2023). https://doi.org/10.1007/s10107-021-01764-6

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