Engineering Optimization: Theory and Practice, Fourth Edition

294 Nonlinear Programming I: One-Dimensional Minimization Methods

5.13.3 Comparison of Methods

It has been shown in Section 5.9 that the Fibonacci method is the most efficient elimina-

tion technique in finding the minimum of a function if the initial interval of uncertainty

is known. In the absence of the initial interval of uncertainty, the quadratic interpo-

lation method or the quasi-Newton method is expected to be more efficient when the

derivatives of the function are not available. When the first derivatives of the function

being minimized are available, the cubic interpolation method or the secant method are

expected to be very efficient. On the other hand, if both the first and second derivatives

of the function are available, the Newton method will be the most efficient one in

finding the optimal step length,λ∗.

In general, the efficiency and reliability of the various methods are problem depen-

dent and any efficient computer program must include many heuristic additions not

indicated explicitly by the method. The heuristic considerations are needed to handle

multimodal functions (functions with multiple extreme points), sharp variations in the

slopes (first derivatives) and curvatures (second derivatives) of the function, and the

effects of round-off errors resulting from the precision used in the arithmetic opera-

tions. A comparative study of the efficiencies of the various search methods is given in

Ref. [5.10].

5.14 MATLAB Solution of One-Dimensional Minimization Problems

The solution of one-dimensional minimization problems, using the MATLAB program

optimset, is illustrated by the following example.

Example 5.15 Find the minimum of the following function:

f (x)= 0. 65 −

0. 75

1 +x^2

− 0. 65 xtan−^1

(

1

x

)

SOLUTION

Step 1: Write an M-fileobjfun.mfor the objective function.

function f= objfun(x)

f= 0.65–(0.75/(1+x^2))–0.65*x*atan(1/x);

Step 2: Invoke unconstrained optimization program (write this in new MATLAB

file).

clc

clear all

warning off

options = optimset('LargeScale','off');

[x,fval] = fminbnd(@objfun,0,0.5,options)