It is well known that complementarity functions play an important role in dealing with complementarity problems. In this paper, we propose a few new classes of complementarity functions for nonlinear complementarity problems and second-order cone complementarity problems. The constructions of such new complementarity functions are based on discrete generalization which is a novel idea in contrast to the continuous generalization of FischerBurmeister function. Surprisingly, these new families of complementarity functions possess continuous differentiability even though they are discrete-oriented extensions. This feature enables that some methods like derivative-free algorithm can be employed directly for solving nonlinear complementarity problems and second-order cone complementarity problems. This is a new discovery to the literature and we believe that such new complementarity functions

We make a unified analysis of interior proximal methods of solving convex second-order cone programming problems. These methods use a proximal distance with respect to second-order cones which can be produced with an appropriate closed proper univariate function in three ways. Under some mild conditions, the sequence generated is bounded with each limit point being a solution, and global rates of convergence estimates are obtained in terms of objective values. A class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution. These results are illustrated with some examples. In addition, we also study the central paths associated with these distance-like functions, and for the linear SOCP we discuss their relations with the sequence generated by the interior proximal methods. From this, we obtain improved convergence

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x H, a spectral decomposition is introduced, and for any function f: R R, we define a corresponding vector-valued function f H (x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.

The system of absolute value equation, denoted by AVE, is a non-differentiable NP-hard problem. Many approaches have been proposed during the past decade and most of them focus on reformulating it as complementarity problem and then solve it accordingly. Another approach is to recast the AVE as a system of nonsmooth equations and then tackle with the nonsmooth equations. In this paper, we follow this path. In particular, we rewrite it as a system of smooth equations and propose four new smoothing functions along with a smoothing-type algorithm to solve the system of equations. The main contribution of this paper focuses on numerical comparisons which suggest a better choice of smoothing function along with the smoothing-type algorithm.

We introduce an entropy-like proximal algorithm for the problem of minimizing a closed proper convex function subject to symmetric cone constraints. The algorithm is based on a distance-like function that is an extension of the Kullback-Leiber relative entropy to the setting of symmetric cones. Like the proximal algorithms for convex programming with nonnegative orthant cone constraints, we show that, under some mild assumptions, the sequence generated by the proposed algorithm is bounded and every accumulation point is a solution of the considered problem. In addition, we also present a dual application of the proposed algorithm to the symmetric cone linear program, leading to a multiplier method which is shown to possess similar properties as the exponential multiplier method (Tseng and Bertsekas in Math. Program. 60:119, 1993) holds.