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Deep Learning the Efficient Frontier of Convex Vector Optimization Problems

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  • Zachary Feinstein
  • Birgit Rudloff

Abstract

In this paper, we design a neural network architecture to approximate the weakly efficient frontier of convex vector optimization problems (CVOP) satisfying Slater's condition. The proposed machine learning methodology provides both an inner and outer approximation of the weakly efficient frontier, as well as an upper bound to the error at each approximated efficient point. In numerical case studies we demonstrate that the proposed algorithm is effectively able to approximate the true weakly efficient frontier of CVOPs. This remains true even for large problems (i.e., many objectives, variables, and constraints) and thus overcoming the curse of dimensionality.

Suggested Citation

  • Zachary Feinstein & Birgit Rudloff, 2022. "Deep Learning the Efficient Frontier of Convex Vector Optimization Problems," Papers 2205.07077, arXiv.org, revised May 2024.
    • Handle: RePEc:arx:papers:2205.07077
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    References listed on IDEAS

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    1. Andreas Löhne & Birgit Rudloff & Firdevs Ulus, 2014. "Primal and dual approximation algorithms for convex vector optimization problems," Journal of Global Optimization, Springer, vol. 60(4), pages 713-736, December.
    2. Gabriele Eichfelder & Leo Warnow, 2021. "Proximity measures based on KKT points for constrained multi-objective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 63-86, May.
    3. Matthias Ehrgott & Lizhen Shao & Anita Schöbel, 2011. "An approximation algorithm for convex multi-objective programming problems," Journal of Global Optimization, Springer, vol. 50(3), pages 397-416, July.
    4. Zachary Feinstein & Birgit Rudloff, 2017. "A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle," Journal of Global Optimization, Springer, vol. 68(1), pages 47-69, May.
    5. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
    6. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
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