Abstract
In this work, we show the consistency of an approach for solving robust optimization problems using sequences of sub-problems generated by ergodic measure preserving transformations. The main result of this paper is that the minimizers and the optimal value of the sub-problems converge, in some sense, to the minimizers and the optimal value of the initial problem, respectively. Our result particularly implies the consistency of the scenario approach for nonconvex optimization problems. Finally, we show that our method can also be used to solve infinite programming problems.
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Acknowledgments
First, the author would like to acknowledge the helpful discussions and great comments about this work given by Professor Marco A. López Cérda, which improved notably the quality of the presented manuscript. Second, the author is grateful to the anonymous reviewers for their valuable suggestions and comments about the work, which increases the presentation of the current version of the manuscript.
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This work was partially supported by grants Fondecyt Regular 1190110 and Fondecyt Regular 1200283.
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Pérez-Aros, P. Ergodic Approach to Robust Optimization and Infinite Programming Problems. Set-Valued Var. Anal 29, 409–423 (2021). https://doi.org/10.1007/s11228-020-00567-9
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DOI: https://doi.org/10.1007/s11228-020-00567-9
Keywords
- Stochastic optimization
- Scenario approach
- Robust optimization
- Infinite programming
- Epi-convergence
- Ergodic theorems.