lecture6 - Today's Outline IE417 Nonlinear Programming...

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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
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Kantorovich Inequality Theorem Let Q ∈ S n × n + + with eigenvalues λ 1 . . . λ n > 0 , then x R n , ( x T x ) 2 ( x T Qx )( x T Q - 1 x ) 4 λ n λ 1 ( λ n + λ 1 ) 2 Proof.
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