Abstract
The eigenvalue complementarity problem (EiCP) is a kind of very useful model, which is widely used in the study of many problems in mechanics, engineering, and economics. The EiCP was shown to be equivalent to a special nonlinear complementarity problem or a mathematical programming problem with complementarity constraints. The existing methods for solving the EiCP are all nonsmooth methods, including nonsmooth or semismooth Newton type methods. In this paper, we reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them. Under mild assumptions, the method is proved globally convergent. Finally, some numerical results and the extensions of the method are also given. The numerical experiments highlight the efficiency of the method.
Highlights
Eigenvalue complementarity problem (EiCP) is proposed in the study of the problems in mechanics, engineering, and economics
We reformulate the EiCP as a system of continuously differentiable equations and give the Levenberg-Marquardt method to solve them
We reformulate the EiCP as a system of continuously differentiable equations that is one of the most interesting themes
Summary
Eigenvalue complementarity problem (EiCP) is proposed in the study of the problems in mechanics, engineering, and economics. The EiCP can be reformulated to be a special complementarity problem or a mathematical programming optimization problem with complementarity constraints and can use nonsmooth or semismooth Newton type method to solve it, such as [5,6,7]. The Levenberg-Marquardt method is one of the widely used methods in solving optimization problems (see, for instance, [8,9,10,11,12,13,14,15]). Use a trust region strategy to replace the line search, the Levenberg-Marquardt method is widely considered to be the progenitor of the trust region method approach for general unconstrained or constrained optimization problems. The advantage of the reformulation is that we solve the equations with continuously differentiable functions for which there are rich powerful solution methods and theory analysis, including the powerful Levenberg-Marquardt method. Given the matrix A ∈ Rn×n and the matrix B ∈ Rn×n, which are positive definite matrix, we consider to find a scalar λ ∈ R and a vector x ∈ Rn \ {0}, such that (A − λB) x ≥ 0, x ≥ 0,
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