Strong Semismoothness of Projection onto Slices of Second‐Order Cone

Second‐order cone (SOC) is a typical subclass of nonpolyhedral symmetric cones and plays a fundamental role in the second‐order cone programming. It is already proven that the metric projection mapping onto SOC is strongly semismooth everywhere. However, whether such property holds for each slice of SOC has not been known yet. In this paper, by virtue of a new property of projection onto the closed and convex set with sufficiently smooth boundary, and some new results about projection onto axis‐weighted SOC, we give an affirmative answer to this problem. Meanwhile, we also show Clarke’s generalized Jacobian and the directional derivative for the projection mapping onto a slice of SOC.