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

By Michael Hintermüller, Kazufumi Ito, Karl Kunisch. SIAM Journal on Optimization, 2002

1,122 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/834e994459e536257f494073316b8860c08bda24

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Citations in the corpus

A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach (2015, methodology): “Here, in contrast to the work originally proposed in [15], the constraint relates two time states (within a time-stepping scheme) rather than spatial constraints.”
A dual mortar approach for 3D finite deformation contact with consistent linearization (2010, methodology): “Yet, it is well known from the mathematical literature on constrained optimization [21, 22] and from applications to contact problems [19, 32] that the max-function is semi-smooth and therefore a Newton method can still be applied.”
Semi-smooth Newton methods for the Signorini problem (2008, methodology): “[6], [7], [8], [9], we shall show that superlinear methods for solving (Sig) can be developed.”
A finite deformation mortar contact formulation using a primal–dual active set strategy (2009, positive): “Following an approach given in [22, 24] we define a so-called complementarity function C j for each slave node j ∈S as C j (z j ,d)=(zn) j −max(0, (zn) j −cng̃ j ), cn>0 (56) This is a nonlinear function of the displacements as both the nodal normal n j in (zn) j =n j ·z j and the nodal weighted gap”
ON REGULARIZATION METHODS FOR THE NUMERICAL SOLUTION OF PARABOLIC CONTROL PROBLEMS WITH POINTWISE STATE CONSTRAINTS (2009, methodology): “Based on the experiences with semismooth Newton methods in [13,30] for elliptic distributed and boundary control problems, and the results in [12], where the active set strategy was interpreted as a semismooth Newton method, we present an active set strategy that can also be understood as a semismoo”
Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model (2016, background): “Keywords: Phase Field; Fluid Filled Fracture; Adaptive Finite Elements; Porous Media; PrimalDual Active Set”
Phase-field modeling of a fluid-driven fracture in a poroelastic medium (2015, methodology): “Here, B is a mass matrix (details in [14]), R(Uh k ) the nonlinear residual from before, c > 0 a constant, and dUk h Newton’s update.”
Simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized-α scheme (2014, methodology): “At each time step, the problem is solved using a semi-smooth Newton process, which can also be interpreted as an active set method as discussed in [14, 15].”
Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control (2013, methodology): “Since this technique is equivalent to a semismooth Newton procedure [24] locally superlinear convergence rates are provided.”
NONSMOOTH ANALYSIS AND OPTIMIZATION (2017, background): “…Hölder’s inequality to create a term involving the Lebesgue measure of the support of the set where the “wrong” linearization is taken (i.e., where max(u(x) +h(x)) 6= max(u(x)) +G(u(x) +h(x))h(x)), which can be shown to go to zero as h→ 0; see [Hintermüller, Ito, and Kunisch 2002, Proposition 4.1].”