Optimization Theory O Professor: Dr. Sahand Daneshvar O Student’s Name: Milad Kermani (125512) M.S student of Mechanical Engineering Department Eastern Mediterranean University, EMU Spring 2013

NONLINEAR PROGRAMMING Golden Section Method Fibonacci Search

Sequential Search Procedure O Dichotomous Search O The Golden Section Method O The Fibonacci Search

Golden Section Method O Aim: Minimizing a strictly quasi-convex * θ function over the interval [a k,b k ].

Initialization Step: O Choose an allowable final length of uncertainty l >0, O [a k,b k ] is the initial interval of uncertainty, O k=1 ( the number of k depends on the points are in the interval). O Calculate:

Cont. O α=0.618 O Evaluate: O Design a table with below components: Kakak bkbk λkλk μkμk θ(λk)θ(λk)θ(μk)θ(μk) …………………

Cont. O According to the value will be obtained for θ(λ k ) & θ(μ k ) have to make decision for next row of table. Follow the processes:

Cont. O Case 1:

Cont. O Case 2:

Example O The length of uncertainty initial interval is 8. (l=8). Reduction this interval of uncertainty is our aim:

Cont. O Evaluate λ 1 and μ 1 and obtain the value of θ for each of these parameters and write down them in the right places of table. Now, the condition of case 2 is happened. Since I want to MINIMIZE the function; thus the θ related to λ 1 is the min one in table.

Cont.

Fibonacci Search O A line search procedure for minimizing a strictly quasi-convex function θ over a closed bounded interval. Fibonacci Sequence {F ν } : F ν+1 =F ν +F ν-1 ν=1,2,… F 0 =F 1 =1 O {F ν }= 1,1,2,3,5,8,13,21,34,55,89,144,233,…

Notice O The most prominent points to remark are the differences in evaluation of λ k and μ k. O The next steps like making a table and other parameters are the same as before. O Just to remind them: L > 0 Allowable final length of uncertainty ε > 0Distinguished constant [a k,b k ]The interval of uncertainty

Initial Steps: O Evaluate: O Evaluation the value of θ for each of λ and μ O Draw a table and follow the previous rules of last table.

Example O The length of uncertainty initial interval is 8. (l=8). Reduction this interval of uncertainty is our aim:

Cont. O n=9 & ε=0.01

Make a table

Thank you for your attention. END