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Chemical Engineering Hand Written_Notes_Part_110

Chemical Engineering Hand Written_Notes_Part_110 - 222 5...

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222 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Φ ( β ) = a + + 2 + 3 d Φ ( β ) /dλ = b + 2 + 3 2 i.e. by solving " β 2 β 3 2 β 3 β 2 # " c d # = " Φ ¡ z ( k ) β s ( k ) ¢ a βb ¡ s ( k ) ¢ T Φ ¡ z ( k ) β s ( k ) ¢ b # The application of necessary condition for optimality yields (9.94) d Φ /dλ = b + 2 + 3 2 = 0 i.e. (9.95) λ = c ± p ( c 2 3 bd ) 3 d One of the two values correspond to the minimum. The su ciency condition for minimum requires (9.96) d 2 Φ /dλ 2 = 2 c + 6 > 0 The fact that d Φ /dλ has opposite sign at λ = 0 and λ = β ensures that the equation 9.94 does not have imaginary roots. Algorithm INITIALIZE: z ( k ) , s ( k ) , h Step 1: Find β β = h WHILE [ d Φ ( β ) /dλ < 0] β = 2 β END WHILE Step 2: Solve for a, b, c and d using z ( k ) , s ( k ) and β Step 3: Find λ using su cient condition for optimality 10. Numerical Methods Based on Optimization Formulation 10.1. Simultaneous Solutions of Linear Algebraic Equations. Con- sider system of linear algebraic equations (10.1) A x = b ; x , b R n where A is a non-singular matrix. De
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