Clarke Generalized Jacobian of the Projection onto the Cone of Positive Semidefinite Matrices

Abstract

This paper studies the differentiability properties of the projection onto the cone of positive semidefinite matrices. In particular, the expression of the Clarke generalized Jacobian of the projection at any symmetric matrix is given.

Clarke Generalized Jacobian of the Projection onto Symmetric Cones

“…Our result generalizes corresponding results of [8] and [12] (from second-order cones and positive semi-definite cones respectively) to symmetric cones. Interesting enough, the expression of the Clarke generalized Jacobian of x + is linked to rank-one matrices.…”

“…This is influenced by the recent work [8,9,12] in the special settings of second-order cones and positive semi-definite cones, where Kanzow, Ferenczi and Fukushima [9] gave an expression for the B-subdifferential of the projection onto second-order cones, Hayashi, Yamashita and Fukushima [8] gave an explicit representation for Clarke generalized Jacobian of the projection onto second-order cones; Malick and Sendov [12] worked out the Clarke generalized Jacobian of the projection onto the cone of symmetric positive semi-definite matrices. We generalize the above results to symmetric cones.…”

“…In particular, Hayashi, Yamashita and Fukushima [8] gave an explicit representation for Clarke generalized Jacobian of the projection onto second-order cones. Malick and Sendov [12] studied the differentiability properties of the projection onto cone of positive semi-definite matrices and worked out its Clarke generalized Jacobian. Recently, Sun and Sun [19] showed that the projection onto symmetric cones, which include the nonnegative orthant, the cone of symmetric positive semi-definite matrices and the second-order cone as special cases, is strongly semismooth everywhere.…”

Generalized Hessians of $C^{1,1}$-Functions and Second-Order Viscosity Subjets

“…The natural semidefinite order there is defined as follows: S R if and only if H, SH H, RH for all H ∈ S n , where ·, · is the Frobenius scalar product on S n . We refer to [15] …”

“…In the literature of nonsmooth analysis there are many works on this topic, see for example [8], [11], [16] and references therein. See also [15], [19], [20] for applications to positive semidefinite optimization, as well as [12], [13] for some generalizations. In this setting, a second-order result -analogous to (3)-reads as follows:…”