The Descent Algorithms For Solving Symmetric Pareto Eigenvalue Problem

The linear and nonlinear complementarity problems widely used in engineering,economy and other fields are an important research topic in the field of optimization and numerical algebra.The eigenvalue complementarity problem,which is a special type of complementary problem,also known as the eigenvalue problem under the cone constraint,originally derived from the study of variational inequality operators based on bifurcation theory.At present,it has important applications in scientific engineering fields such as stability analysis of friction elastic system and dynamics analysis of structural mechanical system.The eigenvalue complementarity problem is a generalization of the classical matrix eigenvalue problem,but many theoretical properties of the classical matrix eigenvalue problem can not be generalized to the eigenvalue complementarity problem,such as similarity invariance,transpose invariance,rotation invariance,and so on.Due to the lack of the necessary theoretical basis,numerical algorithms for solving such problems can only be designed from the perspective of optimization.on the other hand,the number of eigenvalues may increase exponentially with the scale of the problem,it is an NP-difficult problem for solving all complementary eigenvalues of medium or large-scale matrices,which poses great challenges to the calculation of eigenvalues.This thesis focuses on the symmetric complementarity eigenvalues problem on a class of self-dual cones?non-negative cones?,namely the symmetric Pareto eigenvalue problem,which can be described as follows:Given a real symmetric matrix A?R^n×n,find a real number?and a nonzero vector x?Rⁿto satisfyx?0,?A-?I?x?0,x^T?A-?I?x=0.By using the structure and properties of the symmetric matrix,the symmetric Pare-to eigenvalue problem can be transfered equivalently to the constrained optimization problem on the differentiable Rayleigh quotient function.Each stability point of the optimization problem is the complementarity eigenvetor of the symmetric Pareto eigen-value problem.On this basis,this paper deeply studies the basic properties of the symmetric Pareto eigenvalue problem,and then two kinds of descent algorithms for solving the constraint optimization problems are given,and the convergence of the corresponding algorithms is also analysed one class of descent algorithms is an iterative algorithm for solving symmetric Pareto eigenvalue problems by using the modified negative gradi-ent direction with the idea of the steepest descent method.Another type of descent algorithm uses the NCP function and the negative gradient direction as the descent di-rection,and use the accurate linear search instead of the Armijo search to determine the step size.Based on the numerical results,the computational can be improved efficien-cy.In this paper,the algorithm is further extended to solve the generalized eigenvalue complementarity problem derived from the unilateral friction elastic system,and the effectiveness of the algorithm is verified by numerical experiments.