A Fast Alternating Direction Method Of Multipliers For Solving Symmetric Eigenvalue Complementarity Problems

The computation of eigenvalue complementarity problem is very important in physics and engineering.Considering that the embedded matrix is often a symmetric matrix in many cases,a fast alternating direction method of multiplier is proposed to solve symmetric eigenvalue complementarity problems.In Chapter 1,the research background and advances of eigenvalue complementarity problem are briefly summarized.In Chapter 2,we summarize some mathematical symbols and then recall some basic definitions and theorems that will be used in this paper.In Chapter 3,the eigenvalue complementarity problem is characterized as a Rayleigh quotient maximization problem with a simplex constraint for the case that the underlying matrices are symmetric.Because the problem is a constrained fractional programming problem,which is very difficult to be solve.In this paper,the simplex constraint is separated by introducing an auxiliary variable,and the Rayleigh quotient maximization problem is transformed into an optimization model with separable structure,and the alternating direction method of multiplier is used to solve the problem.Considering that there is no explicit solution to one of the subproblems by using the alternating direction method of multiplier directly,this paper linearizes the objective function of the subproblem to obtain a partially linearized alternating direction multiplier method.The advantage of the new method is that each subproblem is simple enough with a closed solution so that the algorithm is easy to implement.Theoretically,the convergence of the algorithm is analyzed under the condition that the objective function satisfies the Kurdyka-Lojasiewicz inequality.Through numerical comparisons,the algorithm in this chapter works better than the classical spectral projection gradient algorithm for solving large-scale problems in terms of taking less computing time.In Chapter 4,in order to reduce the difficulty of solving fractional programming problem,the eigenvalue complementarity problem can be reformulated as the difference of two logarithmic functions over a simplex constraint.Based on resulting model,we present two equivalent characterizations of nonconvex optimization models.Accordingly,we propose a partially linearized alternating direction method of multiplier for the new model.Under certain conditions,the convergence of the method is analyzed.At the same time,the corresponding numerical results are given.Through numerical comparison,the algorithm in this chapter works better than the classical spectral projection gradient algorithm and DC algorithm for solving large-scale problems in terms of taking less computing time.In Chapter 5,we summarize the content of this paper and list our future concerns.