The Primal-Dual Active Set Strategy as a Semismooth Newton Method

TLDR

Slant differentiability is recalled, and the max‑function is shown to be slantly differentiable in Lp‑spaces using a two‑norm concept. The paper addresses complementarity problems arising from constrained optimal control. The study demonstrates that the primal‑dual active‑set strategy and a specific semismooth Newton method produce identical algorithms, yielding new local convergence results and global unconditional convergence via appropriate merit functions.

Abstract

This paper addresses complementarity problems motivated by constrained optimal control problems. It is shown that the primal-dual active set strategy, which is known to be extremely efficient for this class of problems, and a specific semismooth Newton method lead to identical algorithms. The notion of slant differentiability is recalled and it is argued that the $\max$-function is slantly differentiable in Lp-spaces when appropriately combined with a two-norm concept. This leads to new local convergence results of the primal-dual active set strategy. Global unconditional convergence results are obtained by means of appropriate merit functions.