Transcription of Steepest Descent Method - PSU
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Steepest Descent Method - PSU
Steepest Descent Method Kefu Liu Properties of Gradient Vector The gradient vector of a scalar function f ( x1 , x2 ,", xn ) is defined as a column vector T. f f f . f = " =c x1 x2 xn . For example f ( x1 , x2 ) = 25 x12 + x22. at the point x1* = .6, x2* = 4. 2(25) x1* 2(25)(.6) 30 . c = f = * = = . 2 x2 2(4) 8 . The normalized gradient vector c c=. cT c For example, at the point x1* = .6, x2* = 4. 1 30 .96625 . c= = . 302 + 82 8 .2577 . Property 1. The gradient vector represents a direction of maximum rate of increase for the function f (x) at x* . For example, f (.6, 4) = 25(.6)2 + 42 = 25. If we increase x in the direction c by a step size of = .5..6 .96625 . x (1) = x(0) + c = + .5 = . 4 .2577 . The function value becomes 1.
[]30 8 1 30 8(3.75) 0 3.75 T =⎡⎤⎢⎥=− ⎣⎦− ct = Property 3.The maximum rate of change of f (x) at any point is the magnitude of the gradient vector given by x* cc= Tc Steepest descent direction.Let f (x) be a differentiable function with respect to .The direction of steepest descent for
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