The semismooth and smoothing Newton methods for solving Pareto eigenvalue problem

In this paper, we study the numerical behavior of the semismooth and smoothing Newton methods for solving Pareto eigenvalue problem of the formx⩾0,Ax-λBx⩾0,〈x,Ax-λBx〉=0,where (A, B) is a pair of possibly asymmetric matrices of order n. Such an eigenvalue problem arises in mechanics and in other areas of applied mathematics. By using the (smoothing) Fischer-Burmeister NCP function and the normalization condition eTx = 1, the Pareto eigenvalue problem can be converted into a equivalent semismooth (or smoothing) system of equations, where e = (1, … , 1)T. Then a semismooth (or smoothing) Newton algorithm is designed to solve such a semismooth (or smoothing) system of equations. Some numerical results are reported in the paper, which indicates that the proposed algorithms are very effective.