ON THE SOLUTION OF THE SYMMETRIC EIGENVALUE COMPLEMENTARITY PROBLEM BY THE SPECTRAL PROJECTED GRADIENT ALGORITHM JOAQUIM J. J ´ UDICE, MARCOS RAYDAN, SILV ´ ERIO S. ROSA, AND SANDRA A. SANTOS Abstract. This paper is devoted to the Eigenvalue Complementarity Problem (EiCP) with symmetric real matrices. This problem is equivalent to finding a stationary point of a differ- entiable optimization program involving the Rayleigh quotient on a simplex [22]. We discuss a logarithmic function and a quadratic programming formulation to find a complementarity eigen- value by computing a stationary point of an appropriate merit function on a special convex set. A variant of the spectral projected gradient algorithm with a specially designed line search is introduced to solve the EiCP. Computational experience shows that the application of this algorithm to the logarithmic function formulation is a quite efficient way to find a solution to the symmetric EiCP. 1. Introduction Given the matrix A ∈ n×n and the positive definite (PD) matrix B ∈ n×n , the Eigenvalue Complementarity Problem (EiCP) is a problem of the form Find λ> 0 and x ∈ n \{0} such that      w =(λB − A)x, w  0,x  0, x T w =0. (1.1) The EiCP is a particular case of the Mixed Eigenvalue Complementarity Problem (MEiCP J ) that consists of finding a scalar λ> 0 and a vector x ∈ n \{0} such that            w =(λB − A)x, w J  0,x J  0, w T J x J =0, w ¯ J =0, where x J ≡ (x j ,j ∈ J ),w J ≡ (w j ,j ∈ J ),J ⊆{1,...,n} and ¯ J = {1,...n}\ J . Note that the EiCP is obtained when J = {1,...,n}. The MEiCP J is a generalization of the EiCP, that appears more frequently in practical problems of engineering and physics where the computation of eigenvalues is required. Problems involving the resonance frequency of structures and stability of dynamical systems are among these applications and have been discussed in [9]. Extensions of these problems to more general cones have been discussed in [24, 25, 26, 28]. We are interested in the Symmetric EiCP, in which the matrices A and B are both symmetric (i.e., when B is SPD). As is traditional in complementarity problems, the most important conclusions for the EiCP also hold for the MEiCP J . Date : February 25, 2008. Key words and phrases. Complementarity, Projected Gradient Algorithms, Eigenvalue Problems. 1