Clarke Generalized Jacobian of the Projection onto Symmetric Cones

This paper focuses on Clarke generalized Jacobian of the projection onto symmetric cones. First, we recall some basic concepts. Let ̥ : Ω ⊆ X → Y be a locally Lipschitz function on an open set Ω, where X and Y are two finite dimensional inner product spaces over the field R. Let ∇̥(x) denote the derivative of ̥ at x if ̥ is differentiable at x. The Clarke generalized Jacobian of ̥ at x is defined by ∂̥(x) := conv{∂B̥(x)}, where ∂B̥(x) := {limx̄→x,x̄∈D̥ ∇̥(x̄)} is the B-subdifferential of ̥ at x, and D̥ is the set of points of Ω where ̥ is differentiable. We assume that the reader is familiar with the concepts of (strong) semismoothness, and refer to [4, 5, 14, 15, 16] for details.