Some recent advances in projection-type methods for variational inequalities
Journal of Computational and Applied Mathematics - Proceedings of the international conference on recent advances in computational mathematics
A superlinearly convergent projection method for constrained systems of nonlinear equations
Journal of Global Optimization
Monotonicity of NFP mappings associated with variational
AMERICAN-MATH'12/CEA'12 Proceedings of the 6th WSEAS international conference on Computer Engineering and Applications, and Proceedings of the 2012 American conference on Applied Mathematics
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The Josephy--Newton method for solving a nonlinear complementarity problem consists of solving, possibly inexactly, a sequence of linear complementarity problems. Under appropriate regularity assumptions, this method is known to be locally (superlinearly) convergent. To enlarge the domain of convergence of the Newton method, some globalization strategy based on a chosen merit function is typically used. However, to ensure global convergence to a solution, some additional restrictive assumptions are needed. These assumptions imply boundedness of level sets of the merit function and often even (global) uniqueness of the solution. We present a new globalization strategy for monotone problems which is not based on any merit function. Our linesearch procedure utilizes the regularized Newton direction and the monotonicity structure of the problem to force global convergence by means of a (computationally explicit) projection step which reduces the distance to the solution set of the problem. The resulting algorithm is truly globally convergent in the sense that the subproblems are always solvable, and the whole sequence of iterates converges to a solution of the problem without any regularity assumptions. In fact, the solution set can even be unbounded. Each iteration of the new method has the same order of computational cost as an iteration of the damped Newton method. Under natural assumptions, the local superlinear rate of convergence is also achieved.