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Unformatted text preview: lehigh-logo IE417: Nonlinear Programming: Lecture 10 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th February 2006 Jeff Linderoth IE417:Lecture 10 lehigh-logo Last Time: Conjugate Gradient Solving Ax = b Or min ( x ) def = 1 / 2 x T Ax- b T x, with A S n n ++ Conjugate Gradient Algorithm 1 Choose x .r = Ax- b, d =- d , k = 0 2 k =- r T k d k d T k Ad k 3 x k +1 = x k + k d k 4 k +1 = r T k +1 Ad k d T k Ad k 5 d k +1 =- r k +1 + T k +1 d k 6 If r k = 0 , stop. Else k = k + 1 , Go to 2. Jeff Linderoth IE417:Lecture 10 lehigh-logo CG for Unconstrained Optimization min x R n f ( x ) Fletcher-Reeves Conjugate Gradient 1 Given x . d =- f ( x ) , k = 0 2 Compute k . x k +1 = x k + k d k 3 Compute f ( x k +1 ) , If f ( x k +1 ) = 0 , stop . Else k +1 = f ( x k +1 ) T f ( x k +1 ) f ( x k ) T f ( x k ) 4 Compute new direction: d k +1 =- f ( x k +1 ) + k +1 d k Fletcher Reeves method is globally convergence as long as your k satisfies the strong Wolfe conditions (with c 2 < 1 / 2 ) Jeff Linderoth IE417:Lecture 10 lehigh-logo Today: Quasi-Newton Recall: m k ( d ) = f ( x k ) + f ( x k ) T d k + 1 / 2 d T B k d Minimizer of this quadratic function is d k =- B- 1 k f ( x k ) Step: x k +1 = x k + k d k The Question What can B do for you? Given the gradient information that you have recently seen, how would you like your model to behave?...
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This note was uploaded on 02/29/2008 for the course IE 417 taught by Professor Linderoth during the Spring '08 term at Lehigh University .
- Spring '08
- Systems Engineering