Chemical Engineering Hand Written_Notes_Part_109

Chemical Engineering Hand Written_Notes_Part_109 - 220 5....

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220 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES If f(x)is quadratic, it can be shown that D ( k ) converges to A 1 at opti- mum, i.e., D ( n +1) = A 1 The directions generated are A-conjugate There are several variations of this method depending upon choice of β,y, and z. Broyder-Fletcher-Shanno update(BSF). A ( k ) = I x ( k 1) [ I ( k 1) ] T [ x ( k 1) ] T I ( k 1) ¸ A ( k 1) I x ( k 1) [ I ( k 1) ] T [ x ( k 1) ] T I ( k 1) ¸ + x ( k 1) [ x ( k 1) ] T [ x ( k 1) ] T x ( k ) (9.88) 9.5. Leverberg-Marquardt Method. It is known from the expe- rience that the steepest descent method produces large reduction in objec- tive function when z (0) is far away from z , The optimal solution. However, steepest descent method becomes notoriously slow near optimum, on the other hand Newtons method generates ideal search directions near the solution. The Leverberg- Marquardt approach combines as follows. £ H
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Chemical Engineering Hand Written_Notes_Part_109 - 220 5....

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