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Lecture17

# Lecture17 - Advanced Mathematical Programming IE417 Lecture...

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Advanced Mathematical Programming IE417 Lecture 17 Dr. Ted Ralphs

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IE417 Lecture 17 1 Reading for This Lecture Sections 8.6-8.8
IE417 Lecture 17 2 Conjugate Directions If H R n × n is symmetric, the linearly independent vectors d 1 , . . . , d n are called H-conjugate if d T i Hd j = 0 for i 6 = j . Minimizing the quadratic function f ( x ) = c T x + x T Hx . Given x 1 , any x R n can be represented as x 1 + λ j d j . f ( x ) can be rewritten as a function F of λ . F ( λ ) = c T x 1 + X λ j c T d j + ( x 1 + X λ j d j ) T H ( x 1 + X λ j d j ) = X [ c T ( x 1 + λ j d j ) + ( x 1 + λ j d j ) T H ( x 1 + λ j d j )]

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IE417 Lecture 17 3 Comments on Conjugate Directions The function F is separable so we can minimize over each direction sequentially using line search. Hence, we can minimize any quadratic function in n steps. At the k th step, we end up at the minimum of f over the subspace spanned by d 1 , . . . , d k . Also, f ( x k ) T d j = 0 for j = 1 , . . . , k - 1 .
IE417 Lecture 17 4 Quasi-Newton Methods Davidon-Fletcher-Powell Idea 1 : Use a search direction d j = -

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Lecture17 - Advanced Mathematical Programming IE417 Lecture...

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