Chemical Engineering Hand Written_Notes_Part_106

Chemical Engineering Hand Written_Notes_Part_106 - 214...

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214 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES [ z ( k ) ] T As ( k ) =[ z (0) ] T As ( k ) + i 1 X j =1 λ k min s ( j ) T As ( k ) (9.33) =[ z (0) ] T As ( k ) (9.34) (9.35) λ k min = h b + Az (0) i T s ( k ) [ s ( k ) ] T As ( k ) which is identical with β j .Thu s z can be expressed as (9.36) z = z (0) + n 1 X i =1 β j s ( j ) = z (0) + n 1 X i =1 λ ( i ) min s ( i ) This implies that z can be reached in n steps or less. Since above equation holds good for any z (0) , the process converges independent of any choice of starting point z (0) and any arbitrary set of conjugate directions. ¤ Now for development of algorithm we assume that the objective function to be quadratic. (9.37) Φ ( z )=0 . 5 z ( T ) Ax + b T z + c De f ning (9.38) g ( z )= Φ ( z )= Az + b we have g ( k ) = g ( z ( k ) )= Az ( k ) + b (9.39) g ( k 1) = g ( z ( k 1) )= Az ( k 1) + b (9.40) 4 g ( z )= g ( k ) g ( k 1) = 4 g ( k ) = A 4 z ( k )
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Chemical Engineering Hand Written_Notes_Part_106 - 214...

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