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EE4047_Part1 - x f sin 2 1 2 − = x x g x x x f x g x x x...

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City University of Hong Kong Part 1: Optimization - Problems and Classical Methods Optimization Problems Find the “best” solution from all feasible solutions for a problem Best for a function, cost, … Examples: Dimensions of Antenna Wireless network layout Parameters of controllers and systems Facial reconstruction of a criminal
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Condition of minimization Function with single variable x 0 x m x c f(x) x X * Local and Global minimum A function has a local minimum at the value x 0 if there exists a positive value δ such that if A function f(x) has a global minimum at x * if for all values of x f’(x) = 0 for both cases; f’’(x)> 0 for minimum points ( ) ( ) 0 0 , x f x f x x < δ ( ) ( ) * x f x f N.B.: f’(x) represents the differentiation of the function f w.r.t. x
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Example x x x f sin 2 1 ) ( 2 = Newton’s method O x 1 T x 0 P θ tan θ = g’(x 0 ) TA=PA/ tan θ TA = g(x 0 )/g’(x 0 ) x 1 = x 0 -g(x 0 )/g’(x 0 ) x 2 = x 1 -g(x 1 )/g’(x 1 ) : : X r+1 = x r -g(x r )/g’(x r ) x y=g(x) N.B.: g(x) = f’(x) represents the gradient of the function f x 2 A
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Example Find the minimum of We have x
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Unformatted text preview: x f sin 2 1 ) ( 2 − = x x g x x x f x g x x x f sin 1 ) ( cos ) ( ) ( sin 2 1 ) ( 2 + = ′ − = ′ = − = 0.000000006 0.00009461 0.02710331-0.37758256 g(x r ) 1.673612032 1.673653810 1.685450631 1.479425538 g’(x r ) 0.739085133 0.739085134 4 0.739085134 0.739141666 3 0.739141666 0.755222417 2 0.755222417 0.5 1 x r+1 x r Comments ± Fast and simple ± may not be global ± Information about gradient is needed (differentiable) ± Can be extended to n-variable problem Update rule: where λ i is the value of λ that minimize and ( ) r i r r f x x x ∇ − = + λ 1 ( ) ( ) ( ) r r f f x x ∇ − = φ ( )         ∂ ∂ ∂ ∂ ∂ ∂ = ∇ n x f x f x f f , , , 2 1 L x Random Search ± Random generate sufficient number of candidates and find the optimal one ± Comments: ± A large number of candidates needed ± Commonly miss the best one...
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