By Krzysztof C. Kiwiel
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Extra resources for Methods of Descent for Nondifferentiable Optimization
Example text
For nonsmooth and gf(xk)=l mentioned Two examples algorithms above, recently much rese- that do not maintain feasibility of such methods developed m a i n l y and are given below. 7) {t k} Z tk=+~, k=l satisfying tk+i/t k --+ i. Owing to their simplicity, stopping the subgradient criteria for terminating sufficiently is smooth and Therefore are customary timization. require converges More advanced programming, implementations tuning of certain zes, during the calculations chal, 1982; Shor, f are useless is nondifferenin nonsmooth of the subgradient 1982; Nurmi~ski, op- algorithms which regulate but, when properly algorithms when f of f, then Igf(xk)] ~ 10 -6 , which tolerances, (Lemarechal, tuning requires much experimentation, the subgradient x this need not occur when in m a t h e m a t i c a l current iterate For instance, to a m i n i m u m point stopping criteria of the form interactive problem, algorithms have no reliable the iteration when the close to the required solution.
In Section They can be they d i f f e r of the past that will be useful In Section combine of l i n e a r i z a t i o n s case. computed that p r e s e r v e s is d e m o n s t r a t e d concepts when which subproblems. the m e t h o d s of P s h e n i c h n y of an a l g o r i t h m i c of points; terminate approximation. (1978), finding combination which assumption to a m i n i m u m class method to the n o n s m o o t h all p r e v i o u s l y the a l g o r i t h m s ce certain quadratic of L e m a r e c h a l in a way a class and effi- sequence converges algorithm of P s h e n i c h n y ' s subproblems, is a convex no a d d i t i o n a l a minimizing sequence introduce in c o m p l e x i t y are d e s c e n t m e t h o d s direction of using We a convex, (Kiwiel,1983).
Each their its m i n i m u m of f. P a r t i c u l a r f happens neralized cutting then members presented plane (Pshenichny and Danilin,1975) of search cally, instead dratic programming dient, which recursively updated subgradient information In S e c t i o n them with cription bal 2 we derive convergence extensions. a ge- is e s s e n t i a l basic versions procedure 5 we study Stemming More In this convex methods of the A l g o r i t h m section function construct previously we derive f: RN÷R.