Multiple-Choice Test Chapter Golden Section Search Method Optimization COMPLETE SOLUTION SET

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1 Mltiple-Choice Test Chapter 09.0 Golden Section Search Method Optimization COMPLETE SOLUTION SET. Which o the ollowing statements is incorrect regarding the Eqal Interval Search and Golden Section Search methods? (A) Both methods reqire an initial bondary region to start the search (B) The nmber o iterations in both methods are aected by the size o ε (C) Everything else being eqal, the Golden Section Search method shold ind an optimal soltion aster. (D) Everything else being eqal, the Eqal Interval Search method shold ind an optimal soltion aster. Soltion The correct answer is (D). De to the manner in which the intermediate points in the Golden Section Search method are determined, the initial search region size is redced mch qicker than the Eqal Interval Search method and hence converges to an optimal soltion aster.

2 . Which o the ollowing parameters is not reqired to se the Golden Section Search method or optimization? (A) The lower bond or the search region (B) The pper bond or the search region (C) The golden ratio (D) The nction to be optimized Soltion The correct answer is (C). The Golden Section Search method is an optimization algorithm that reqires search bondaries (lower and pper) and a one-dimensional nction to be optimized. The Golden Ratio is simply the ratio o the distance between the intermediary points to the search bondary.

3 . When applying the Golden Section Search method to a nction () to ind its maimm, the ( ) > ( ) condition holds tre or the intermediate points and.which o the ollowing statements is incorrect? (A) The new search region is determined by, ] [ (B) The Intermediate point stays as one o the intermediate points (C) The pper bond stays the same (D) The new search region is determined by, ] Soltion The correct answer is (D). [ l I ( ) > ( ), then the new l,, and are determined as ollows: = l = = 5 = l + ( l ) Thereore, the statement The new search region is determined by, ] is incorrect. [ l

4 . In the graph below, the lower and pper bondary o the search is given by and respectively. I and are the initial intermediary points, which o the ollowing statement is alse? ( ) ( ) ( ) ( ) ( ) (A) The distance between and is eqal to the distance between and (B) The distance between and is approimately 0.68 times the distance between and (C) The distance between and is approimately 0.68 times the distance between and (D) The distance between and is eqal to the distance between and Soltion The correct answer is (B). Reerring to Figre 6 in the chapter (also shown below), we can see that choices A, C and D are tre based on the Golden Ratio, however no sch assertion can be made abot choice B.

5 ( ) ( ) ( ) ( ) b a ( ) a b Figre 6.

6 5. Using the Golden Section Search method, ind two nmbers whose sm is 90 and their prodct is as large as possible. Condct two iterations on the interval [0,90]. (A) 0 and 60 (B) 5 and 5 (C) 8 and 5 (D) 0 and 70 Soltion The correct answer is (C). To model this problem we mst recognize that the two nmbers and y are related to each other as + y = 90 and the nction to be maimized is (, y) = y. We can model the problem as a one-dimensional optimization problem by sbstitting the vale o y in terms o as ( ) = (90 ) = 90 Iteration : Using [0, 90] as the search bondaries 5 = l + ( l ) 5 = 0 + (90) = = ( l ) 5 = 90 (90) =.769 The nction is evalated at the intermediate points as ( 55.6) = and (.769) = This is an interesting case where ( ) = ( ), thereore we can proceed either way. Assme we eliminate the region to the right o and pdate the pper bondary point as =. The lower bondary point l remains nchanged. The irst intermediate point is pdated to assme the vale o and inally the second intermediate point is re-calclated as ollows:

7 5 = ( l ) 5 = 55.6 (55.6 0) =.6 Iteration : The process is repeated in the second iteration with the new vales or the bondary and intermediate points calclated in the previos iteration as shown below. = l = 55.6 =.77 =.6 Again the nction is evalated at the intermediate points as (.77) = 9. and (.6) = Since ( ) > ( ), we eliminate the region to the let o and pdate the lower bondary point as l =. The pper bondary point remains nchanged. At the end o the second iteration the soltion is or is + l = = 8.5 The vale or y is calclated by sbtracting it rom 90 as y = = The iterations will contine ntil the stopping criterion is met when =. 99. Smmary reslts o all the iterations are shown below assming ε<0.05. The theoretical optimal soltion or this problem is when the nmbers are eqal to each other at 5.

8 Iteration l () () ε

9 6. Consider the problem o inding the minimm o the nction shown below. Given the intermediate points in the drawing, what wold be the search region in the net iteration? ( ) ( ) l ( ) ( ) ( ) l (A), ] [ (B), ] [ [ l, [ l, (C) ] (D) ] Soltion The correct answer is (C). As seen in the drawing the optimal soltion is between [ l, ]. Either the region to the right o or the region to the let o will be eliminated. De to location o the optimal soltion we eliminate, ] leaving the new search region as l, ]. [ [