Applications of Mathematics

, Volume 52, Issue 6, pp 453–472 | Cite as

Existence of minimizers and necessary conditions in set-valued optimization with equilibrium constraints


In this paper we study set-valued optimization problems with equilibrium constraints (SOPECs) described by parametric generalized equations in the form 0 ∈ G(x) + Q(x), where both G and Q are set-valued mappings between infinite-dimensional spaces. Such models particularly arise from certain optimization-related problems governed by set-valued variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish general results on the existence of optimal solutions under appropriate assumptions of the Palais-Smale type and then derive necessary conditions for optimality in the models under consideration by using advanced tools of variational analysis and generalized differentiation.


variational analysis nonsmooth and set-valued optimization equilibrium constraints existence of optimal solutions necessary optimality conditions generalized differentiation 


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  1. [1]
    T. Q. Bao, P. Gupta, B. S. Mordukhovich: Necessary conditions in multiobjective optimization with equilibrium constraints. J. Optim. Theory Appl. 135 (2007).Google Scholar
  2. [2]
    T. Q. Bao, B. S. Mordukhovich: Variational principles for set-valued mappings with applications to multiobjective optimization. Control Cybern. 36 (2007), 531–562.MathSciNetGoogle Scholar
  3. [3]
    J. M. Borwein, Q. J. Zhu: Techniques of Variational Analysis. CMS Books in Math., Vol. 20. Springer-Verlag, New York, 2005.Google Scholar
  4. [4]
    F. Facchinei, J.-S. Pang: Finite-Dimensional Variational Inequalities and Complementary Problems, Vol. I, Vol. II. Springer-Verlag, New York, 2003.Google Scholar
  5. [5]
    J. Jahn: Vector Optimization. Theory, Applications and Extensions. Series Oper. Res. Springer-Verlag, Berlin, 2004.Google Scholar
  6. [6]
    Z.-Q. Luo, J.-S. Pang, and D. Ralph: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, 1997.MATHGoogle Scholar
  7. [7]
    B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. I. Basic Theory. Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 330. Springer-Verlag, Berlin, 2006.Google Scholar
  8. [8]
    B. S. Mordukhovich: Variational Analysis and Generalized Differentiation. II. Applications. Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 331. Springer-Verlag, Berlin, 2006.Google Scholar
  9. [9]
    B. S. Mordukhovich, J. V. Outrata: On second-order subdifferentials and their applications. SIAM J. Optim. 12 (2001), 139–169.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    B. S. Mordukhovich, J. V. Outrata: Coderivative analysis of quasivariational inequalities with applications to stability and optimization. SIAM J. Optim. To appear.Google Scholar
  11. [11]
    B. S. Mordukhovich, J. V. Outrata, and M. Červinka: Equilibrium problems with complementarity constraints: case study with applications to oligopolistic markets. Optimization. To appear.Google Scholar
  12. [12]
    J. V. Outrata: Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. 24 (1999), 627–644.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    J. V. Outrata: A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38 (2000), 1623–1638.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    J. V. Outrata: A note on a class of equilibrium problems with equilibrium constraints. Kybernetika 40 (2004), 585–594.MathSciNetGoogle Scholar
  15. [15]
    J. V. Outrata, M. Kočvara, and J. Zowe:: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht, 1998.MATHGoogle Scholar
  16. [16]
    S. M. Robinson: Generalized equations and their solutions. I. Basic theory. Math. Program. Study 10 (1979), 128–141.MATHGoogle Scholar
  17. [17]
    R.T. Rockafellar, R. J.-B. Wets: Variational Analysis. Grundlehren Series (Fundamental Principles of Mathematical Sciences), Vol. 317. Springer-Verlag, Berlin, 1998.Google Scholar
  18. [18]
    X. Y. Zheng, K. F. Ng: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17 (2006), 1154–1175.MATHCrossRefMathSciNetGoogle Scholar

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© Mathematical Institute, Academy of Sciences of Czech Republic 2007

Authors and Affiliations

  1. 1.Department of MathematicsWayne State UniversityDetroitUSA

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