A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem

Thi, Hoai An Le; Moeini, Mahdi; Dinh, Tao Pham; Júdice, Joaquim J.

doi:10.1007/s10589-010-9388-5

Cited by 40 publications

(8 citation statements)

References 29 publications

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“…Many practical algorithms have been developed during the last years for the solution of this problem and some of its extensions [5,9,25,27,28,51,57,68,74,75,81,84]. Given a matrix A ∈ R n×n and a Positive Definite (PD) matrix B ∈ R n×n (i.e., x T Bx > 0 for all x ∈ R n − {0}), the EiCP consists of finding a complementary eigenvalue λ ∈ R 1 and an associated eigenvector x ∈ R n − {0} satisfying the following conditions…”

Section: Applications and Formulations Of Nonconvex Optimization Probmentioning

confidence: 99%

Optimization With Linear Complementarity Constraints

Júdice

2014

Pesqui. Oper.

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ABSTRACT. A Mathematical Program with Linear Complementarity Constraints (MPLCC)is an optimization problem where a continuously differentiable function is minimized on a set defined by linear constraints and complementarity conditions on pairs of complementary variables. This problem finds many applications in several areas of science, engineering and economics and is also an important tool for the solution of some NP-hard structured and nonconvex optimization problems, such as bilevel, bilinear and nonconvex quadratic programs and the eigenvalue complementarity problem. In this paper some of the most relevant applications of the MPLCC and formulations of nonconvex optimization problems as MPLCCs are first presented.Algorithms for computing a feasible solution, a stationary point and a global minimum for the MPLCC are next discussed. The most important nonlinear programming methods, complementarity algorithms, enumerative techniques and 0 − 1 integer programming approaches for the MPLCC are reviewed. Some comments about the computational performance of these algorithms and a few topics for future research are also included in this survey.

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Section: Applications and Formulations Of Nonconvex Optimization Probmentioning

confidence: 99%

Optimization With Linear Complementarity Constraints

Júdice

2014

Pesqui. Oper.

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“…Modeling the EiCP as an optimization problem is a problem‐solving approach, which has been dealt with in the literature. In the work of Queiroz et al, the symmetric EiCP was reduced to the problem of finding stationary points of the Rayleigh quotient function on a simplex: Some local deterministic methods such as the projected gradient method and the difference of convex functions (DC) programming were then proposed to solve such EiCPs as interesting options for solving convex constrained problems. However, this reduction is no longer valid for asymmetric EiCPs.…”

Section: Introductionmentioning

confidence: 99%

Power iteration and inverse power iteration for eigenvalue complementarity problem

Abdi

Shakeri

2019

Numerical Linear Algebra App

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Summary In this paper, an inverse complementarity power iteration method (ICPIM) for solving eigenvalue complementarity problems (EiCPs) is proposed. Previously, the complementarity power iteration method (CPIM) for solving EiCPs was designed based on the projection onto the convex cone K. In the new algorithm, a strongly monotone linear complementarity problem over the convex cone K is needed to be solved at each iteration. It is shown that, for the symmetric EiCPs, the CPIM can be interpreted as the well‐known conditional gradient method, which requires only linear optimization steps over a well‐suited domain. Moreover, the ICPIM is closely related to the successive quadratic programming (SQP) via renormalization of iterates. The global convergence of these two algorithms is established by defining two nonnegative merit functions with zero global minimum on the solution set of the symmetric EiCP. Finally, some numerical simulations are included to evaluate the efficiency of the proposed algorithms.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

A Kind of Stochastic Eigenvalue Complementarity Problems

Wang

2018

Mathematical Problems in Engineering

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With the development of computer science, computational electromagnetics have also been widely used. Electromagnetic phenomena are closely related to eigenvalue problems. On the other hand, in order to solve the uncertainty of input data, the stochastic eigenvalue complementarity problem, which is a general formulation for the eigenvalue complementarity problem, has aroused interest in research. So, in this paper, we propose a new kind of stochastic eigenvalue complementarity problem. We reformulate the given stochastic eigenvalue complementarity problem as a system of nonsmooth equations with nonnegative constraints. Then, a projected smoothing Newton method is presented to solve it. The global and local convergence properties of the given method for solving the proposed stochastic eigenvalue complementarity problem are also given. Finally, the related numerical results show that the proposed method is efficient.

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A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem

Cited by 40 publications

References 29 publications

Optimization With Linear Complementarity Constraints

Optimization With Linear Complementarity Constraints

Power iteration and inverse power iteration for eigenvalue complementarity problem

A Kind of Stochastic Eigenvalue Complementarity Problems

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