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On a Quantitative Semicontinuity Property of Variational Systems with Applications to Perturbed Quasidifferentiable Optimization

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Constructive Nonsmooth Analysis and Related Topics

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 87))

Abstract

Lipschitz lower semicontinuity is a quantitative stability property for set-valued maps with relevant applications to perturbation analysis of optimization problems. The present paper reports on an attempt of studying such property, by starting with a related result valid for variational systems in metric spaces. Elements of nonsmooth analysis are subsequently employed to express and apply such result and its consequences in more structured settings. This approach leads to obtain a solvability, stability, and sensitivity condition for perturbed optimization problems with quasidifferentiable data.

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Notes

  1. 1.

    Actually, this is not the original definition, but an alternative one, as resulting from Proposition 1.2 in [25]. The notion of prox-regularity was introduced in [24] in a finite-dimensional setting. It is worth noting that, quite recently, the notion of prox-regular set has been extended to the uniformly convex Banach space setting in [3]. Since such an extension requires a certain amount of technicalities and the case of Hilbert spaces is already significant, consequent generalizations of results here presented to such a broader setting will be disregarded here.

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Acknowledgements

The revised version of this paper took benefit from the careful reading and the useful remarks by two anonymous referees. The author thanks both of them.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Via Cozzi, 53, 20125, Milano, Italy

    A. Uderzo

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  1. A. Uderzo
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Correspondence to A. Uderzo .

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Editors and Affiliations

  1. Saint Petersburg State University, Saint Petersburg, Russia

    Vladimir F. Demyanov

  2. Department of Industrial & Systems Engin, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  3. Higher School of Economics, National Research University, Nizhny Novgorod, Russia

    Mikhail Batsyn

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Uderzo, A. (2014). On a Quantitative Semicontinuity Property of Variational Systems with Applications to Perturbed Quasidifferentiable Optimization. In: Demyanov, V., Pardalos, P., Batsyn, M. (eds) Constructive Nonsmooth Analysis and Related Topics. Springer Optimization and Its Applications, vol 87. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8615-2_8

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