An efficient adaptive trust-region method for systems of nonlinear equations

Abstract

We present an adaptive trust-region algorithm to solve systems of nonlinear equations. Using the nonmonotone technique of Grippo, Lampariello and Lucidi, we introduce a new adaptive radius to decrease the total number of iterations and function evaluations. In contrast with the pervious methods, the new adaptive radius ensures that the size of radius is not too large or too small. We show that the sequence generated by the proposed adaptive radius is decreasing, so it prevents the production of too large radius as possible. Furthermore, it is shown that this sequence is reduced slowly, so it prevents the production of the intensely small radius. The global and quadratic convergence of the proposed approach are proved. Preliminary numerical results of our algorithm are also reported which indicate the promising behaviour of the new procedure to solve systems of nonlinear equations.

Keywords:

• nonlinear equations
• trust-region framework
• nonmonotone technique
• adaptive radius
• convergence theory

2010 AMS Subject Classifications::

• 90C30
• 93E24
• 34A34