lecture9

# lecture9 - Problems IE417 Nonlinear Programming Lecture 9...

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lehigh-logo IE417: Nonlinear Programming: Lecture 9 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 14th February 2006 Jeff Linderoth IE417:Lecture 9 lehigh-logo Problems Homework notes... For optimizers, there is no such thing as a positive definite matrix that is non-symmetric ! Hessians are always positive semidefinite x T Ax = 1 / 2 x T ( A + A T ) x , and A + A T is symmetric Jeff Linderoth IE417:Lecture 9 lehigh-logo From the Home Office in Macungie, PA Top Ten Algorithms of the Century 1 Monte Carlo Method 2 Simplex Method 3 Conjugate Gradient Method 4 Matrix Decompositions and Factorizations 5 Optimizing FORTRAN Compiler 6 QR Algorithm 7 Quicksort 8 Fast Fourier Transform 9 Integer Relation Detection Algorithm 10 Fast Multipole Algorithm Jeff Linderoth IE417:Lecture 9 lehigh-logo Today: Conjugate Gradient If A ∈ S n × n ++ , then solving Ax = b is equivalent to min x R n φ ( x ) def = 1 / 2 x T Ax - b T x φ ( x ) = r ( x ) = Ax - b Conjugate Gradient algorithm generates conjugate directions Conjugate Directions Non-zero vectors { d 0 , d 1 , . . . d } are conjugate directions with re- spect to a matrix A if d T i Ad j = 0 i, j Note: A = I orthogonal Jeff Linderoth IE417:Lecture 9

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lehigh-logo Conjugate Vectors are Linearly Independent Lemma If { d 1 , . . . d n } are conjugate with respect to A R n × n , then { d 1 , . . . d n } are linearly independent. Proof. If α k = 0 k is the only solution to α 1 d 1 + . . . α n d n = 0 , then { d 1 , . . . d n } are linearly independent. For each k it is simple to establish that α k = 0 by applying the conjugacy property to the system d k A ( α 1 d 1 + . . . α n d n ) = 0 Jeff Linderoth IE417:Lecture 9 lehigh-logo Conjugate Directions Method Let { d 0 , d 1 , . . . , d
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