no. 2
Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 271-293.

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.

DOI : 10.1051/cocv:2006005
Classification : 90C33, 90C25, 47J20, 49J53
Mots clés : multivalued complementarity problem, copositive mappings, asymptotic analysis, outer semicontinuity, graphical convergence 
@article{COCV_2006__12_2_271_0,
     author = {Flores-Baz\'an, Fabi\'an and L\'opez, Rub\'en},
     title = {Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {271--293},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {2},
     year = {2006},
     doi = {10.1051/cocv:2006005},
     mrnumber = {2209354},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2006005/}
}
TY  - JOUR
AU  - Flores-Bazán, Fabián
AU  - López, Rubén
TI  - Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2006
SP  - 271
EP  - 293
VL  - 12
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2006005/
DO  - 10.1051/cocv:2006005
LA  - en
ID  - COCV_2006__12_2_271_0
ER  - 
%0 Journal Article
%A Flores-Bazán, Fabián
%A López, Rubén
%T Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2006
%P 271-293
%V 12
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2006005/
%R 10.1051/cocv:2006005
%G en
%F COCV_2006__12_2_271_0
Flores-Bazán, Fabián; López, Rubén. Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 271-293. doi : 10.1051/cocv:2006005. http://www.numdam.org/articles/10.1051/cocv:2006005/

[1] J.-P. Aubin and A. Cellina, Differential Inclusions. Springer, Berlin (1984). | MR | Zbl

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

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

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

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

[6] A. Daniilidis and N. Hadjisavvas, Coercivity conditions and variational inequalities. Math. Program. 86 (1999) 433-438. | Zbl

[7] F. Flores-Bazán, Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case. SIAM J. Optim. 11 (2000) 675-690. | Zbl

[8] F. Flores-Bazán, Existence theory for finite dimensional pseudomonotone equilibrium problems. Acta Appl. Math. 77 (2003) 249-297. | Zbl

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

[10] C.B. García, Some classes of matrices in linear complementarity theory. Math. Program. 5 (1973) 299-310. | Zbl

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

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

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

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

[15] S. Karamardian, The complementarity problem. Math. Program. 2 (1972) 107-129. | Zbl

[16] S. Karamardian, An existence theorem for the complementarity problem. J. Optim. Theory Appl. 19 (1976) 227-232. | Zbl

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

[18] J.J. Moré, Classes of functions and feasibility conditions in nonlinear complementarity problems. Math. Program. 6 (1974) 327-338. | Zbl

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

[20] J. Parida and A. Sen, Duality and existence theory for nondifferenciable programming. J. Optim. Theory Appl. 48 (1986) 451-458. | Zbl

[21] J. Parida and A. Sen, A class of nonlinear complementarity problems for multifunctions. J. Optim. Theory Appl. 53 (1987) 105-113. | Zbl

[22] J. Parida and A. Sen, A variational-like inequality for multifunctions with applications. J. Math. Anal. Appl. 124 (1987) 73-81. | Zbl

[23] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer, Berlin (1998). | MR | Zbl

[24] R. Saigal, Extension of the generalized complementarity problem. Math. Oper. Res. 1 (1976) 260-266. | Zbl

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

Cité par Sources :