Preprint A130/2002
A New Proximal-Based Globalization Strategy for the Josephy-Newton Method for Variational Inequalities

B. F. Svaiter | Solodov, M. V.

**Keywords: **
Variational inequality | Josephy-Newton method | proximal point method | globalization.

We propose a new approach to globalizing the
Josephy-Newton algorithm for solving the monotone variational
inequality problem.
Known globalization strategies rely either on minimization
of a suitable merit function, or on a projection-type approach.
The technique proposed here is based on a linesearch
in the regularized Josephy-Newton direction which
finds a trial point and a proximal point subproblem
(i.e., subproblem with suitable parameters),
for which this trial point is an acceptable approximate solution.
We emphasize that this requires only checking a certain approximation
criterion, and in particular, does not entail actually solving any
nonlinear proximal point subproblems.
The method converges globally under very mild assumptions. Furthermore,
an easy modification of the method secures the
local superlinear rate of convergence under standard conditions.