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Asymptotic analysis, existence and sensitivity results for aclass of multivalued complementarity problems

Published online by Cambridge University Press:  22 March 2006

Fabián Flores-Bazán  and
Rubén López
Fabián Flores-Bazán
Affiliation:
Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Concepción, Chile; fflores@ing-mat.udec.cl
Rubén López
Affiliation:
Facultad de Ingeniería, Universidad Católica de la Santísima Concepción, Concepción, Chile; rlopez@ucsc.cl
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Abstract

In this work we study the multivalued complementarity problem on the non-negative orthant. This is carried out by describing the asymptotic behavior of the sequence of approximate solutions to its multivalued variational inequality formulation. By introducing new classes of multifunctions we provide several existence (possibly allowing unbounded solution set), stability as well as sensitivity results which extend and generalize most of the existing ones in the literature. We also present some kind of robustness results regarding existence of solution with respect to certain perturbations. Topological properties of the solution-set multifunction are established and some notions of approximable multifunctions are also discussed. In addition, some estimates for the solution set and its asymptotic cone are derived, as well as the existence of solutions for perturbed problems is studied.

Keywords

Multivalued complementarity problemcopositive mappingsasymptotic analysisouter semicontinuitygraphical convergence.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 2 , April 2006 , pp. 271 - 293
Copyright
© EDP Sciences, SMAI, 2006

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References

J.-P. Aubin and A. Cellina, Differential Inclusions. Springer, Berlin (1984).
J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990).
A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2003).
R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, New York (1992).
Crouzeix, J.P., Pseudomonotone variational inequality problems: Existence of solutions. Math. Program. 78 (1997) 305–314.
Daniilidis, A. and Hadjisavvas, N., Coercivity conditions and variational inequalities. Math. Program. 86 (1999) 433–438. CrossRef
Flores-Bazán, F., Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case. SIAM J. Optim. 11 (2000) 675–690. CrossRef
Flores-Bazán, F., Existence theory for finite dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77 (2003) 249–297. CrossRef
Flores-Bazán, F. and López, R., The linear complementarity problem under asymptotic analysis. Math. Oper. Res. 30 (2005) 73–90. CrossRef
García, C.B., Some classes of matrices in linear complementarity theory. Math. Program. 5 (1973) 299–310. CrossRef
Gowda, S.M., Complementarity problems over locally compact cones. SIAM J. Control Optim. 27 (1989) 836–841. CrossRef
Gowda, S.M. and Pang, J.-S., The basic theorem of complementarity revisited. Math. Program. 58 (1993) 161–177. CrossRef
Gowda, S.M. and Pang, J.-S., Some existence results for multivalued complementarity problems. Math. Oper. Res. 17 (1992) 657–669. CrossRef
Isac, G., The numerical range theory and boundedness of solutions of the complementarity problem. J. Math. Anal. Appl. 143 (1989) 235–251. CrossRef
Karamardian, S., The complementarity problem. Math. Program. 2 (1972) 107–129. CrossRef
Karamardian, S., An existence theorem for the complementarity problem. J. Optim. Theory Appl. 19 (1976) 227–232. CrossRef
Mangasarian, O.L. and McLinden, L., Simple bounds for solutions of monotone complementarity problems and convex programs. Math. Program. 32 (1985) 32–40. CrossRef
Moré, J.J., Classes of functions and feasibility conditions in nonlinear complementarity problems. Math. Program. 6 (1974) 327–338. CrossRef
Moré, J.J., Coercivity conditions in nonlinear complementarity problems. SIAM Rev. 17 (1974) 1–16. CrossRef
Parida, J. and Sen, A., Duality and existence theory for nondifferenciable programming. J. Optim. Theory Appl. 48 (1986) 451–458. CrossRef
Parida, J. and Sen, A., A class of nonlinear complementarity problems for multifunctions. J. Optim. Theory Appl. 53 (1987) 105–113. CrossRef
Parida, J. and Sen, A., A variational-like inequality for multifunctions with applications. J. Math. Anal. Appl. 124 (1987) 73–81. CrossRef
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer, Berlin (1998).
Saigal, R., Extension of the generalized complementarity problem. Math. Oper. Res. 1 (1976) 260–266. CrossRef
Zhao, Y., Existence of a solution to nonlinear variational inequality under generalized positive homogeneity. Oper. Res. Lett. 25 (1999) 231–239. CrossRef