Direct Search Methods - Numerical Methods - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Multi Dimensional Direct Search

Methods

Multi Dimensional Direct Search

Methods Method-Overview

• Obvious approach is to enumerate all possible

solutions and find the min or the max.

• Very generally applicable but computationally

complex

• Direct search methods are open
• A good initial estimate of the solution is

required

• The objective function need not be

differentiable

Example

The cross-sectional area A of a gutter with base lengthb and edge length ofl is given by

( 2 cosθ ) sin θ

A = b + b + l l

.

Assuming that the width of material to be bent into the gutter shape is 6, find the angle θ

and edge lengthl which maximizes the cross-sectional area of the gutter.

l b l θ (^) θ

Solution

Recognizing that the base length b can be expressed

as , we can re-write the area function as

Use as the initial estimate of the solution and use

Golden Search method to determine optimal solution in each

dimension.

To use the golden search method we will use 0 and 3 as the

lower and upper bounds for the search region

( 0 , π/ 4 ) b = 6 − 2 l

f ( l ,θ ) =( 6 − 2 l + l cosθ) l sin θ

Solution Cont.

Iteration 1 along (0,1)

The maximum area of 4.8823 is obtained at point ( 2. 6459 , 0. 87 )

Iteration x (^) l x (^) u x 1 x 2 f(x 1 ) f(x 2 ) ε 1 0.0000 1.5714 0.9712 0.6002 4.8084 4.3215 1. 2 0.6002 1.5714 1.2005 0.9712 4.1088 4.8084 0. 3 0.6002 1.2005 0.9712 0.8295 4.8084 4.8689 0. 4 0.6002 0.9712 0.8295 0.7419 4.8689 4.7533 0. 5 0.7419 0.9712 0.8836 0.8295 4.8816 4.8689 0. 6 0.8295 0.9712 0.9171 0.8836 4.8672 4.8816 0. 7 0.8295 0.9171 0.8836 0.8630 4.8816 4.8820 0. 8 0.8295 0.8836 0.8630 0.8502 4.8820 4.8790 0. 9 0.8502 0.8836 0.8708 0.8630 4.8826 4.8820 0.

Solution Cont.

 Since this is a two-dimensional search problem, the

two searches along the two dimensions completes

the first iteration.

 In the next iteration we return to the first dimension

for which we conducted a search and start the

second iteration with a search along this dimension.

 After the fifth cycle, the optimal solution of (2.0016,

10420) with an area of 5.1960 is obtained.

 The optimal solution to the problem is exactly 60

degrees which is 1.0472 radians and an edge and

base length of 2 inches. The area of the gutter at this

point is 5.1962.