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.
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This research has been partially supported by grants MTM2005-08572-C03-02, from MEC (Spain) and FEDER (E.U.), and ACOMP06/203, from Generalitat Valenciana (Spain).
F.J. Gómez-Senent acknowledges a special permission for studies (‘Licencia por Estudios’) from Consejería de Educación y Cultura de la Región de Murcia (Spain).
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Cánovas, M.J., Gómez-Senent, F.J. & Parra, J. On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization. Set-Valued Anal 16, 511–538 (2008). https://doi.org/10.1007/s11228-007-0052-x
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DOI: https://doi.org/10.1007/s11228-007-0052-x
Keywords
- Strong Lipschitz stability
- Metric regularity
- Lipschitz modulus
- Optimal set mapping
- Linear semi-infinite programming