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.
Mots clés : nonlinear complementarity problem, variational inequality, asymptotic analysis, sensitivity analysis
@article{COCV_2008__14_4_744_0, 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}, pages = {744--758}, publisher = {EDP-Sciences}, volume = {14}, number = {4}, year = {2008}, doi = {10.1051/cocv:2008003}, mrnumber = {2451793}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2008003/} }
TY - JOUR AU - López, Rubén TI - Some new existence, sensitivity and stability results for the nonlinear complementarity problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 744 EP - 758 VL - 14 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2008003/ DO - 10.1051/cocv:2008003 LA - en ID - COCV_2008__14_4_744_0 ER -
%0 Journal Article %A López, Rubén %T Some new existence, sensitivity and stability results for the nonlinear complementarity problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 744-758 %V 14 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2008003/ %R 10.1051/cocv:2008003 %G en %F COCV_2008__14_4_744_0
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] Set-Valued Analysis. Birkhäuser, Boston (1990). | MR | Zbl
and ,[2] Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Berlin (2003). | MR | Zbl
and ,[3] The Linear Complementarity Problem. Academic Press, New York (1992). | MR | Zbl
, and ,[4] Pseudomonotone variational inequality problems: Existence of solutions. Math. Program. 78 (1997) 305-314. | MR | Zbl
,[5] Sensitivity analysis in variational inequalities. Math. Oper. Res. 13 (1988) 421-434. | MR | Zbl
,[6] Some perturbation results for the linear complementarity problem. Math. Program. 23 (1982) 181-192. | MR | Zbl
,[7] Total stability of variational inequalities. Technical Report 09-98, Dipartimento di Informatica e Sistematica, Università Degli Stuti di Roma “La Sapienza” (1998).
and ,[8] Finite-Dimensional Variational Inequalities and Complementarity Problems I. Springer, New York (2003). | Zbl
and ,[9] The linear complementarity problem under asymptotic analysis. Math. Oper. Res. 30 (2005) 73-90. | MR | Zbl
and ,[10] Characterizing Q-matrices beyong L-matrices. J. Optim. Theory Appl. 127 (2005) 447-457. | MR | Zbl
and ,[11] Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems. ESAIM: COCV 12 (2006) 271-293. | EuDML | Numdam | MR | Zbl
and ,[12] Complementarity problems over locally compact cones. SIAM J. Control Optim. 27 (1989) 836-841. | MR | Zbl
,[13] On solution stability of the linear complementarity problems. Math. Oper. Res. 17 (1992) 77-83. | MR | Zbl
and ,[14] Some existence results for multivalued complementarity problems. Math. Oper. Res. 17 (1992) 657-669. | MR | Zbl
and ,[15] The basic theorem of complementarity revisited. Math. Program. 58 (1993) 161-177. | MR | Zbl
and ,[16] On the boundedness and stability to the affine variational inequality problem. SIAM J. Control Optim. 32 (1994) 421-441. | MR | Zbl
and ,[17] On the Lipschitzian properties of polyhedral multifunctions. Math. Program. 74 (1996) 267-278. | MR | Zbl
and ,[18] Application of degree theory in stability of the complementarity problem. Math. Oper. Res. 12 (1987) 368-376. | MR | Zbl
,[19] Finite-dimensional variational and nonlinear complementarity problems: A survey of theory, algorithms and applications. Math. Program. 48 (1990) 161-220. | MR | Zbl
and ,[20] Point-to-set maps in mathematical programming. SIAM Rev. 15 (1973) 591-603. | MR | Zbl
,[21] The numerical range theory and boundedness of solutions of the complementarity problem. J. Math. Anal. Appl. 143 (1989) 235-251. | MR | Zbl
,[22] Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75 (1992) 281-295. | MR | Zbl
,[23] Generalized complementarity problem. J. Optim. Theory Appl. 8 (1971) 161-168. | MR | Zbl
,[24] Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18 (1976) 445-454. | MR | Zbl
,[25] An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). | MR | Zbl
and ,[26] Sensitivity analysis for variational inequalities and complementarity problems. Ann. Oper. Res. 27 (1990) 143-174. | MR | Zbl
,[27] Characterizations of bounded solutions of linear complementarity problems. Math. Program. Study 19 (1982) 153-166. | MR | Zbl
,[28] Simple bounds for solutions of monotone complementarity problems and convex programs. Math. Program. 32 (1985) 32-40. | MR | Zbl
and ,[29] A monotone complementarity problem with feasible solutions but no complementarity solutions. Math. Program. 12 (1977) 131-132. | MR | Zbl
,[30] On the parametric nonlinear complementarity problem. Math. Program. Study 7 (1978) 142-150. | MR | Zbl
,[31] Coercivity conditions in nonlinear complementarity problems. SIAM Rev. 17 (1974) 1-16. | MR | Zbl
,[32] Complementarity problems, in Nonconvex Optimization and its Applications: Handbook of Global Optimization, R. Horst and P.M. Pardalos Eds., Kluwer, Dordrecht (1995). | MR | Zbl
,[33] Some continuity properties of polyhedral multifunctions. Math. Program. Study 14 (1981) 206-214. | MR | Zbl
,[34] Variational Analysis. Springer, Berlin (1998). | MR | Zbl
and ,[35] Sensitivity analysis for complementarity problems. J. Optim. Theory Appl. 48 (1986) 191-204. | MR | Zbl
,[36] Continuity properties of solutions of vector optimization. Nonlinear Anal. 64 (2006) 2496-2506. | MR | Zbl
and ,[37] Existence of a solution to nonlinear variational inequality under generalized positive homogeneity. Oper. Res. Lett. 25 (1999) 231-239. | MR | Zbl
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