Semismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution

Unified Flash Calculations with Isenthalpic and Isochoric Constraints

“…Smoothing techniques for the nonsmooth complementarity conditions are also frequently applied. Recent discussions on solution strategies for this non-linear system are found in [17,18].…”

Adaptive inexact smoothing Newton method for a nonconforming discretization of a variational inequality

“…when the smoothing estimator is ζ sm -times smaller than the discretization estimator. As for the amount of smoothing, we will proceed following [26] and diminish it by a fixed factor ζ ∈]0, 1[ on each smoothing step. We are now ready to present our adaptive inexact smoothing Newton algorithm that includes the above adaptive criteria for stopping the iterative solvers.…”

“…For an enlightening summary of numerical methods solving problem (1.1), we refer to the books of Ferris et al [20], Facchinei and Pang [21,22], Bonnans et al [23], Ito and Kunisch [24], and Ulbrich [25]. Recently, we have proposed in [26] an adaptive smoothing Newton method for the resolution of nonlinear discrete problems in the form (1.1).…”

“…In this work, we consider a system of PDEs with complementarity constraints in an infinite-dimensional framework. Our goal is to estimate the overall error between the unknown PDE solution and a numerical approximation at each resolution step in an adaptive algorithm inspired from [26]. The guiding principle of the considered approach, following [26], is to approximate the complementarity constraints in (1.1b) by a system of smooth (differentiable) nonlinear equations C µ (X) = 0, where C µ : R n → R m is a smooth (differentiable) approximation of a nondifferentiable complementarity function (C-function) C : R n → R m with a parameter µ > 0.…”

“…Our goal is to estimate the overall error between the unknown PDE solution and a numerical approximation at each resolution step in an adaptive algorithm inspired from [26]. The guiding principle of the considered approach, following [26], is to approximate the complementarity constraints in (1.1b) by a system of smooth (differentiable) nonlinear equations C µ (X) = 0, where C µ : R n → R m is a smooth (differentiable) approximation of a nondifferentiable complementarity function (C-function) C : R n → R m with a parameter µ > 0. This reformulation brings us to approximate problem (1.1) at each smoothing step j ≥ 1, with parameter µ j > 0, by finding a vector X j ∈ R n such that…”