On the asymmetric eigenvalue complementarity problem

Abstract

In this paper, we discuss the Eigenvalue Complementarity Problem (EiCP) where at least one of its defining matrices is asymmetric. A sufficient condition for the existence of a solution to the EiCP is established. The EiCP is shown to be equivalent to finding a global minimum of an appropriate merit function on a convex set Ω defined by linear constraints. A sufficient condition for a stationary point of this function on Ω to be a solution of the EiCP is presented.A branch-and-bound procedure is developed for finding a global minimum of this merit function on Ω. In addition, a sequential enumerative algorithm for the computation of the minimum and the maximum eigenvalues is also discussed. Computational experience is included to highlight the efficiency and efficacy of the proposed methodologies to solve the asymmetric EiCP.

A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem

“…Computational experiments show that DCA is quite efficient and in most of the cases DCA outperforms SPGA, one among the best existing algorithms for EiCP. We intend to investigate in a near future the application of DCA to the asymmetric EICP [16]. On the other hand, the design of an efficient procedure for solving the quadratically constrained quadratic programming subproblems required by Algorithm 1 is also one of our current research areas.…”

Reconstructing a matrix from a partial sampling of Pareto eigenvalues

“…By substituting (20) and (21) into the equation (19) one gets the specific form of the linear system that has to be solved; we omit the details.…”

A Kind of Stochastic Eigenvalue Complementarity Problems

“…In the last few years, the eigenvalue complementarity problem has drawn increasing attention, in many literature systems, such as [8][9][10][11][12][13] and the references therein. Among them, in [8], the authors study an eigenvalue complementarity problem and find its origins in the solution of a contract problem in mechanics.…”

“…The authors transform this problem into a differentiable optimization program involving the Rayleigh quotient on a simplex and find its stationary point by the spectral projected gradient algorithm. In [10][11][12][13], many methods are proposed to solve the eigenvalue complementarity problems, such as Levenberg-Marquardt method and the derivative-free projection method. In [14], the stability of dynamic system is studied.…”