This paper modifies the convergence conditions of a ''back-tracking''
global Newton method announced in Goldstein ( 1991 ), making them shar
per and easier to apply. A new version of the Kantorovich inequalities
is presented that is simple to state and prove. An application is mad
e to the centering problem for polytopes. Based on an idea of Ye (1989
), an algorithm is given for the feasibility problem of linear inequal
ities.