IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v171y2016i2d10.1007_s10957-016-0897-y.html
   My bibliography  Save this article

Second-Order Analysis of Piecewise Linear Functions with Applications to Optimization and Stability

Author

Listed:
  • B. S. Mordukhovich

    (Wayne State University)

  • M. E. Sarabi

    (Wayne State University)

Abstract

This paper is devoted to second-order variational analysis of a rather broad class of extended-real-valued piecewise liner functions and their applications to various issues of optimization and stability. Based on our recent explicit calculations of the second-order subdifferential for such functions, we establish relationships between nondegeneracy and second-order qualification for fully amenable compositions involving piecewise linear functions. We then provide a second-order characterization of full stable local minimizers in composite optimization and constrained minimax problems.

Suggested Citation

  • B. S. Mordukhovich & M. E. Sarabi, 2016. "Second-Order Analysis of Piecewise Linear Functions with Applications to Optimization and Stability," Journal of Optimization Theory and Applications, Springer, vol. 171(2), pages 504-526, November.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:2:d:10.1007_s10957-016-0897-y
    DOI: 10.1007/s10957-016-0897-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-0897-y
    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/s10957-016-0897-y?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. B. S. Mordukhovich & T. T. A. Nghia & R. T. Rockafellar, 2015. "Full Stability in Finite-Dimensional Optimization," Mathematics of Operations Research, INFORMS, vol. 40(1), pages 226-252, February.
    2. Stephen M. Robinson, 1980. "Strongly Regular Generalized Equations," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 43-62, February.
    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. B. S. Mordukhovich & M. E. Sarabi, 2017. "Stability Analysis for Composite Optimization Problems and Parametric Variational Systems," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 554-577, February.
    2. Duong Thi Kim Huyen & Jen-Chih Yao & Nguyen Dong Yen, 2019. "Sensitivity Analysis of a Stationary Point Set Map Under Total Perturbations. Part 1: Lipschitzian Stability," Journal of Optimization Theory and Applications, Springer, vol. 180(1), pages 91-116, January.
    3. M. Durea & R. Strugariu, 2011. "On parametric vector optimization via metric regularity of constraint systems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(3), pages 409-425, December.
    4. Liang Chen & Anping Liao, 2020. "On the Convergence Properties of a Second-Order Augmented Lagrangian Method for Nonlinear Programming Problems with Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 248-265, October.
    5. Giorgio, 2019. "On Second-Order Optimality Conditions in Smooth Nonlinear Programming Problems," DEM Working Papers Series 171, University of Pavia, Department of Economics and Management.
    6. 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.
    7. Fabiana R. Oliveira & Orizon P. Ferreira & Gilson N. Silva, 2019. "Newton’s method with feasible inexact projections for solving constrained generalized equations," Computational Optimization and Applications, Springer, vol. 72(1), pages 159-177, January.
    8. Xi Yin Zheng & Jiangxing Zhu, 2022. "Stable global well-posedness and global strong metric regularity," Journal of Global Optimization, Springer, vol. 83(2), pages 359-376, June.
    9. Guo, Qiangqiang & Ban, Xuegang (Jeff), 2023. "A multi-scale control framework for urban traffic control with connected and automated vehicles," Transportation Research Part B: Methodological, Elsevier, vol. 175(C).
    10. J. Han & D. Sun, 1997. "Newton and Quasi-Newton Methods for Normal Maps with Polyhedral Sets," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 659-676, September.
    11. Nguyen Qui, 2014. "Stability for trust-region methods via generalized differentiation," Journal of Global Optimization, Springer, vol. 59(1), pages 139-164, May.
    12. Michael Patriksson & R. Tyrrell Rockafellar, 2003. "Sensitivity Analysis of Aggregated Variational Inequality Problems, with Application to Traffic Equilibria," Transportation Science, INFORMS, vol. 37(1), pages 56-68, February.
    13. U. Felgenhauer, 1999. "Regularity Properties of Optimal Controls with Application to Discrete Approximation," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 97-110, July.
    14. Ilker Birbil, S. & Gürkan, G. & Listes, O.L., 2004. "Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints : Sample-Path Analysis," Discussion Paper 2004-25, Tilburg University, Center for Economic Research.
    15. Yong-Jin Liu & Li Wang, 2016. "Properties associated with the epigraph of the $$l_1$$ l 1 norm function of projection onto the nonnegative orthant," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 84(1), pages 205-221, August.
    16. A. F. Izmailov & M. V. Solodov, 2015. "Newton-Type Methods: A Broader View," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 577-620, February.
    17. J. V. Outrata, 1999. "Optimality Conditions for a Class of Mathematical Programs with Equilibrium Constraints," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 627-644, August.
    18. Ashkan Mohammadi & Boris S. Mordukhovich & M. Ebrahim Sarabi, 2020. "Superlinear Convergence of the Sequential Quadratic Method in Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 731-758, September.
    19. A. L. Dontchev, 1998. "A Proof of the Necessity of Linear Independence Condition and Strong Second-Order Sufficient Optimality Condition for Lipschitzian Stability in Nonlinear Programming," Journal of Optimization Theory and Applications, Springer, vol. 98(2), pages 467-473, August.
    20. M. Bianchi & G. Kassay & R. Pini, 2015. "Stability Results of Variational Systems Under Openness with Respect to Fixed Sets," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 92-108, January.

    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:171:y:2016:i:2:d:10.1007_s10957-016-0897-y. 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.