Preprint A123/2002
Superlinearly Convergent Algorithms for Solving Singular Equations and Smooth Reformulations of Complementarity Problems

M.V. Solodov | Izmailov, A.F.

**Keywords: **
nonlinear equations | singularity | regularity | complementarity | reformulation | superlinear convergence

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 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 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.