Optimization Methods in Communications Engineering
Explore the principles of nonlinear programming with a focus on the Golden Section Method, an elimination method used in optimization processes. Learn about various techniques like Unrestricted Search, Fibonacci Method, and Quadratic Interpolation. Understand the historical significance of the Golden Ratio in mathematics and architecture.
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Technical Engineering College / Al-Najaf Al-Mustansiriyah University Technical College / Al-Najaf Communications Techniques Eng. Dpt. Communications Techniques Eng. Dpt. College of Engineering Electrical Engineering Department Ph.D. courses /Comm. Eng./2018-2019 Optimization Computer Networks-4thClass-2011/2012 Computer Networks-4thClass-2015/2016 Lect. 3: One Dimension Minimization Methods
Ch. 5 Nonlinear Programming I a)Elimination Methods 1. Unrestricted Search i.Search with fixed step. ii.Search with accelerated step size. 2. Exhaustive Search 3. Dichotomous Search 4. Interval Halving Method. 5. Fibonacci Method 6. Golden Section Method. b)Interpolation Methods 1. Requiring no Derivatives. i. Quadratic Interpolation Method. 2. Requiring Derivatives. i. Cubic Interpolation Method. ii. Direct Root Methods. a.Newton. b.Quasi-Newton. c.Secant.
Technical Engineering College / Al-Najaf Al-Mustansiriyah University College of Engineering Technical College / Al-Najaf Communications Techniques Eng. Dpt. Communications Techniques Eng. Dpt. Electrical Engineering Department PhD courses /Comm. Eng./2018-2019 Computer Networks-4thClass-2011/2012 Computer Networks-4thClass-2015/2016 Main Topics Golden section elimination method
Golden Section Method This method is an elimination method The section to be eliminated in each iteration depends on a ratio that is relative to the well known (golden section ratio) from which the method takes its name. It likes Fibonacci method except that in this method the total number of experiments is not required prior to starting the optimization process. In the golden section method we start with the assumption that we are going to conduct a large number of experiments (n=N where N ), but practically the total number of experiments can be decided during the computation depending on the desired accuracy.
Golden Section Method Fibonacci Golden Section This method is ?? 1 ?? 1 ?? ?2= ?0 ?2= lim ?0 ?? ? n=N where N ?? 2 ?? ?? 2 ?? 1 ?? 1 ?? ?3= lim ?0= lim ?0 =?? 2 ? ? ?2 ?0 ?? ?? ? ?? (? 2)?? 1 ?? 2 ?? 1 ?? 1 ?? = ?? ??=?? (? 1) ?? (? 1) ?? ? 1 ?? 1 ?? ?? ? ?? (? 2)?? 1 ?0 ??= lim ?0 ?? ?? lim ?0 ? ? The same way: ?? ? ?? (? 2)?0 k ? = lim ?? = lim ?? ? ?
c Golden Section Method Now if we define a ratio ? as: ?? ?? 1 ? = lim ? Thus: ? = 1 +1 ?? ?? 1 = 1+?? 2 ?2 ? 1 = 0 ??=?? 1+ ?? 2 ? ?? 1 ? = 1.618
c Golden Section Method ? = 1.618 The ratio ? has a historical background. Ancient Greek architects believed that a building having the sides d and b satisfying the relation. ? + ? ? =? ?= ? This ratio matches what is called the golden mean in Euclid s geometry which sates when the a section is sectored into two unequal parts and the ratio of the whole Section to the larger part is equal to the ratio of the larger to the smaller, the division is called the golden section and the ratio is called the golden mean
c Golden Section Method ? (?? 1 =?0 = lim )2?0 ?2 ?2 ?2= 0.382?0 ?? ?? ? ?? (? 2)?? 1 =?? 1 = lim ?? ?? ?2= 0.382?? 1 ? ?? ?? 1 ? = lim ? ? 1 ? 1 ? = 1.618 ?? 1 ?? 1 ? ??= 0.681? 1?0 ?? lim ?0 ??= ?
c Golden Section Method In the interval ?0?,? Find ?1= ? + ?2 Where ?2 , ?2= ?-?2 ?2= 0.382 ?0 Procedure ?0 = Evaluate f(x) at ?1??? ?2(?1and ?2) Find the new interval ?2 Since f is uni-modal, eliminate one of the three resulting parts Find ??by adding or subtracting ?? to or from one of the ends of the interval ?? 1. =?? 1 ?? ? = 3 ?2 ? = ? + 1 No Evaluate ??and compare it to the other point evaluation value which was left after elimination process in the previous iteration Yes Reach accuracy Find ?? End
c Golden Section Method Example: Solutiongolden sectioin 2.pdf
