Exact formula for the second-order tangent set of the second-order cone complementarity set

TL;DR

This paper derives an exact formula for the second-order tangent set of the nonconvex second-order cone complementarity set, establishing second-order directional differentiability and applying it to optimality conditions.

Contribution

It provides the first explicit formula for the second-order tangent set of the SOC complementarity set, linking it to the second-order directional derivative of the projection operator.

Findings

The SOC complementarity set is second-order directionally differentiable.

An explicit formula for the second-order tangent set of the SOC complementarity set is established.

Second-order necessary optimality conditions are derived for related optimization problems.

Abstract

The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover, unlike the vector complementarity set, the SOC complementarity set is not even the union of finitely many polyhedral convex sets. Despite these difficulties, we succeed in showing that like the vector complementarity set, the SOC complementarity set is second-order directionally differentiable and an exact formula for the second-order tangent set of the SOC complementarity set can be given. We derive these results by establishing the relationship between the second-order tangent set of the SOC complementarity set and the second-order directional derivative of the projection operator over the second-order cone, and calculating the second-order directional derivative of the projection operator over the second-order cone. As an application, we derive second-order necessary optimality conditions for the mathematical program with second-order cone complementarity constraints.