Optimization TheoryO Professor:

Dr. Sahand DaneshvarO Student’s Name:

Milad Kermani (125512)M.S student of Mechanical Engineering Department

Eastern Mediterranean University, EMU Spring 2013

NONLINEAR PROGRAMMIN

GGolden Section Method

Fibonacci Search

Sequential Search Procedure

O Dichotomous Search

O The Golden Section Method

O The Fibonacci Search

Golden Section MethodO Aim:

Minimizing a strictly quasi-convex * θ function over the interval [ak,bk].

Initialization Step: O Choose an allowable final length of

uncertainty l >0,O [ak,bk] 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=0.618O Evaluate:

O Design a table with below components:K ak bk λk μk θ(λk) θ(μk)

… … … … … … …

Cont.

O According to the value will be obtained for

θ(λk) & θ(μk) 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 λ1 and μ1 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 λ1 is the min one in table.

Cont.

Fibonacci SearchO A line search procedure for

minimizing a strictly quasi-convex function θ over a closed bounded interval.Fibonacci Sequence {Fν} :

Fν+1=Fν+Fν-1

ν=1,2,…F0=F1=1

O {Fν}= 1,1,2,3,5,8,13,21,34,55,89,144,233,…

NoticeO The most prominent points to

remark are the differences in evaluation of λk and μk .

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 uncertaintyε > 0 Distinguished constant[ak,bk] 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.

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