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A cutting-plane method to nonsmooth multiobjective optimization problems

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  • Vieira, D.A.G.
  • Lisboa, A.C.

Abstract

The cutting-plane optimization methods rely on the idea that any subgradient of the objective function or the active/violated constraints defines a halfspace to be excluded from a set that contains an optimal solution: the localizing set. This algorithm converges towards a global minimum of any pseudoconvex subdifferentiable function. A naive extension for multiobjective optimization would be using simultaneously some subgradients of all objective functions for a given feasible point. However, as demonstrated in this paper, this approach can lead to a convergence towards non-optimal points. This paper introduces an optimization strategy for cutting-plane methods to cope with multiobjective problems without any scalarization procedure. The proposed strategy guarantees that its optimal solution is a Pareto Optimal solution of the original problem, which is also no worse than the starting point, and that any Pareto Optimal solution can be sampled. Moreover, the auxiliary problem is infeasible only if the original problem is also infeasible. The new strategy inherits the original theoretical guarantees of cutting planes methods and it can be applied to build other strategies.

Suggested Citation

  • Vieira, D.A.G. & Lisboa, A.C., 2019. "A cutting-plane method to nonsmooth multiobjective optimization problems," European Journal of Operational Research, Elsevier, vol. 275(3), pages 822-829.
    • Handle: RePEc:eee:ejores:v:275:y:2019:i:3:p:822-829
    DOI: 10.1016/j.ejor.2018.12.047
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    References listed on IDEAS

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    Cited by:

    1. Kabgani, Alireza & Soleimani-damaneh, Majid, 2022. "Semi-quasidifferentiability in nonsmooth nonconvex multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 35-45.
    2. Li-Ping Pang & Fan-Yun Meng & Jian-Song Yang, 2023. "A class of infeasible proximal bundle methods for nonsmooth nonconvex multi-objective optimization problems," Journal of Global Optimization, Springer, vol. 85(4), pages 891-915, April.

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