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

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 L p-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.

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

“…Since only finitely many constellations with active sets are possible we can deduce that G −1 is uniformly bounded in this neighborhood. Hence convergence results in [16,27] now give the local convergence result.…”

“…We apply the algorithm to a finite element discretization of an implicit Eulerdiscretization of (P m ). Using that the primal-dual active set method can be reformulated as a semi-smooth Newton method [27] we show local convergence of our algorithm. Finally, in Section 5 we present numerical simulations for the non-local as well as for the local Allen-Cahn variational inequality with three and more phases.…”

“…The mapping y → max(0, y) from R to R is slantly differentiable and a possible slanting function is given by G : R → R with G(y) = 1 for y > 0 and G(y) = 0 for y ≤ 0, see [27]. Hence for the above function H one derives the slanting function…”

“…For the numerical approximation of solutions u of (P m ) we introduce a primaldual active set method or equivalently a semi-smooth Newton method [7,27]. Both are well known in the context of optimization with partial differential equations as constraints.…”

A primal-dual active-set algorithm for bilaterally constrained total variation deblurring and piecewise constant Mumford-Shah segmentation problems

“…The function F 3 is strongly semi-smooth and the derivatives ∂F 2 /∂u and ∂F 3 /∂λ are defined in the sense of slant differentiability which is a generalized derivative, see [12]. We can solve the above system directly to obtain the updates.…”