Abstract

Given a multifunction F:R+n↪RnF:\R^n_+\hookrightarrow\R^n and q∈Rnq\in\R^n, the multivalued complementarity problem (MCP) on the positive orthant consists in finding xˉ≥0, yˉ∈F(xˉ): yˉ+q≥0, ⟨yˉ+q,xˉ⟩=0.\bar x\geq 0,~\bar y\in F(\bar x):~\bar y+q\geq 0,~ \langle \bar y+q,\bar x\rangle=0. Such a formulation appears in many applications in Science and Engineering and therefore was the object of many investigations in the last three decades. Most of the works existing in the literature deal with the case when FF is pseudomonotone (in the Karamardian sense) or quasimonotone, and only a few assume copositivity. In this work we introduce the notion of asymptotic multifunction with respect to a class of re-scaling functions including those with slow growth, and the notion of asymptotic multifunction associated to a sequence of multifunctions rather to a single one. Based on these two concepts we establish new existence theorems for the MCP for a class of multifunctions larger than copositive without assuming positive (sub)homogeneity as in a previous work. In addition, some stability and sensitivity results, as well as a robustness property, are provided. Thus, in this regard, we unify and generalize some of the results previously established.

Author information

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Fabián Flores-Bazán

Dep. de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile

fflores@ing-mat.udec.cl

Keywords

  • Complementarity problem
  • copositive multifunction
  • asymptotically bounded multifunction
  • asymptotic analysis

Mathematics Subject Classification

90C33, 49J53

References

  1. M. Bianchi, N. Hadjisavvas, S. Schaible: Minimal coercivity conditions and exceptional families of elements in quasimonotone variational inequalities, J. Optim. Theory Appl. 122 (2004) 1–17.https://doi.org/10.1023/B:JOTA.0000041728.12683.89Google Scholar
  2. S. C. Billups, K. G. Murty: Complementarity problems, J. Comput. Appl. Math. 124 (2000) 303–318.https://doi.org/10.1016/S0377-0427(00)00432-5Google Scholar
  3. R. W. Cottle, J. S. Pang, R. E. Stone: The Linear Complementarity Problem, Academic Press, Boston (1992).Google Scholar
  4. J.-P. Crouzeix: Pseudomonotone variational inequality problems: existence of solutions, Math. Program., Ser. A 78 (1997) 305–314.https://doi.org/10.1007/BF02614358Google Scholar
  5. A. Daniilidis, N. Hadjisavvas: Coercivity conditions and variational inequalities, Math. Program., Ser. A 86 (1999) 433–438.https://doi.org/10.1007/s101070050097Google Scholar
  6. F. Flores-Bazán: Existence theory for finite dimensional pseudomonotone equilibrium problems, Acta Appl. Math. 77 (2003) 249–297.https://doi.org/10.1023/A:1024971128483Google Scholar
  7. F. Flores-Bazán, R. López: Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems, ESAIM, Control Optim. Calc. Var. 12 (2006) 271–293.https://doi.org/10.1051/cocv:2006005Google Scholar
  8. C. B. García: Some classes of matrices in linear complementarity theory, Math. Program. 5 (1973) 299–310.https://doi.org/10.1007/BF01580135Google Scholar
  9. S. M. Gowda: Complementarity problems over locally compact cones, SIAM J. Control Optimization 27 (1989) 836–841.https://doi.org/10.1137/0327044Google Scholar
  10. S. M. Gowda, J.-S. Pang: Some existence results for multivalued complementarity problems, Math. Oper. Res. 17 (1992) 657–669.https://doi.org/10.1287/moor.17.3.657Google Scholar
  11. H. Han, Z. H. Huang, S. C. Fang: Solvability of variational inequality problems, J. Optim. Theory Appl. 122 (2004) 501–520.https://doi.org/10.1023/B:JOTA.0000042593.74934.b7Google Scholar
  12. P. T. Harker, J. S. Pang: Finite dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Program., Ser. B 48 (1990) 161–220.https://doi.org/10.1007/BF01582255Google Scholar
  13. Z. H. Huang: Generalization of an existence theorem for variational inequalities, J. Optim. Theory Appl. 118 (2003) 567–585.https://doi.org/10.1023/B:JOTA.0000004871.55273.15Google Scholar
  14. S. Karamardian: The complementarity problem, Math. Program. 2 (1972) 107–129.https://doi.org/10.1007/BF01584538Google Scholar
  15. S. Karamardian: An existence theorem for the complementarity problem, J. Optim. Theory Appl. 19 (1976) 227–232.https://doi.org/10.1007/BF00934094Google Scholar
  16. J. J. Moré: Classes of functions and feasibility conditions in nonlinear complementarity problems, Math. Program. 6 (1974) 327–338.https://doi.org/10.1007/BF01580248Google Scholar
  17. J. J. Moré: Coercivity conditions in nonlinear complementarity problems, SIAM Rev. 17 (1974) 1–16.https://doi.org/10.1137/1016001Google Scholar
  18. J. Parida, A. Sen: Duality and existence theory for nondifferenciable programming, J. Optim. Theory. Appl. 48 (1986) 451–458.https://doi.org/10.1007/BF00940571Google Scholar
  19. J. Parida, A. Sen: A class of nonlinear complementarity problems for multifunctions, J. Optim. Theory. Appl. 53 (1987) 105–113.https://doi.org/10.1007/BF00938819Google Scholar
  20. J. Parida, A. Sen: A variational-like inequality for multifunctions with applications, J. Math. Anal. Appl. 124 (1987) 73–81.https://doi.org/10.1016/0022-247X(87)90025-4Google Scholar
  21. R. T. Rockafellar, R. J.-B. Wets: Variational Analysis, Springer, Berlin (1998).https://doi.org/10.1007/978-3-642-02431-3Google Scholar
  22. R. Saigal: Extension of the generalized complementarity problem, Math. Oper. Res. 1 (1976) 260–266.https://doi.org/10.1287/moor.1.3.260Google Scholar
  23. Y. Zhao: Existence of a solution to nonlinear variational inequality under generalized positive homogeneity, Oper. Res. Lett. 25 (1999) 231–239.https://doi.org/10.1016/S0167-6377(99)00024-3Google Scholar

Suggested citation

F. Flores-Bazán. “Asymptotically Bounded Multifunctions and the MCP beyond Copositivity.” Journal of Convex Analysis 17 (2010), No. 1, 253–276. https://doi.org/10.68381/jca17019

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.