On the solution and applications of generalized equations using Newton's method

Ioannis K. Argyros¹

¹ Department of Mathematical Sciences Cameron University Lawton, OK 73505, U.S.A.

Applicationes Mathematicae, Tome 31 (2004) no. 2, pp. 229-242

Résumé

We provide local and semilocal convergence results for Newton's method when used to solve generalized equations. Using Lipschitz as well as center-Lipschitz conditions on the operators involved instead of just Lipschitz conditions we show that our Newton–Kantorovich hypotheses are weaker than earlier sufficient conditions for the convergence of Newton's method. In the semilocal case we provide finer error bounds and a better information on the location of the solution. In the local case we can provide a larger convergence radius. Our results apply to generalized equations involving single as well as multivalued operators, which include variational inequalities, nonlinear complementarity problems and nonsmooth convex minimization problems.

Détail

DOI : 10.4064/am31-2-7

Keywords: provide local semilocal convergence results newtons method solve generalized equations using lipschitz center lipschitz conditions operators involved instead just lipschitz conditions newton kantorovich hypotheses weaker earlier sufficient conditions convergence newtons method semilocal provide finer error bounds better information location solution local provide larger convergence radius results apply generalized equations involving single multivalued operators which include variational inequalities nonlinear complementarity problems nonsmooth convex minimization problems