Some new existence, sensitivity and stability results for the nonlinear complementarity problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 744-758.

In this work we study the nonlinear complementarity problem on the nonnegative orthant. This is done by approximating its equivalent variational-inequality-formulation by a sequence of variational inequalities with nested compact domains. This approach yields simultaneously existence, sensitivity, and stability results. By introducing new classes of functions and a suitable metric for performing the approximation, we provide bounds for the asymptotic set of the solution set and coercive existence results, which extend and generalize most of the existing ones from the literature. Such results are given in terms of some sets called coercive existence sets, which we also employ for obtaining new sensitivity and stability results. Topological properties of the solution-set-mapping and bounds for it are also established. Finally, we deal with the piecewise affine case.

DOI : https://doi.org/10.1051/cocv:2008003
Classification : 90C31,  90C33,  47J20,  49J40,  49J45
Mots clés : nonlinear complementarity problem, variational inequality, asymptotic analysis, sensitivity analysis
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author = {L\'opez, Rub\'en},
title = {Some new existence, sensitivity and stability results for the nonlinear complementarity problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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mrnumber = {2451793},
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url = {http://www.numdam.org/articles/10.1051/cocv:2008003/}
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López, Rubén. Some new existence, sensitivity and stability results for the nonlinear complementarity problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 4, pp. 744-758. doi : 10.1051/cocv:2008003. http://www.numdam.org/articles/10.1051/cocv:2008003/

[1] J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhäuser, Boston (1990). | MR 1048347 | Zbl 0713.49021

[2] A. Auslender and M. Teboulle, Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2003). | MR 1931309 | Zbl 1017.49001

[3] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, New York (1992). | MR 1150683 | Zbl 0757.90078

[4] J.P. Crouzeix, Pseudomonotone variational inequality problems: Existence of solutions. Math. Program. 78 (1997) 305-314. | MR 1466134 | Zbl 0887.90167

[5] S. Dafermos, Sensitivity analysis in variational inequalities. Math. Oper. Res. 13 (1988) 421-434. | MR 961802 | Zbl 0674.49007

[6] R. Doverspike, Some perturbation results for the linear complementarity problem. Math. Program. 23 (1982) 181-192. | MR 657078 | Zbl 0484.90087

[7] 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).

[8] F. Facchinei and J.S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2003). | Zbl 1062.90001

[9] F. Flores-Bazán and R. López, The linear complementarity problem under asymptotic analysis. Math. Oper. Res. 30 (2005) 73-90. | MR 2125138 | Zbl 1082.90122

[10] F. Flores-Bazán and R. López, Characterizing Q-matrices beyong L-matrices. J. Optim. Theory Appl. 127 (2005) 447-457. | MR 2186133 | Zbl 1116.90100

[11] F. Flores-Bazán and R. López, Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems. ESAIM: COCV 12 (2006) 271-293. | EuDML 249671 | Numdam | MR 2209354 | Zbl 1275.90105

[12] M.S. Gowda, Complementarity problems over locally compact cones. SIAM J. Control Optim. 27 (1989) 836-841. | MR 1001922 | Zbl 0679.90082

[13] M.S. Gowda and J.S. Pang, On solution stability of the linear complementarity problems. Math. Oper. Res. 17 (1992) 77-83. | MR 1148779 | Zbl 0773.90078

[14] M.S. Gowda and J.S. Pang, Some existence results for multivalued complementarity problems. Math. Oper. Res. 17 (1992) 657-669. | MR 1177729 | Zbl 0777.90069

[15] M.S. Gowda and J.S. Pang, The basic theorem of complementarity revisited. Math. Program. 58 (1993) 161-177. | MR 1216489 | Zbl 0778.90074

[16] M.S. Gowda and J.S. Pang, On the boundedness and stability to the affine variational inequality problem. SIAM J. Control Optim. 32 (1994) 421-441. | MR 1261147 | Zbl 0800.93967

[17] M.S. Gowda and R. Sznajder, On the Lipschitzian properties of polyhedral multifunctions. Math. Program. 74 (1996) 267-278. | MR 1407688 | Zbl 0854.49010

[18] C.D. Ha, Application of degree theory in stability of the complementarity problem. Math. Oper. Res. 12 (1987) 368-376. | MR 888983 | Zbl 0616.90082

[19] P.T. Harker and J.S. Pang, Finite-dimensional variational and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Program. 48 (1990) 161-220. | MR 1073707 | Zbl 0734.90098

[20] W.W. Hogan, Point-to-set maps in mathematical programming. SIAM Rev. 15 (1973) 591-603. | MR 345641 | Zbl 0256.90042

[21] G. Isac, The numerical range theory and boundedness of solutions of the complementarity problem. J. Math. Anal. Appl. 143 (1989) 235-251. | MR 1019459 | Zbl 0722.47008

[22] G. Isac, Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75 (1992) 281-295. | MR 1191588 | Zbl 0795.90071

[23] S. Karamardian, Generalized complementarity problem. J. Optim. Theory Appl. 8 (1971) 161-168. | MR 321540 | Zbl 0218.90052

[24] S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18 (1976) 445-454. | MR 472053 | Zbl 0304.49026

[25] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). | MR 567696 | Zbl 0457.35001

[26] J. Kyparisis, Sensitivity analysis for variational inequalities and complementarity problems. Ann. Oper. Res. 27 (1990) 143-174. | MR 1088991 | Zbl 0723.90075

[27] O.L. Mangasarian, Characterizations of bounded solutions of linear complementarity problems. Math. Program. Study 19 (1982) 153-166. | MR 669730 | Zbl 0487.90088

[28] O.L. Mangasarian and L. Mclinden, Simple bounds for solutions of monotone complementarity problems and convex programs. Math. Program. 32 (1985) 32-40. | MR 787742 | Zbl 0567.90093

[29] N. Megiddo, A monotone complementarity problem with feasible solutions but no complementarity solutions. Math. Program. 12 (1977) 131-132. | MR 472038 | Zbl 0353.90084

[30] N. Megiddo, On the parametric nonlinear complementarity problem. Math. Program. Study 7 (1978) 142-150. | MR 472079 | Zbl 0379.90094

[31] J.J. Moré, Coercivity conditions in nonlinear complementarity problems. SIAM Rev. 17 (1974) 1-16. | MR 336496 | Zbl 0253.65033

[32] 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). | MR 1377087 | Zbl 0833.90114

[33] S.M. Robinson, Some continuity properties of polyhedral multifunctions. Math. Program. Study 14 (1981) 206-214. | MR 600130 | Zbl 0449.90090

[34] R.T. Rockafellar and R.J. Wets, Variational Analysis. Springer, Berlin (1998). | MR 1491362 | Zbl 0888.49001

[35] R.L. Tobin, Sensitivity analysis for complementarity problems. J. Optim. Theory Appl. 48 (1986) 191-204. | MR 825392 | Zbl 0557.49004

[36] S.W. Xiang and Y.H. Zhou, Continuity properties of solutions of vector optimization. Nonlinear Anal. 64 (2006) 2496-2506. | MR 2215822 | Zbl 1113.90145

[37] Y. Zhao, Existence of a solution to nonlinear variational inequality under generalized positive homogeneity. Oper. Res. Lett. 25 (1999) 231-239. | MR 1733051 | Zbl 0955.49004

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