# Modified Line Search Method for Global Optimization

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2 Figure. Example of lie search method for the fuctio f(x)=x 2 cosiderig: (a) p k =(-) k+ ad s k =2+3/2 k+ ; (b) p k =- ad s k =/2 k+ ; (c) p k =- ad s k =3/2 k+ ; (d) p k =- ad s k =5/2 k+ We preferred this way of fidig a adequate value for s k due to the fact that at each iteratio the purpose is to improve the value of the fuctio by optimizig the ewly obtaied poit. Sice sometimes it ca be time cosumig to fid the right value for s k, we applied the radom procedure to geerate aother step util the value of the fuctio i the ewly obtaied poit is improved. This way, we esure that we are movig i a better positio which ca help i fidig the global optimum poit. The modified lie search method (pseudo code) is described below: Begi Geerate N poits i, i=,, N over the search space. k:=; Repeat For i= to N do repeat p k :=radom; if odd(i) the p k :=(-) p k ; s k :=radom; util f( i + p k s k )<f( i ); k:=k+; for all i i := i + p k s k Util coditio Prit the best solutio. Ed The MLS may be ru for a specified umber of iteratios or whe the best solutio is foud. The algorithm may be also termiated if the solutios foud are close to the optimal value with the kow optimal value. I Figure 2, we illustrate how the MLS works for 0 iteratios. 3. Techiques Used for Comparisos The results obtaied by MLS are compared with the results obtaied by lie search ad Geetic Algorithms for all the cosidered test fuctios. The obtaied results are also compared with Differetial Evolutio but oly for two of the cosidered test fuctios [5]. Proceedigs of the First Asia Iteratioal Coferece o Modellig & Simulatio (AMS'07)

4 Figure 3. Covergece toward the optimum solutios of the algorithms MLS, GA ad LS: (a) Sphere test fuctio; (b) Dixo ad Price test fuctio; (c) Ackley test fuctio; (d) Griewak test fuctio. The followig test fuctios were cosidered: Sphere fuctio (f ) 2 x i f(x)= i= Number of dimesios: ; Rage of iitial poits: - 0 xi 0 for i=...; Global miimum: x* = (0, 0,...,0), f(x*) = 0 Dixo ad Price fuctio (f 2 ) f(x)= 2 2 i( 2xi xi ) + ( x + i= Number of dimesios: ; Rage of iitial poits: - ) 0 xi 0 for i=...; Global miimum: z=2 i-, f(x*) = 0 2 x i z z = 2, Proceedigs of the First Asia Iteratioal Coferece o Modellig & Simulatio (AMS'07)

5 Fuctio Algorithm No of dimesios No of iitial poits (for MLS) ad populatio size for GA No of iteratios Optimum foud Actual optimum MLS , f f 2 f 3 f 4 GA , DE , LS , MLS , GA , DE LS , MLS , GA , DE , LS , MLS , GA , DE LS , Table. Parameters used ad results obtaied by the cosidered techiques for all the four test fuctios. Ackley fuctio (f 3 ) x i e i = i = f(x)=20 + e 20 - Number of dimesios: ; Rage of iitial poits: xi 5.2 for i=...; Global miimum: x* = (0, 0,...,0), f(x*) = 0 Griewak fuctio (f 4 ) f(x)= 2 xi 4000 i= i= e xi cos + i cos(2π ) Number of dimesios: ; Rage of iitial poits: - 0 xi 0 for i=...; Global miimum: x* = (0, 0,...,0), f(x*) = 0 x i 3.. Results ad discussios Table depicts the values of the parameters used for each techique ad the results obtaied for the four test fuctios. I Figure 3, the covergece of the test fuctios towards the optimum poit is depicted. Comparisos betwee MLS, GA ad LS are performed. As evidet from Table ad from Figure 3, MLS obtaied the best results for all the test fuctios (except for Dixo ad Price fuctio where the stadard LS performed well). Also, there is a big differece betwee results obtaied by MLS ad the results obtaied by the other techiques used (example: GA ad DE). Proceedigs of the First Asia Iteratioal Coferece o Modellig & Simulatio (AMS'07)

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