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Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization

Abstract

The paper concerns multiobjective linear optimization problems in \(\mathbb {R}^{n}\) that are parameterized with respect to the right-hand side perturbations of inequality constraints. Our focus is on measuring the variation of the feasible set and the Pareto front mappings around a nominal element while paying attention to some specific directions. This idea is formalized by means of the so-called epigraphical multifunction, which is defined by adding a fixed cone to the images of the original mapping. Through the epigraphical feasible and Pareto front mappings we describe the corresponding vector subdifferentials and employ them to verifying Lipschitzian stability of the perturbed mappings with computing the associated Lipschitz moduli. The particular case of ordinary linear programs is analyzed, where we show that the subdifferentials of both multifunctions are proportional subsets. We also provide a method for computing the optimal value of linear programs without knowing any optimal solution. Some illustrative examples are also given in the paper.

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Acknowledgements

This research has been partially supported by grants MTM2014-59179-C2-(1,2)-P and PGC2018-097960-B-C2(1,2) from MINECO/MICINN, Spain, and ERDF, “A way to make Europe”, European Union.

Research of the second author is also partially supported by the Australian Research Council (ARC) Discovery Grants Scheme (Project Grant # DP180100602).

Research of third author was partially supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research grant #15RT04, and by Australian Research Council under grant DP-190100555.

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Correspondence to M. A. López.

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Dedicated by his coauthors to Prof. Marco A. López on his 70th birthday.

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Cánovas, M.J., López, M.A., Mordukhovich, B.S. et al. Subdifferentials and Stability Analysis of Feasible Set and Pareto Front Mappings in Linear Multiobjective Optimization. Vietnam J. Math. 48, 315–334 (2020). https://doi.org/10.1007/s10013-020-00402-6

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Keywords

  • Epigraphical set-valued mappings
  • Feasible set mappings
  • Lipschitz moduli
  • Linear programming
  • Optimal value functions
  • Multiobjective optimization

Mathematics Subject Classification (2010)

  • 49J53
  • 90C31
  • 15A39
  • 90C05
  • 90C29