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Optimization Theory

Optimization Theory. Professor: Dr. Sahand Daneshvar 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.

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Optimization Theory

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  1. Optimization Theory • Professor:Dr. SahandDaneshvar • Student’s Name: MiladKermani (125512)M.S student of Mechanical Engineering Department Eastern Mediterranean University, EMU Spring 2013

  2. NONLINEAR PROGRAMMING Golden Section Method Fibonacci Search

  3. Sequential Search Procedure • Dichotomous Search • The Golden Section Method • The Fibonacci Search

  4. Golden Section Method • Aim: Minimizing a strictly quasi-convex * θfunction over the interval [ak,bk].

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

  6. Cont. • α=0.618 • Evaluate: • Design a table with below components:

  7. Cont. • According to the value will be obtained for θ(λk) & θ(μk) have to make decision for next row of table. Follow the processes:

  8. Cont. • Case 1:

  9. Cont. • Case 2:

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

  11. Cont. • Evaluate λ1and μ1and 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 θrelated to λ1 is the min one in table.

  12. Cont.

  13. Fibonacci Search • A line search procedure for minimizing a strictly quasi-convex function θ over a closed bounded interval.Fibonacci Sequence {Fν} : Fν+1=Fν+Fν-1ν=1,2,… F0=F1=1 • {Fν}= 1,1,2,3,5,8,13,21,34,55,89,144,233,…

  14. Notice • The most prominent points to remark are the differences in evaluation of λkand μk . • The next steps like making a table and other parameters are the same as before. • Just to remind them:L > 0 Allowable final length of uncertaintyε > 0 Distinguished constant[ak,bk] The interval of uncertainty

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

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

  17. Cont. • n=9 & ε=0.01

  18. Make a table

  19. Thank you for your attention. END

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