Superlinearly Convergent Algorithms for Solving Singular Equations and Smooth Reformulations of Complementarity Problems

We propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular. Compared to other techniques for computing singular solutions, a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis. Important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation-based reformulations of the nonlinear complementarity problems. Reformulations of complementarity problems serve both as illustration of and motivation for our approach, and one of them we consider in detail. We show that the proposed method possesses local superlinear/quadratic convergence under reasonable assumptions. We further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent Newton-type algorithms for solving complementarity problems, which are typically based on nonsmooth reformulations. Therefore our approach appears to be an interesting complement to the existing ones.