Thesis Summary:

This thesis is concerned with the study of the equilibrium problem by using asymptotic analysis. As a model of study we consider the complementarity problem (CP). It is well-known that the (CP) is equivalent to a variational inequality problem (VIP). We employ the latter problem in order to study (CP). Under some continuity assumptions the (VIP) has solutions as soon as is defined on a bounded set. The object of this thesis is to deal with the (VIP) defined on unbounded sets (as in the (CP)). To this end we apply the asymptotic analysis: we approximate the (VIP) with problems defined on bounded sets and that have solutions and then we study the asymptotic properties of the normalized approximate solutions of such problems. With the aid of the obtained information, the reformulated Gowda-Pang existence theorem and by introducing several new classes of mappings, we obtain new existence, stability and sensitivity results. Moreover, we obtain bounds for the solutions sets and the asymptotic cones of the solution sets. The multivalued, piecewise polyhedral, and linear complementarity cases are studied in detail, the results from the literature are recovered and new results are given. The Lemke’s algorithm allows the resolution of the linear complementarity problem in a finite number of steps. For large size problems we have iterative algorithms. Among the latter the splitting method plays an important role. In order for this algorithm to be well-defined some of the matrices involved must be Q-matrices. We study such matrices in detail and characterize them within a new class of matrices we introduced, which enjoys good properties.

ISI Publications from the Thesis

Fabián FLORES-BAZáN, Ruben LOPEZ: Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems. ESAIM: Control Optimization and Calculus of Variations, Vol. 12, pp. 271-293, (2006)

Fabián FLORES-BAZáN, Ruben LOPEZ: Characterizing Q-matrices beyond L-matrices. Journal of Optimization Theory and Applications, vol. 127, pp. 447-457, (2005)

Fabián FLORES-BAZáN, Ruben LOPEZ: The linear complementarity problem under asymptotic analysis. Mathematics of Operation Research, vol. 30, 1, pp. 73-90. (2005)