lec17_10202006 - 10.34, Numerical Methods Applied to...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 f xx Hessian Matrix = 2 2 2 1 2 2 2 1 2 2 1 2 x f x x f x x f x f H V 2 f in BEERS but second derivative can also mean: f xx Laplacian Tr{H } = 2 2 2 2 1 2 x f x f + - scalar V 2 f in Physics Texts V ·( V f ) Constrained Optimization Equality Constraints: min x f (x ) such that g (x ) = 0 May be able to invert this statement as: x N = G ( x 1 , x 2 , …, x N-1 ) Then we can state min as: min f (x 1 , x 2 , …, x N-1 , G ( x 1 , x 2 , …, x N-1 )) Notice the x N is gone. Constrained becomes unconstrained. Solve with previous methods. Other way to do this: Lagrange Multipliers Unconstrained 0 = mn x n x f at the minimum - constrained problems do not work that way! o BOUNDARIES GET IN THE WAY Constrained: min . min . const x n const x n x g x f = λ V f | const. min = λ V g| x const. min Gradient of f equals 0 in directions parallel to constraint but not perpendicular
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/27/2011 for the course CHEMICAL E 10.302 taught by Professor Clarkcolton during the Fall '04 term at MIT.

Page1 / 3

lec17_10202006 - 10.34, Numerical Methods Applied to...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online