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We introduce an Alternating Direction Method of Multipliers (ADMM) for finding a solution of the nonsymmetric Eigenvalue Complementarity Problem (EiCP). A simpler version of this method is proposed for the symmetric EiCP, that is, for the computation of a Stationary Point (SP) of a Standard Fractional Quadratic Program. The algorithm is also extended for the computation of an SP of a Standard Quadratic Program (StQP). Convergence analyses of these three versions of ADMM are presented. The main computational effort of ADMM is the solution of a Strictly Convex StQP, which can be efficiently solved by a Block Principal Pivoting algorithm. Furthermore, this algorithm provides a stopping criterion for ADMM that improves very much its efficacy to compute an accurate solution of the EiCP. Numerical results indicate that ADMM is in general very efficient for solving symmetric EiCPs in terms of the number of iterations and computational effort, but is less efficient for the solution of nonsymmetric EiCPs. However, ADMM is able to provide a good initial point for a fast second-order method, such as the so-called Semismooth Newton method. The resulting hybrid ADMM and SN algorithm seems to be quite efficient in practice for the solution of nonsymmetric EiCPs. [ABSTRACT FROM AUTHOR]

We study splitting methods for solving the Eigenvalue Complementarity Problem (EiCP). We introduce four variants, which depend on the properties (symmetry, nonsymmetry, positive definite, negative definite, indefinite) of the matrices included in the definition of EiCP. Convergence analyses for each one of these versions of the splitting method are discussed. Special choices for the splitting matrices associated with these versions are recommended and tested on the solution of small and large symmetric and nonsymmetric EiCPs. These experiments show that the four versions of the splitting method work well at least for some choices of the splitting matrices. Furthermore, these versions of the splitting methods seem to be competitive with the most efficient state-of-the-art algorithms for the solution of EiCP. [ABSTRACT FROM AUTHOR]

In this paper, the solution of the symmetric quadratic eigenvalue complementarity problem (QEiCP) is addressed. The QEiCP has a solution provided the so-called co-regular and co-hyperbolic properties hold and is said to be symmetric if all the matrices involved in its definition are symmetric. We show that under the two conditions stated above the symmetric QEiCP can be reduced to the problem of computing a stationary point of an appropriate nonlinear program (NLP). We also investigate the reduction of the QEiCP to a simpler eigenvalue complementarity problem (EiCP). This transformation enables us to show that the co-regular and co-hyperbolic properties are not necessary for the existence of a solution to the QEiCP. Furthermore the QEiCP is shown to be equivalent to the problem of finding a stationary point of a quadratic fractional program (QFP) under special conditions on the matrices of the QEiCP. The use of the so-called spectral projected-gradient (SPG) algorithm for dealing with the programs NLP and QFP is also investigated. Some considerations about the implementation of this algorithm are discussed. Computational experience is included to highlight the efficiency of the algorithm for finding a solution of the QEiCP by exploring the NLPs mentioned above. [ABSTRACT FROM PUBLISHER]

In this paper, we address the solution of the symmetric eigenvalue complementarity problem (EiCP) by treating an equivalent reformulation of finding a stationary point of a fractional quadratic program on the unit simplex. The spectral projected-gradient (SPG) method has been recommended to this optimization problem when the dimension of the symmetric EiCP is large and the accuracy of the solution is not a very important issue. We suggest a new algorithm which combines elements from the SPG method and the block active set method, where the latter was originally designed for box constrained quadratic programs. In the new algorithm the projection onto the unit simplex in the SPG method is replaced by the much cheaper projection onto a box. This can be of particular advantage for large and sparse symmetric EiCPs. Global convergence to a solution of the symmetric EiCP is established. Computational experience with medium and large symmetric EiCPs is reported to illustrate the efficacy and efficiency of the new algorithm. [ABSTRACT FROM AUTHOR]

The eigenvalue complementarity problem (EiCP) differs from the traditional eigenvalue problem in that the primal and dual variables belong to a closed and convex coneKand its dual, respectively, and satisfy a complementarity condition. In this paper we investigate the solution of the second-order cone EiCP (SOCEiCP) whereKis the Lorentz cone. We first show that the SOCEiCP reduces to a special Variational Inequality Problem on a compact set defined byKand a normalization constraint. This guarantees that SOCEiCP has at least one solution, and a new enumerative algorithm is introduced for finding a solution to this problem. The method is based on finding a global minimum of an appropriate nonlinear programming (NLP) formulation of the SOCEiCP using a special branching scheme along with a local nonlinear optimizer that computes stationary points on subsets of the feasible region of NLP associated with the nodes generated by the algorithm. A semi-smooth Newton's method is combined with this enumerative algorithm to enhance its numerical performance. Our computational experience illustrates the efficacy of the proposed techniques in practice. [ABSTRACT FROM AUTHOR]

The solution of the Conic Quadratic Eigenvalue Complementarity Problem (CQEiCP) is first investigated without assuming symmetry on the matrices defining the problem. A new sufficient condition for existence of solutions of CQEiCP is presented, extending to arbitrary pointed, closed and convex cones a condition known to hold when the cone is the nonnegative orthant. We also address the symmetric CQEiCP where all its defining matrices are symmetric. We show that, assuming that two of its defining matrices are positive definite, this symmetric CQEiCP reduces to the computation of a stationary point of an appropriate merit function on a convex set. Furthermore, we discuss the use of the so called Spectral Projected Gradient (SPG) algorithm for solving CQEiCP when the cone of interest is the second-order cone (SOCQEiCP). A new algorithm is designed for the computation of the projections required by the SPG method to deal with SOCQEiCP. Numerical results are included to illustrate the efficiency of the SPG method and the new projection technique in practice. [ABSTRACT FROM AUTHOR]