EE4047_Part1 - Part 1 Optimization Problems and Classical...

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1 City University of Hong Kong City University of Hong Kong Part 1: Optimization - Problems and Classical Methods Optimization Problems square6 Find the “best” solution from all feasible solutions for a problem square6 Best for a function, cost, … square6 Examples: square6 Dimensions of Antenna square6 Wireless network layout square6 Parameters of controllers and systems square6 Facial reconstruction of a criminal square6
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2 Condition of minimization square6 Function with single variable x 0 x m x c f(x) x X * Local and Global minimum square6 A function has a local minimum at the value x 0 if there exists a positive value δ such that if square6 A function f(x) has a global minimum at x * if for all values of x square6 f’(x) = 0 for both cases (max or min) ; square6 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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3 Example x x x f sin 2 1 ) ( 2 - = f’ ( x ) 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 g(x) N.B.: g(x) = f’(x) represents the gradient of the function f x 2 A
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4 Example square6 Find the minimum of square6 We have x x x f sin
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