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.