A new approach for solving nonlinear algebraic systems with complementarity conditions. Application to compositional multiphase equilibrium problems

Abstract

We present a new method to solve general systems of equations containing complementarity conditions, with a special focus on those arising in the thermodynamics of multicomponent multiphase mixtures at equilibrium. Indeed, the unified formulation introduced by Lauser et al. [Adv. Water Res. 34 (2011), 957-966] has recently emerged as a promising way to automatically handle the appearance and disappearance of phases in porous media compositional multiphase flows. From a mathematical viewpoint and after discretization in space and time, this leads to a system consisting of algebraic equations and nonlinear complementarity equations. Such a system exhibit serious convergence difficulties for the existing semismooth and smoothing methods. This observation led us to design a new strategy called NPIPM (Non-Parametric Interior-Point Method ). Inspired from interior-point methods in optimization, the technique we propose avoids any parameter management while ensuring good theoretical convergence results. These are validated by extensive numerical tests, in which we compare NPIPM to the Newton-min method.

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

“…Now we employ a nonparametric interior-point method to problem (1.1). More precisely, we consider the method introduced in [44] where a systematic strategy is used to steer the sequence of smoothing parameters towards zero.…”

“…where θ is a small positive real parameter, chosen once and for all. This equation prevents µ from rushing to zero in just one iteration, and ensures quadratic convergence, see [44]. The unknown of system (4.1) is now the enlarged vector X = (X, µ) T ∈ R n+1 .…”

“…Recall that our goal is to make µ equal to 0 in the limit while ensuring the positivity of the updated iterate. Another choice for the additional equation (4.2) added to system (4.1) was developed and introduced in a recent work, see [45,Section 3]. The proposed equation does not require to truncate the Newton direction, and couples µ and X in a tighter way.…”

“…The entire method is described by the following adaptive algorithm, which drives the smoothing parameter µ j to zero as µ j := αµ j−1 at each smoothing iteration. Other common empirical ways to progressively reduce µ j can be found, e.g., in [45]. The adaptive inexact smoothing Newton algorithm is the following:…”

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

“…It can be combined with a path-following strategy to properly update the regularization parameter, see, e.g., [15,16]. Inspired from the interior-point methods [17,18], another approach is the non-parametric interiorpoint method proposed recently in [19]. 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].…”