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On Lipschitz Semicontinuity Properties of Variational Systems with Application to Parametric Optimization

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  • A. Uderzo

    (University of Milano-Bicocca)

Abstract

In this paper, two properties of recognized interest in variational analysis, known as Lipschitz lower semicontinuity and calmness, are studied with reference to a general class of variational systems, i.e. to solution mappings to parameterized generalized equations. In the consideration of the metric nature of such properties, some related sufficient conditions are established, which are expressed via nondegeneracy conditions on derivative-like objects appropriate for a metric space analysis. For certain classes of generalized equations in Asplund spaces, it is shown how such conditions can be formulated by using the Fréchet coderivative of the field and the derivative of the base. Applications to the stability analysis of parametric constrained optimization problems are proposed.

Suggested Citation

  • A. Uderzo, 2014. "On Lipschitz Semicontinuity Properties of Variational Systems with Application to Parametric Optimization," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 47-78, July.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:1:d:10.1007_s10957-013-0455-9
    DOI: 10.1007/s10957-013-0455-9
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    References listed on IDEAS

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    1. Jean-Pierre Aubin, 1984. "Lipschitz Behavior of Solutions to Convex Minimization Problems," Mathematics of Operations Research, INFORMS, vol. 9(1), pages 87-111, February.
    2. Frank H. Clarke, 1976. "A New Approach to Lagrange Multipliers," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 165-174, May.
    3. T. Chuong & A. Kruger & J.-C. Yao, 2011. "Calmness of efficient solution maps in parametric vector optimization," Journal of Global Optimization, Springer, vol. 51(4), pages 677-688, December.
    4. Francisco Aragón Artacho & Boris Mordukhovich, 2011. "Enhanced metric regularity and Lipschitzian properties of variational systems," Journal of Global Optimization, Springer, vol. 50(1), pages 145-167, May.
    5. N. D. Yen, 1997. "Stability of the Solution Set of Perturbed Nonsmooth Inequality Systems and Application," Journal of Optimization Theory and Applications, Springer, vol. 93(1), pages 199-225, April.
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