Computing Methods in Optimization Problems. Proceedings of a by A. V. Balakrishnan, Lucien W. Neustadt

By A. V. Balakrishnan, Lucien W. Neustadt

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Extra resources for Computing Methods in Optimization Problems. Proceedings of a Conference Held at University of California, Los Angeles January 30–31, 1964

Example text

4 515. 0 504. 0 494. 2 475. 1 468. 1 461. 0 440. 0 433. 0 426. 0 419. 0 412. 0 405. 0 0 0. 036 0. 116 0. 264 0. 309 0. 342 0. 365 0. 383 0. 412 0. 417 0. 425 0. 429 0 0. 116 0. 265 0. 312 0. 344 0. 368 0. 385 0. 425 0. 429 T h e n a t^ o f 2 s e e s w a s t r i e d and t h i s p r o v e d to b e m u c h m o r e difficult b e c a u s e o f the s t r i n g e n t l i m i t a t i o n in the a v a i l a b l e fast s t o r a g e . T h e f i r s t a t t e m p t w a s with an i s o ­ t h e r m a l p r o f i l e o f 4 5 0 ° C and an e o f 0.

T h u s , the p r o b l e m o f m a x i m i z a t i o n with r e s p e c t t o 2n v a r i a b l e s s i m u l t a n e o u s l y has b e e n r e ­ d u c e d t o that o f η m a x i m i z a t i o n s with r e s p e c t t o t w o p a r a ­ meters. n T h e r e a r e n o w t w o m e t h o d s o f p r o c e e d i n g : o n e i s the t a b u l a r m e t h o d u s e d b y A r i s [ 1 9 6 1 ] and the o t h e r i s an a n ­ a l y t i c a l m e t h o d due t o P i s m e n and Ioffe [ 1 9 6 3 ] . Unfortunately the t a b u l a r m e t h o d , f o r the p r o b l e m d i s c u s s e d h e r e , r e q u i r e s the t a b u l a t i o n o f t w o v a r i a b l e s f o r e a c h t a n k .

FIRST METHOD OF SOLUTION: DISCRETE APPROXIMATION The f i r s t a p p r o a c h w a s t o w r i t e the E q u a t i o n s ( 1 . 3) in finite d i f f e r e n c e f o r m to o b t a i n x 1 lr - x = l,r-l 1 -t r (k lr + k 2r 0 + k_ ) x . 3r lr X 2r~ X 2,r-1 t (k, x . - k . x . ) r lr lr 4r 2r X 3r" X 3,r-1 t (k. 1) 0 It i s a p p a r e n t that t h e s e e q u a t i o n s a r e p h y s i c a l l y e q u i v a l e n t to a p p r o x i m a t i n g the tube b y a s e r i e s o f c o m p l e t e l y s t i r r e d t a n k s ; t h e y r e p r e s e n t a m a s s b a l a n c e o n the r** t a n k .

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