A modification of Armijo's step-size rule for negative curvature

F o r the p r o b l e m : rain f ( x ) s u b j e c t to x ~ R ° C E n w h e r e R ° is an open set , A r m i j o in [1] p r o p o s e d the fo l lowing a lgo r i thm in c o n j u n c t i o n wi th C a u c h y ' s m e t h o d of s t e e p e s t de scen t . G iven a po in t Xk C R °, and the g r a d i e n t v e c t o r Vf(Xk), find i (k) , the sma l l e s t in teger f rom i = 0,1 . . . . . such tha t Xk -Vf (Xk) 2 -i E R °, and fEXk -V f(Xk ) 2-il -f (Xk) <--eellV f(xOll2"2 i w h e r e 0 < ~ < 1 is a p r e a s s i g n e d cons tan t . H e s h o w e d tha t a c c u m u l a t i o n po in t s of {Xk} in R ° are s t a t i ona ry po in t s if Vf(Xk) is L i p s c h i t z con t inuous . A d i r ec t ion Sk is sa id to be a n o n a s c e n t d irect ion at Xk if s~ Vf(xk) -< 0. If a t a po in t Xk the n x n H e s s i a n ma t r ix V2f(xk) is not pos i t i ve def ini te , a v e c t o r dk is said to be a direct ion o f nonpos i t i v e curva ture if drk V 2 f ( x k ) & <-O.