lec18_10232006 - 10.34 Numerical Methods Applied to...

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10.34, Numerical Methods Applied to Chemical Engineering Professor William H. Green Lecture #18: Optimization. Sensitivity Analysis. Introduction: Boundary Value Problems (BVPs). Summary: Optimization with Constraints min x f(x ) such that c m (x ) - s m = 0 s m 0 m = 1 … N inequalities min x f(x ) + ξ (c-s) 2 s m = 0 m > N inequalities penalty method, second term ξ (c-s) 2 is optional KKT conditions: at constrained (local) minimum: Augmented V x f – Σ m ( λ m V x c m ) = 0 Æ Lagrangian c m – s m = 0 (LA) λ m c m = 0 {see book} s m 0 m = 1 … N inequalities s m = 0 equalities = = = m m m m m m m s c s c c f x F s c x λ 0 ) ( N e w t o n Æ SQP If everything is linear: Æ SIMPLEX (i.e. many business problems) g(x ) = 0 Æ x N = G (x 1 , …, x N-1 ) Unconstrained Æ trust region Newton-type BFGS gigantic Æ conjugate gradient In Chemical Engineering, the problems often involve models with differential equations: cost return f(x ) = ( ) i f i o i i x t Y x t Y w ) ; ( ) ; ( knobs what we need what we produce (can adjust) feed composition Need Jacobian of G with respect to Y; need in stiff solver to solve.
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lec18_10232006 - 10.34 Numerical Methods Applied to...

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