Lecture-5 - Lecture-5 Quadratic Functions Quadratic...

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1 Lecture-5 Quadratic Functions Quadratic Functions x b Qx x x f T T - = 2 1 ) ( b Qx x f - = ) ( f b Qx x if of point stationary a is it then , of solution unique a is * = Q is symmetric, Hessian of f If the linear system Qx=b can not be solved, then function does not have a stationary point, it is unbounded
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2 Quadratic Functions x b Qx x x f T T - = 2 1 ) ( b Qx x f - = ) ( ) ( ) ( ) ( 2 1 ) ( p x b p x Q p x p x f T T a a a a + - + + = + Q is symmetric, Hessian of f According to definition, for any vector x and p: Quadratic Functions Qp p b Qx p x f p x f p b Qp p p Q x Qx p x b Qx x p b x b Qp p p Q x Qx p Qx x p b x b p x Q p Q x p x f T T T T T T T T T T T T T T T T T T 2 2 2 2 1 ) ( ) ( ) ( ) ( 2 1 2 1 ) ( 2 1 ) )( ( 2 1 ) ( a a a a a a a a a a a a a a a + - + = + - + + + - = - - + + + = - - + + = + If x * is stationary point Qp p b Qx p x f p x f T T 2 * * * 2 1 ) ( ) ( ) ( a a a + - + = + Qp p x f p x f T 2 * * 2 1 ) ( ) ( a a + = + ) ( ) ( ) ( 2 1 ) ( p x b p x Q p x p x f T T a a a a + - + + = +
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3 Quadratic Functions Qp p x f p x f T 2 * * 2 1 ) ( ) ( a a + = + j T j j Qu u x f u x f 2 * * 2 1 ) ( ) ( a a + = + j j j u
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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-5 - Lecture-5 Quadratic Functions Quadratic...

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