10 34 Numerical Methods Applied to Chemical Engineering Professor William H Green Lecture 17 Constrained Optimization Notation second derivative of f x We normally mean fxx 2 f x 2 H 2 1 f x 2 x1 Hessian Matrix 2 f x1 x 2 2 f x 22 V2f in BEERS but second derivative can also mean 2 f 2 f Tr H scalar x12 x 22 Laplacian fxx V2f in Physics Texts V Vf Constrained Optimization Equality Constraints minx f x such that g x 0 May be able to invert this statement as xN G x1 x2 xN 1 Then we can state min as min f x1 x2 xN 1 G x1 x2 xN 1 Notice the xN is gone Constrained becomes unconstrained Solve with previous methods Other way to do this Lagrange Multipliers f x n Unconstrained 0 at the minimum x mn constrained problems do not work that way o Constrained BOUNDARIES GET IN THE WAY f x n x const min g x n Vf const min Vg x const min x const min Gradient of f equals 0 in directions parallel to constraint but not perpendicular Create a new function L x f x g x is unknown before you do the problem VxL 0 L g x 0 at constrained min at constrained min Second derivatives not necessarily all positive Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY Augmented Lagrangian LA f x g x 1 2 0 g x 2 minx LA given initial guess 0 0 xmin 0 VxLA xmin 0 0 Vf 0 xmin 0 Vg xmin 0 1 0 0 g x Vg x Vf 0 g x min 0 Vg x 1 As 0 shrinks 1 gets large magnifying g x 2 term and thus holding the constraint 2 0 more strictly minx LA using 1 get a new xmin In quantum mechanics corresponds to orbital energies Most of the time does not have a physical meaning 0 is a mathematical trick More Than One Constraint Suppose you have 1 constraints L f x g1 x 0 make sure these are compatible g2 x 0 i e there is a feasible space set g3 x 0 of x that satisfies all constraints igi x i VL 0 Vf iVgi i Inequality Constraints very common min f x s t g x 0 h x 0 Active inequality constraints h xmin 0 Inactive inequality constraints h xmin 0 Usually we do not know whether h s are active or inactive before doing a problem but must leave in during optimization process to allow finding of solution Vf KjVhj K 0 when hj is active also hj 0 and kj 0 Kjhj x const min 0 when hj is inactive 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 17 Page 2 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY if inactive hj 0 and kj 0 Vhj can be anything it does not affect the problem Karash Kahn Tucker KKT conditions L f x igi x kjhj x VL xmin 0 h xmin 0 g xmin 0 Kj 0 Kjhj 0 To handle active inactive constraints add slack variables hj x 0 hj x Sj 0 Sj 0 Augmented Method LA Optimal Sj max hj x k Kj k 0 LA f x igi kjhj k kj k 2 1 2 0 gi2 hj2 0 kj 2 F x VLA 0 Use Newton s Method with Broyden to approximate the Hessian matrix Trying to solve JLA x VLA Use Newton s method to find x Jacobian is messy 2 L x x i j old C n x j old C m x j 0 T x x f x old old S C x old 2L If we want to minpf p 2 p x x i j 1 such that c m x T p Vf x p x old old p j c m x old 0 m 1 N constraints j x old can easily get p same as x above quadratic program Sequential Quadratic Programming SQP In MATLAB fmincon 10 34 Numerical Methods Applied to Chemical Engineering Prof William Green Lecture 17 Page 3 of 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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