Joaquim Júdice (Universidade de Coimbra)

The Eingenvalue Complementarity Problem: Theory and Algorithms

Given a real matrix A and a real (symmetric or asymmetric) Positive Definite matrix B, the Eigenvalue Complementarity Problem (EiCP) is an extension of the well-known Generalized Eigenvalue Problem GEiP(A,B) where some of the variables of the problem are required to be nonnegative and to satisfy a complementarity constraint. This problem finds interesting applications in contact problems. A few nonlinear programming formulations are introduced for the symmetric EiCP, such that stationary points of the corresponding objective functions on appropriate convex sets lead to solutions of the problem. When at least one of the matrices A or B is asymmetric, the EiCP reduces to a Finite-Dimensional Variational Inequality and to a Global Optimization Problem. Projected gradient methods and an enumerative algorithm are introduced for finding a solution to the EiCP. The computation of several complementary eigenvalues and of the maximum and minimum of these eigenvalues is also discussed. Computational experience is reported to illustrate the efficiency of the algorithms to deal with the EiCP.