Supplement to ``A Semismooth Newton Method for Fast, Generic Convex Programming''

The proof relies on the proof of Lemma 3.6, below. Let z, δ ∈ R, and let δ → 0. Suppose z + δ converges to a point that falls into one of the first three cases given in Section 2. Then, from the statement and proof of Lemma 3.6, an element JPK∗exp (z + δ) of the generalized Jacobian of the projection onto the dual of the exponential cone at z + δ, is just a matrix with fixed entries, since projections onto convex sets are continuous. If z + δ converges to a point that falls into the fourth case, then brute force, e.g., using symbolic manipulation software, reveals that an element of the generalized Jacobian (i.e., the inverse of the specific 4x4 matrix D given in (S.6), below) is also a constant matrix, even as z 1 , z ? 2 , ν ? → 0; for completeness, we give D−1 in (S.26), at the end of the supplement. Thus in all the cases, the Jacobian is a constant matrix, which is enough to establish that the limit in (15) exists.