Set-Valued Analysis

, Volume 16, Issue 5, pp 511–538

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

Article

DOI: 10.1007/s11228-007-0052-x

Cite this article as:
Cánovas, M.J., Gómez-Senent, F.J. & Parra, J. Set-Valued Anal (2008) 16: 511. doi:10.1007/s11228-007-0052-x

Abstract

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n ⩽ 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem’s coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.

Keywords

Strong Lipschitz stability Metric regularity Lipschitz modulus Optimal set mapping Linear semi-infinite programming 

Mathematics Subject Classifications (2000)

90C34 49J53 90C31 90C05 

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • M. J. Cánovas
    • 1
  • F. J. Gómez-Senent
    • 1
  • J. Parra
    • 1
  1. 1.Operations Research CenterMiguel Hernández University of ElcheElcheSpain

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