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Unformatted text preview: IE417: Nonlinear Programming: Lecture 6 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 2nd February 2006 Jeff Linderoth IE417:Lecture 6 Today’s Outline Review Kantorovich CUTEr Jeff Linderoth IE417:Lecture 6 Stuff We Learned Last Time Focusing on Quadratic Functions: f ( x ) = 1 2 x T Qx- b T x Steepest Descent Has a Linear Convergence Rate Steps of Proof: Closed form for step length of exact minimizer: α k = ∇ f ( x k ) T ∇ f ( x k ) ∇ f ( x k ) T Q ∇ f ( x k ) So step is: x k +1 = x k- ∇ f ( x k ) T ∇ f ( x k ) ∇ f ( x k ) T Q ∇ f ( x k ) ∇ f ( x k ) Jeff Linderoth IE417:Lecture 6 More Proof Steps of Proof Q-norm measures suboptimality: 1 2 x- x * Q = f ( x )- f ( x * ) Exercise 3.7 (Plug and Chug): x k +1- x k 2 Q = 1- ( ∇ f ( x k ) T ∇ f ( x k )) 2 ( ∇ f ( x k ) T Q ∇ f ( x k ))( ∇ f ( x k ) T Q- 1 ∇ f ( x k )) Kantorovich Inequality: Jeff Linderoth IE417:Lecture 6 Kantorovich Inequality Theorem Let Q ∈ S n × n +...
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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