Lecture-9 - Lecture-9 Conjugate Direction Algorithm...

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1 Lecture-9 Conjugate Direction Algorithm (Solution of Linear System or Minimization of A Quadratic Function) Conjugate Gradient • Linear conjugate gradient: for solving linear systems Ax= b with PD matrix, A. – Exact solution in n steps (Hestenes & Stiefel, 1950s) – Approximate solution in fewer than n steps • Non-linear conjugate gradient: for solving large- scale non-linear optimization problems. – Fletcher and Reeves, 1964 – Polk-Ribiere, 1969
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2 Conjugate Gradient Or minimize the following function: b Ax = A is symmetric PD. (1) x b Ax x x T T - = 2 1 ) ( f (2) ) ( ) ( x r b x = - = f r ( x ) is the residual j i 0 2200 = j T i Ap p { } 1 1 0 , , , - = n p p p S K The set S is conjugate wrt A if Linear Independence 0 then 0 if 1 2 1 0 1 1 1 1 0 0 = = = = = + + - - - n n n p p p s s s s s s s K K { } 1 1 0 , , , - = n p p p S K S is linearly independent Conjugate set is also linearly independent. j i 0 2200 = j T i p
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3 Conjugate Direction Method k k k k p x x a + = + 1 j i 0 2200 = j T i Ap p Line search 1D minimizer of a quadratic function x b Ax x x T T - = 2 1 ) ( f k T k k T k k Ap p p f a - = Convergence Rate of Steepest Descent ( 29 ( 29 ( 29 ( 29 ( 29 ) ( 0 0 0 ) 2 1 ( k T k k T k k k k T T k k T k k T k T k k T k T k k T k k T k T k k T k k T k T k k k k T k k T k k k k f Q f f f Qg g g b Q x Qg g g b Qg x g b Qg x Qg g g b Qg g Qg x g b Qg g x g x b g x Q g x d d g x f d d = - = - = - = = + + - = + - - = = - - - - = - a a a a a a a a a a a a k k T k k T k k k f f Q f f f x x
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This note was uploaded on 06/12/2011 for the course COT 6505 taught by Professor Shah during the Spring '07 term at University of Central Florida.

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Lecture-9 - Lecture-9 Conjugate Direction Algorithm...

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