Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem

Wenli Liu,Ranran Cui,Xiaoni Chi,Qili Yang,Guoqiang Wang

doi:10.1155/2021/6674520

Abstract

In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

Highlights

A weighted second-order cone (SOC) complementarity function and its smoothing function are presented. en, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency
Introduction e weighted second-order cone complementarity problem (WSOCCP) is, for a given weight vector w ∈ K and a continuously differentiable function F: Rn × Rn × Rm ⟶ Rn+m, to find vectors (x, s, y) ∈ Rn × Rn × Rm such that x∘s F(x, s, w, y) where ∘ represents the Jordan product and K is the Cartesian product of second-order cone, that is, K Kn1 ×
Where μ ∈ R is a smoothing parameter. e main contribution of this paper is to show the Jacobian consistency of the smoothing function (7) and estimate the distance between the subgradient of the weighted SOC complementarity function (5) and the gradient of its smoothing function (7). ese properties will be critical to solve weighted SOC complementarity problems by smoothing methods

Summary

Preliminaries

We briefly recall some definitions and results about the Euclidean Jordan algebra [11] associated with the SOC Kn and subdifferentials [12]. Given an element x (x0; x1) ∈ R × Rn− 1, we define the symmetric matrix. It is easy to verify that x ∘ s L(x)s for any s ∈ Rn. L(x) is positive definite (and invertible) if and only if x ∈ intKn. For each x (x0; x1) ∈ R × Rn− 1, let λ1, λ2 and u(1), u(2) be the spectral values and the associated spectral vectors of x, given by λi x0 +(− 1)i x1 , u(i). ConvzBG(z), where DG denotes the set of points at which G is differentiable. ZG(z) 􏼈G′(z)􏼉 if G is continuously differentiable at z

Smoothing Function

Jacobian Consistency

Conclusions