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

International audienceWe consider nonlinear algebraic systems with complementarity constraints stemming from numerical discretizations of nonlinear complementarity problems. The particularity is that they are non-differentiable, so that classical linearization schemes like the Newton method cannot be applied directly. To approximate the solution of such nonlinear systems, an iterative linearization algorithm like the semismooth Newton-min or an interior-point algorithm can be used. Alternatively, the non-differentiable nonlinearity can be smoothed, which allows a direct application of the Newton method. Corresponding linear systems can be solved only approximately using an iterative linear algebraic solver, leading to inexact approaches. In this work, we design a general framework to systematically steer these different ingredients. We first derive an a posteriori error estimate given by the norm of the considered system's residual. We then, relying on smoothing, design a simple strategy of tightening the smoothing parameter. We finally distinguish the smoothing, linearization, and algebraic error components, which enables us to formulate an adaptive algorithm where the linear and nonlinear solvers are stopped when the corresponding error components do not affect significantly the overall error. Numerical experiments indicate that the proposed algorithm, possibly in combination with the GMRES algebraic solver, ensures important savings in terms of the number of iterations and execution time. It appears rather promising in comparison with the other methods, namely since its performance seems remarkably stable over a range of academic and industrial problems