Chemical Engineering Hand Written_Notes_Part_83

# Chemical Engineering Hand Written_Notes_Part_83 - 168 5...

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168 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Thus, above equation reduces to F ( z + z ) F ( z ) = 1 2! N X i =1 N X j =1 2 F ( z + λ z ) ∂z i ∂z j z i z j (2.15) (0 < λ < 1) This implies that sign of F ( a + z ) F ( a ) at extreme point z is same as sign of R.H.S. Since the 2’nd partial derivative 2 F ∂z i ∂z j ¸ is continuous in the neighborhood of z = z , its value at z = z + λ z will have same sign as its value at z = z for all su ciently small z . If the quantity (2.16) N X i =1 N X j =1 2 F ( z + λ z ) ∂z i ∂z j z i z j ' ( z ) T [ 2 F ( z )] z 0 for all z , then z = z will be a local minimum. In other words, if Hessian ma- trix [ 2 F ( z )] positive semi-de fi nite, then z = z will be a local minimum. If the quantity (2.17) N X i =1 N X j =1 2 F ( z + λ z ) ∂z i ∂z j z i z j ' ( z ) T [ 2 F ( z )] z 0 for all z , then z = z will be a local maximum. In other words, if Hessian matrix [ 2 F ( z )]

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