Introduction to Optimization - Numerical Methods in Engineering with MATLAB - page 409

and the geometricconstraints are

L 1 cos θ 1 + L 2 cos θ 2 + L 3 cos θ 3 = B

L 1 sin θ 1 + L 2 sin θ 2 + L 3 sin θ 3 = H

The principle of minimum potentialenergy states that the equilibrium config-
uration of the systemis the onethatsatisfies geometricconstraints and mini-
mizes the potentialenergy. Determine the equilibrium values of

θ 1 , θ 2 and θ 3

given that L 1 = 1 . 2m, L 2 = 1 . 5m, L 3 = 1 . 0 m, B = 3 . 5m, H = 0, W 1 = 20kNand

W 2 = 30kN.

MATLAB Functions

x = fmnbnd(@func,a,b) returns x that minimizes the function func of a single
variable. The minimumpoint must be bracketedin (a,b) . The algorithmused
is Brent's method thatcombines golden section search with quadratic interpo-
lation. It is moreefficientthan goldSearch that uses just the golden section
search.
x = fminsearch(@func,xStart) returns the vector of independent variables that
minimizes the multivariate function func . The vector xStart contains the
starting values of x . The algorithmis the Nelder-Mead method , also known
as the downhill simplex , which is reliable, but much less efficientthan Powell's
method.

Both of these functionscan becalledwith variouscontrol optionsthat set op-
timization parameters(e.g., the error tolerance) and control the display of results.
There are also additionaloutput parametersthat may be usedinthe function call, as
illustratedinthe following example (the dataistaken from Example 10.4):

>> [x,fmin,output] = fminsearch(@fex10

_

4,[1 5])

x=

0.7331 7.5878

fmin =
18.6929
output =
iterations: 38
funcCount: 72
algorithm: 'Nelder-Mead simplex direct search'