Chemical Engineering Hand Written_Notes_Part_103

Chemical Engineering Hand Written_Notes_Part_103 - 208 5...

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208 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES idea in simplex method is to evaluate f ( x ) at ( n + 1) vertices of a simplex and use this information to move towards the optimum. Let x ( m ) be such that (9.4) x ( m ) = max i (1 , .... n + 1) f ( x ( i ) ) and x ( c ) represent centroid of the remaining points de fi ned as (9.5) x ( c ) = 1 n n +1 X i =1 ,i 6 = m x ( i ) Then, we fi nd a new point x ( new ) new such that by moving in direction of x ( c ) x ( m ) as follows (9.6) x ( new ) = x ( m ) + λ ¡ x ( c ) x ( m ) ¢ The choice of λ = 1 x ( new ) = x ( c ) and λ = 0 x ( new ) = x ( m ) . In order to maintain the regularity of simplex, the re fl ection should be symmetric and λ = 2 is selected for achieving symmetry. The property of simplex used by algorithm is that a new simplex- can be generated on any face of the old one by projecting any chosen vertex a suitable distance through the centroid of the remaining vertices of old simplex. The new Simplex is then formed by replacing old vertices with newly generated projected point.
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