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

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  • Vu, Duc Thach Son
  • Ben Gharbia, Ibtihel
  • Haddou, Mounir
  • Tran, Quang Huy

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. (2011) 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. Due to the nonsmoothness of the latter, semismooth and smoothing methods commonly used for solving such a system are often slow or may not converge at all. This observation led us to design a new strategy called NPIPM (NonParametric Interior-Point Method). Inspired from interior-point methods in optimization, the technique we propose has the advantage of avoiding any parameter management while enjoying theoretical global convergence. This is validated by extensive numerical tests, in which we compare NPIPM to the Newton-min method, the standard reference for almost all reservoir engineers and thermodynamicists.

Suggested Citation

  • Vu, Duc Thach Son & Ben Gharbia, Ibtihel & Haddou, Mounir & Tran, Quang Huy, 2021. "A new approach for solving nonlinear algebraic systems with complementarity conditions. Application to compositional multiphase equilibrium problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1243-1274.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:1243-1274
    DOI: 10.1016/j.matcom.2021.07.015
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    References listed on IDEAS

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    1. Mounir Haddou & Patrick Maheux, 2014. "Smoothing Methods for Nonlinear Complementarity Problems," Journal of Optimization Theory and Applications, Springer, vol. 160(3), pages 711-729, March.
    2. A. Auslender & R. Cominetti & M. Haddou, 1997. "Asymptotic Analysis for Penalty and Barrier Methods in Convex and Linear Programming," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 43-62, February.
    3. Jean-Pierre Dussault & Mathieu Frappier & Jean Charles Gilbert, 2019. "A lower bound on the iterative complexity of the Harker and Pang globalization technique of the Newton-min algorithm for solving the linear complementarity problem," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(4), pages 359-380, December.
    4. Axel Dreves & Francisco Facchinei & Andreas Fischer & Markus Herrich, 2014. "A new error bound result for Generalized Nash Equilibrium Problems and its algorithmic application," Computational Optimization and Applications, Springer, vol. 59(1), pages 63-84, October.
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