IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v169y2016i3d10.1007_s10957-016-0914-1.html
   My bibliography  Save this article

Optimality Conditions for Semi-Infinite and Generalized Semi-Infinite Programs Via Lower Order Exact Penalty Functions

Author

Listed:
  • Xiaoqi Yang

    (The Hong Kong Polytechnic University)

  • Zhangyou Chen

    (South-West Jiao Tong University)

  • Jinchuan Zhou

    (Shandong University of Technology)

Abstract

In this paper, we will study optimality conditions of semi-infinite programs and generalized semi-infinite programs by employing lower order exact penalty functions and the condition that the generalized second-order directional derivative of the constraint function at the candidate point along any feasible direction for the linearized constraint set is non-positive. We consider three types of penalty functions for semi-infinite program and investigate the relationship among the exactness of these penalty functions. We employ lower order integral exact penalty functions and the second-order generalized derivative of the constraint function to establish optimality conditions for semi-infinite programs. We adopt the exact penalty function technique in terms of a classical augmented Lagrangian function for the lower-level problems of generalized semi-infinite programs to transform them into standard semi-infinite programs and then apply our results for semi-infinite programs to derive the optimality condition for generalized semi-infinite programs. We will give various examples to illustrate our results and assumptions.

Suggested Citation

  • Xiaoqi Yang & Zhangyou Chen & Jinchuan Zhou, 2016. "Optimality Conditions for Semi-Infinite and Generalized Semi-Infinite Programs Via Lower Order Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 984-1012, June.
    • Handle: RePEc:spr:joptap:v:169:y:2016:i:3:d:10.1007_s10957-016-0914-1
    DOI: 10.1007/s10957-016-0914-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-016-0914-1
    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/s10957-016-0914-1?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. O. Stein, 2004. "On Constraint Qualifications in Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 647-671, June.
    2. Oliver Stein, 2001. "First-Order Optimality Conditions for Degenerate Index Sets in Generalized Semi-Infinite Optimization," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 565-582, August.
    3. X. Q. Yang & Z. Q. Meng, 2007. "Lagrange Multipliers and Calmness Conditions of Order p," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 95-101, February.
    4. Still, G., 1999. "Generalized semi-infinite programming: Theory and methods," European Journal of Operational Research, Elsevier, vol. 119(2), pages 301-313, December.
    5. J. J. Rückmann & A. Shapiro, 1999. "First-Order Optimality Conditions in Generalized Semi-Infinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 677-691, June.
    6. Xi Yin Zheng & Xiaoqi Yang, 2007. "Lagrange Multipliers in Nonsmooth Semi-Infinite Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 32(1), pages 168-181, February.
    7. X. X. Huang & X. Q. Yang, 2003. "A Unified Augmented Lagrangian Approach to Duality and Exact Penalization," Mathematics of Operations Research, INFORMS, vol. 28(3), pages 533-552, August.
    8. J. J. Ye & S. Y. Wu, 2008. "First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 419-434, May.
    9. Angelia Nedić & Asuman Ozdaglar, 2008. "Separation of Nonconvex Sets with General Augmenting Functions," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 587-605, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Olga Kostyukova & Tatiana Tchemisova, 2017. "Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 76-103, October.

    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. Kanzi, N. & Nobakhtian, S., 2010. "Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems," European Journal of Operational Research, Elsevier, vol. 205(2), pages 253-261, September.
    2. Stein, Oliver, 2012. "How to solve a semi-infinite optimization problem," European Journal of Operational Research, Elsevier, vol. 223(2), pages 312-320.
    3. J. J. Ye & S. Y. Wu, 2008. "First Order Optimality Conditions for Generalized Semi-Infinite Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 419-434, May.
    4. Kaiwen Meng & Xiaoqi Yang, 2015. "First- and Second-Order Necessary Conditions Via Exact Penalty Functions," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 720-752, June.
    5. Stuart M. Harwood & Paul I. Barton, 2017. "How to solve a design centering problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 86(1), pages 215-254, August.
    6. Stein, Oliver & Still, Georg, 2002. "On generalized semi-infinite optimization and bilevel optimization," European Journal of Operational Research, Elsevier, vol. 142(3), pages 444-462, November.
    7. Nader Kanzi, 2013. "Lagrange multiplier rules for non-differentiable DC generalized semi-infinite programming problems," Journal of Global Optimization, Springer, vol. 56(2), pages 417-430, June.
    8. O. I. Kostyukova & T. V. Tchemisova & S. A. Yermalinskaya, 2010. "Convex Semi-Infinite Programming: Implicit Optimality Criterion Based on the Concept of Immobile Indices," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 325-342, May.
    9. Geletu, Abebe & Hoffmann, Armin, 2004. "A conceptual method for solving generalized semi-infinite programming problems via global optimization by exact discontinuous penalization," European Journal of Operational Research, Elsevier, vol. 157(1), pages 3-15, August.
    10. C. Y. Wang & X. Q. Yang & X. M. Yang, 2013. "Nonlinear Augmented Lagrangian and Duality Theory," Mathematics of Operations Research, INFORMS, vol. 38(4), pages 740-760, November.
    11. O. Stein, 2004. "On Constraint Qualifications in Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 647-671, June.
    12. O. Kostyukova & T. Tchemisova, 2012. "Implicit optimality criterion for convex SIP problem with box constrained index set," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 475-502, July.
    13. Alexander Mitsos & Angelos Tsoukalas, 2015. "Global optimization of generalized semi-infinite programs via restriction of the right hand side," Journal of Global Optimization, Springer, vol. 61(1), pages 1-17, January.
    14. Hatim Djelassi & Moll Glass & Alexander Mitsos, 2019. "Discretization-based algorithms for generalized semi-infinite and bilevel programs with coupling equality constraints," Journal of Global Optimization, Springer, vol. 75(2), pages 341-392, October.
    15. Volker Maag, 2015. "A collision detection approach for maximizing the material utilization," Computational Optimization and Applications, Springer, vol. 61(3), pages 761-781, July.
    16. Yaohua Hu & Carisa Kwok Wai Yu & Xiaoqi Yang, 2019. "Incremental quasi-subgradient methods for minimizing the sum of quasi-convex functions," Journal of Global Optimization, Springer, vol. 75(4), pages 1003-1028, December.
    17. 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.
    18. Birbil, S.I. & Bouza, G. & Frenk, J.B.G. & Still, G.J., 2003. "Equilibrium Constrained Optimization Problems," ERIM Report Series Research in Management ERS-2003-085-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    19. Goberna, M.A. & Guerra-Vazquez, F. & Todorov, M.I., 2013. "Constraint qualifications in linear vector semi-infinite optimization," European Journal of Operational Research, Elsevier, vol. 227(1), pages 12-21.
    20. S. K. Zhu & S. J. Li, 2014. "Unified Duality Theory for Constrained Extremum Problems. Part II: Special Duality Schemes," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 763-782, 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:169:y:2016:i:3:d:10.1007_s10957-016-0914-1. 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.