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Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization

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Abstract

This paper was originally motivated by the problem of providing a point-based formula (only involving the nominal data, and not data in a neighborhood) for estimating the calmness modulus of the optimal set mapping in linear semi-infinite optimization under perturbations of all coefficients. With this aim in mind, the paper establishes as a key tool a basic result on finite-valued convex functions in the \(n\)-dimensional Euclidean space. Specifically, this result provides an upper limit characterization of the boundary of the subdifferential of such a convex function. When applied to the supremum function associated with our constraint system, this characterization allows us to derive an upper estimate for the aimed calmness modulus in linear semi-infinite optimization under the uniqueness of nominal optimal solution.

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Acknowledgments

The authors would like to acknowledge Professor Marco A. López for his support and valuable hints and comments, specially in relation to Sect. 3. We would also like to thank the anonymous referee for his/her comments and suggestions about further connections between the calmness moduli of \(\vartheta \) and \(\mathcal {S}\), which will be helpful for future research. This research has been partially supported by Grant MTM2011-29064-C03-03 from MINECO, Spain. The research of the second author is also partially supported by Fondecyt Project No 1110019, ECOS-Conicyt project No C10E08, and Math-Amsud No. 13MATH-01 2013

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Authors and Affiliations

  1. Center of Operations Research, Miguel Hernández University of Elche, 03202 , Elche, Alicante, Spain

    M. J. Cánovas, J. Parra & F. J. Toledo

  2. Departamento de Ingeniería Matemático, Centro de Modelamiento Matemático (CMM), Universidad de Chile, Santiago, Chile

    A. Hantoute

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  1. M. J. Cánovas
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  2. A. Hantoute
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  3. J. Parra
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  4. F. J. Toledo
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Correspondence to J. Parra.

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Cánovas, M.J., Hantoute, A., Parra, J. et al. Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization. Optim Lett 9, 513–521 (2015). https://doi.org/10.1007/s11590-014-0767-1

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  • DOI: https://doi.org/10.1007/s11590-014-0767-1

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