IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v162y2014i1d10.1007_s10957-013-0455-9.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-013-0455-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-013-0455-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Vu Thi Huong & Jen-Chih Yao & Nguyen Dong Yen, 2017. "On the Stability and Solution Sensitivity of a Consumer Problem," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 567-589, November.
    2. Nooshin Movahedian, 2017. "Bounded Lagrange multiplier rules for general nonsmooth problems and application to mathematical programs with equilibrium constraints," Journal of Global Optimization, Springer, vol. 67(4), pages 829-850, April.
    3. Enrico Bellino, 2010. "Comment To ‘Commodity Content . . .’ By Fujimoto And Opocher," Metroeconomica, Wiley Blackwell, vol. 61(4), pages 749-753, November.
    4. Gui-Hua Lin & Mei-Ju Luo & Jin Zhang, 2016. "Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints," Journal of Global Optimization, Springer, vol. 66(3), pages 487-510, November.
    5. Kuang Bai & Yixia Song & Jin Zhang, 2023. "Second-Order Enhanced Optimality Conditions and Constraint Qualifications," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1264-1284, September.
    6. Y. Y. Zhou & X. Q. Yang, 2009. "Duality and Penalization in Optimization via an Augmented Lagrangian Function with Applications," Journal of Optimization Theory and Applications, Springer, vol. 140(1), pages 171-188, January.
    7. Florian Scheuer & Alexander Wolitzky, 2016. "Capital Taxation under Political Constraints," American Economic Review, American Economic Association, vol. 106(8), pages 2304-2328, August.
    8. Alexander Y. Kruger & Nguyen H. Thao, 2015. "Quantitative Characterizations of Regularity Properties of Collections of Sets," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 41-67, January.
    9. Huynh Van Ngai & Nguyen Huu Tron & Michel Théra, 2014. "Metric Regularity of the Sum of Multifunctions and Applications," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 355-390, February.
    10. Abhishek Singh & Debdas Ghosh & Qamrul Hasan Ansari, 2024. "Inexact Newton Method for Solving Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 201(3), pages 1333-1363, June.
    11. Casey Rothschild & Florian Scheuer, 2016. "Optimal Taxation with Rent-Seeking," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 83(3), pages 1225-1262.
    12. Jin Zhang & Xide Zhu, 2022. "Linear Convergence of Prox-SVRG Method for Separable Non-smooth Convex Optimization Problems under Bounded Metric Subregularity," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 564-597, February.
    13. Yuntong Wang, 2014. "Envelope Theorem without Differentiability," Working Papers 1404, University of Windsor, Department of Economics.
    14. C. H. Jeffrey Pang, 2011. "Generalized Differentiation with Positively Homogeneous Maps: Applications in Set-Valued Analysis and Metric Regularity," Mathematics of Operations Research, INFORMS, vol. 36(3), pages 377-397, August.
    15. Zachary Bethune & Tai-Wei Hu & Guillaume Rocheteau, 2018. "Optimal Credit Cycles," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 27, pages 231-245, January.
    16. Thai Doan Chuong & Do Sang Kim, 2016. "Hölder-Like Property and Metric Regularity of a Positive-Order for Implicit Multifunctions," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 596-611, May.
    17. Ewald, Christian Oliver & Nolan, Charles, 2024. "On the adaptation of the Lagrange formalism to continuous time stochastic optimal control: A Lagrange-Chow redux," Journal of Economic Dynamics and Control, Elsevier, vol. 162(C).
    18. Michaël Gaydu & Michel Geoffroy & Yvesner Marcelin, 2016. "Prederivatives of convex set-valued maps and applications to set optimization problems," Journal of Global Optimization, Springer, vol. 64(1), pages 141-158, January.
    19. Adrian Petruşel & Gabriela Petruşel & Jen-Chih Yao, 2019. "Pseudo-contractivity and Metric Regularity in Fixed Point Theory," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 5-18, January.
    20. Nguyen Ngoc Luan & Do Sang Kim & Nguyen Dong Yen, 2022. "Two Optimal Value Functions in Parametric Conic Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 574-597, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:162:y:2014:i:1:d:10.1007_s10957-013-0455-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.