Chapter Multidimensional Direct Search Method

Transcription

1 Chapter Mutidimensiona Direct Search Method After reading this chapter, you shoud be abe to:. Understand the fundamentas of the mutidimensiona direct search methods. Understand how the coordinate cycing search method works 3. Sove muti-dimensiona optimization probems using the coordinate cycing search method Optimization Techniques Methods for finding optima soutions in mutidimensiona spaces are not too different than their cousins used in finding optima soutions in a singe dimension. The trade-off between genera appicabiity versus computationa compeity aso eists in mutidimensiona optimization. The mutidimensiona direct search methods we wi cover in this chapter, ike the one-dimensiona Goden Section Search method ( does not require a differentiabe function. These methods are sometimes referred to as Zeroth Order Agorithms because it is not required to differentiate the optimization function. Probaby the most obvious soution to an optimization probem in mutidimensiona space is to systematicay evauate every possibe soution and seect the maimum or the minimum depending on our objective. This is a very generay appicabe approach and may even be usefu if the soution space is reativey sma. However, as the dimensions of the probem space, (number of independent variabes), increase, the computationa compeity of this soution approach quicky becomes unmanageabe. Therefore, we are interested in methods that inteigenty search through the soution space to find an optima soution without enumerating a possibe soutions. It is important to note that some of the popuar optimization techniques you may have heard of such as simuated anneaing, tabu search, neura networks and genetic agorithms a fa under this famiy of optimization techniques. What is the Coordinate Cycing Search Method and How Does it Work? The coordinate cycing search method, starts from an initia point and ooks for an optima soution aong each coordinate direction iterativey. For eampe, using a function f (, y) with two independent variabes and y, and starting at point ( 0, y0) ; the first iteration wi move aong direction (, 0), unti an optima soution is found for the function f (, y0 ). The net search invoves searching aong the direction (0,) to determine the optima vaue for the function f (, y) where is the soution found in the previous search. Once searches in a directions are competed, the process is repeated in the net cyce. The search wi

2 Chapter continue unti convergence occurs or a predetermined error imit is met. The search aong each coordinate direction can be conducted by using anyone of the one-dimensiona search techniques previousy covered. A visua representation of how the search converges is shown beow in Figure. Optima point third cyce second cyce Initia point search first cyce ength Figure Visua Representation of a Mutidimensiona Search Eampe Consider Figure beow. The cross-sectiona area A of a gutter with a base ength b and an edge ength of is given by A = ( b + b + cos ) sin Assuming that the width of the materia to be bent into the gutter shape is 6 inches, find the ange and edge ength which maimizes the cross-sectiona area of the gutter.

3 Mutidimensiona Direct Search Method b Figure Cross section of the gutter Soution Recognizing that the base ength b can be epressed as b = 6, we can re-write the area function to be optimized in terms of two independent variabes giving f (, ) = (6 + cos ) sin. Let us consider an initia point (0, π ). We wi use the Goden Section Search method to 6 determine the optima soution aong direction (,0) namey the independent variabe corresponding to the ength of each side. To use the Goden Section Search method, we wi use 0 and 3 as the ower and upper bounds, respectivey for the search region (Can you determine why we are using 3 as the upper bound?) and ook for the optima soution of the function f (,0.5360) with a convergence imit of ε < Tabe beow shows the iterations of the Goden Section Search method in the (,0) direction. The maimum area of in is obtained at point (.6459,0.5360). Tabe Summary of the Goden Section Search iterations aong direction (,0) for Eampe. Here = and f ( i ) = ( 6 + cos( ) sin( ) ) Iteration u f ( ) f (

4 Chapter To search aong the (0,) direction corresponding to the ange, we again use the Goden Section Search method, but in this case using the function f (.6459, ). Tabe beow shows the iterations of the Goden Section Search method in the (0,) direction. Note that at the new optima point (.6459,0.8668), the approimation of the maimum area is improved to 4.883in. Tabe Summary of the Goden Section Search iterations aong direction (0,). Here =.6459 and f ( i ) = ( cos ).6549 sin Iteration u f ( ) f ( After competing these two iterations, we use the optima point to start a new cyce. Tabe 3 shows the first set of iterations for the second cyce. Tabe 3 Summary of the Goden Section Search iterations aong direction (,0) Iteration u f ( ) f ( and ( ) ( ( ) ( )) Here = f i = 6 + cos sin Note that we sti use the initia intervas chosen for i and u vaues throughout the cyces. Since this is a two-dimensiona search probem, the two searches aong the two dimensions competes the first cyce. In the net cyce, we return to the first dimension for which we conducted a search, namey, and start the second cyce with a search aong this dimension.

5 Mutidimensiona Direct Search Method Namey, ook for the optima soution of the function f (,0.8668). Each cyce consists of enough iterations to satisfy the predetermined convergence imit. After the fifth cyce, the optima soution of (.006,.040) with an area of 5.960in 0 is obtained. The optima soution to the probem happens at eacty 60 which is.047 radians, having an edge and base ength of in.the area of the gutter at this point is 5.96in. Therefore foding the sheet meta in such a way that the base is in and the sides 0 are in at an ange of 60 maimizes the area of the gutter. OPTIMIZATION Topic Mutidimensiona Direct Search Method Summary Tetbook notes for the mutidimensiona direct search method Major A engineering majors Authors Ai Yacin Date December 9, 0 Web Site