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Some new existence, sensitivity and stability results for thenonlinear complementarity problem

Published online by Cambridge University Press:  18 January 2008

Rubén López
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 nonlinear complementarity problem on thenonnegative orthant. This is done by approximating its equivalentvariational-inequality-formulation by a sequence of variationalinequalities with nested compact domains. This approach yieldssimultaneously existence, sensitivity, and stability results. Byintroducing new classes of functions and a suitable metric forperforming the approximation, we provide bounds for the asymptoticset of the solution set and coercive existence results, which extendand generalize most of the existing ones from the literature. Suchresults are given in terms of some sets called coercive existencesets, which we also employ for obtaining new sensitivity andstability results. Topological properties of thesolution-set-mapping and bounds for it are also established.Finally, we deal with the piecewise affine case.

Keywords

Nonlinear complementarity problemvariational inequalityasymptotic analysissensitivity analysis
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 4 , October 2008 , pp. 744 - 758
Copyright
© EDP Sciences, SMAI, 2008

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References

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.
Dafermos, S., Sensitivity analysis in variational inequalities. Math. Oper. Res. 13 (1988) 421–434. CrossRef
Doverspike, R., Some perturbation results for the linear complementarity problem. Math. Program. 23 (1982) 181–192. CrossRef
F. Facchinei and J.S. Pang, Total stability of variational inequalities. Technical Report 09–98, Dipartimento di Informatica e Sistematica, Università Degli Stuti di Roma “La Sapienza” (1998).
F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2003).
Flores-Bazán, F. and López, R., The linear complementarity problem under asymptotic analysis. Math. Oper. Res. 30 (2005) 73–90. CrossRef
Flores-Bazán, F. and López, R., Characterizing Q-matrices beyong L-matrices. J. Optim. Theory Appl. 127 (2005) 447–457. CrossRef
Flores-Bazán, F. and López, R., Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems. ESAIM: COCV 12 (2006) 271–293. CrossRef
Gowda, M.S., Complementarity problems over locally compact cones. SIAM J. Control Optim. 27 (1989) 836–841. CrossRef
Gowda, M.S. and Pang, J.S., On solution stability of the linear complementarity problems. Math. Oper. Res. 17 (1992) 77–83. CrossRef
Gowda, M.S. and Pang, J.S., Some existence results for multivalued complementarity problems. Math. Oper. Res. 17 (1992) 657–669. CrossRef
Gowda, M.S. and Pang, J.S., The basic theorem of complementarity revisited. Math. Program. 58 (1993) 161–177. CrossRef
Gowda, M.S. and Pang, J.S., On the boundedness and stability to the affine variational inequality problem. SIAM J. Control Optim. 32 (1994) 421–441. CrossRef
Gowda, M.S. and Sznajder, R., On the Lipschitzian properties of polyhedral multifunctions. Math. Program. 74 (1996) 267–278. CrossRef
Application, C.D. Ha of degree theory in stability of the complementarity problem. Math. Oper. Res. 12 (1987) 368–376.
Harker, P.T. and Pang, J.S., Finite-dimensional variational and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Program. 48 (1990) 161–220. CrossRef
Hogan, W.W., Point-to-set maps in mathematical programming. SIAM Rev. 15 (1973) 591–603. CrossRef
Isac, G., The numerical range theory and boundedness of solutions of the complementarity problem. J. Math. Anal. Appl. 143 (1989) 235–251. CrossRef
Isac, G., Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75 (1992) 281–295.
Karamardian, S., Generalized complementarity problem. J. Optim. Theory Appl. 8 (1971) 161–168. CrossRef
Karamardian, S., Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18 (1976) 445–454. CrossRef
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980).
Kyparisis, J., Sensitivity analysis for variational inequalities and complementarity problems. Ann. Oper. Res. 27 (1990) 143–174. CrossRef
Mangasarian, O.L., Characterizations of bounded solutions of linear complementarity problems. Math. Program. Study 19 (1982) 153–166. 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
Megiddo, N., A monotone complementarity problem with feasible solutions but no complementarity solutions. Math. Program. 12 (1977) 131–132. CrossRef
Megiddo, N., On the parametric nonlinear complementarity problem. Math. Program. Study 7 (1978) 142–150. CrossRef
Moré, J.J., Coercivity conditions in nonlinear complementarity problems. SIAM Rev. 17 (1974) 1–16. CrossRef
J.S. Pang, Complementarity problems, in Nonconvex Optimization and its Applications: Handbook of Global Optimization, R. Horst and P.M. Pardalos Eds., Kluwer, Dordrecht (1995).
Robinson, S.M., Some continuity properties of polyhedral multifunctions. Math. Program. Study 14 (1981) 206–214. CrossRef
R.T. Rockafellar and R.J. Wets, Variational Analysis. Springer, Berlin (1998).
Tobin, R.L., Sensitivity analysis for complementarity problems. J. Optim. Theory Appl. 48 (1986) 191–204. CrossRef
Xiang, S.W. and Zhou, Y.H., Continuity properties of solutions of vector optimization. Nonlinear Anal. 64 (2006) 2496–2506. CrossRef
Zhao, Y., Existence of a solution to nonlinear variational inequality under generalized positive homogeneity. Oper. Res. Lett. 25 (1999) 231–239. CrossRef