One-parametric class of smoothing functions for (WSOCCP) and its Jacobian consistency

Abstract

In this paper, we are interested in the study of weighted second-order cone complementarity problem (WSOCCP) and the smoothing function associated with it. A weighted second-order cone complementarity problem is to find a pair of vectors ( x , y ) that belong to the intersection of a manifold and a second-order cone such that the product of the vectors in a certain algebra, x ∘ y equals a given weight vector w. It is obvious to see that if w is zero, we get a second-order cone complementarity problem (SOCCP). Thus, (WSOCCP) provides the essential framework for a broader class of problems arising from real applications (science and engineering) more conveniently. This paper demonstrates the Jacobian consistency of the one-parametric class of smoothing functions. Furthermore, we provide an estimation of the distance between the subgradient of the one-parametric class of the weighted SOC complementarity functions and the gradient of its associated smoothing function. This estimation facilitates to adjust an appropriate parameter in smoothing methods.