A semismooth Newton method for tensor eigenvalue complementarity problem

Abstract

In this paper, we consider the tensor eigenvalue complementarity problem which is closely related to the optimality conditions for polynomial optimization, as well as a class of differential inclusions with nonconvex processes. By introducing an NCPfunction, we reformulate the tensor eigenvalue complementarity problem as a system of nonlinear equations. We show that this function is strongly semismooth but not differentiable, in which case the classical smoothing methods cannot apply. Furthermore, we propose a damped semismooth Newton method for tensor eigenvalue complementarity problem. A new procedure to evaluate an element of the generalized Jocobian is given, which turns out to be an element of the B-subdifferential under mild assumptions. As a result, the convergence of the damped semismooth Newton method is guaranteed by existing results. The numerical experiments also show that our method is efficient and promising.

A semismooth Newton method for support vector classification and regression

“…is the ǫ-insensitive loss function which we call ǫ-L2-loss function associated with (x i , y i ). The parameter ǫ > 0 is given so that the loss is zero if |ω T x i − y i | ≤ ǫ. Ho and Lin [16], and Gu et al [14] refer to SVR using (7) as L2-loss SVR and ǫ-SVR respectively. We refer to it as ǫ-L2-loss SVR.…”

“…We refer to it as ǫ-L2-loss SVR. One can easily verify that the functions of (5) and (7) are continuously differentiable but not twice differentiable. An illustration of the loss functions is in Figure 1.…”

“…On the other hand, in optimization community, the semismooth Newton method has been well studied, and has been successfully used in many applications, especially in solving modern optimization problems, such as the nearest correlation matrix problem [29,31], the nearest Euclidean distance matrix problem [23], the tensor eigenvalue complementarity problem [7], solving the system of absolute value equations [10], the solution of quasi-variational inequations [11], as well as linear and convex quadratic semidefinite program-ming problems [44]. The concept of semismoothness was introduced by Mifflin [28], and was popularized by Qi and Sun [33].…”

Improved dc programming approaches for solving the quadratic eigenvalue complementarity problem

“…During the past several years, many theoretical results [2,3,4,5], applications [6,7,8] and extensions [9,10,11,12,13,14] of EiCP have been discussed and a number of efficient algorithms have been proposed for the solution of this problem [15,16,17,18,19,20,21,22]. Contrary to the EiCP, QEiCP may have no solution even when the matrix A of the leading λ-term is PD [23].…”

Spectral projected gradient methods for generalized tensor eigenvalue complementarity problems

“…The properties of Pareto eigenvalues and their connection to polynomial optimization are studied in [37]. Recently, as a special type of nonlinear complementarity problems, the tensor complementarity problem is inspiring more and more research in the literature [2,6,8,9,13,15,26,38,39,40]. A shifted projected power method for TEiCP was proposed in [9], in which they need an adaptive shift to force the objective to be (locally) convex to guarantee the convergence of power method.…”