Chapter 11, Problem 5RP

Explanation of Solution

Finding an optimal solution using Golden Section Search:

The given NLP is as follows:

$\begin{array}{l}Max\text{ 3}x-x^{\text{2}}\\ Suchthat,\\ 0\le x\le 5\end{array}$

Here, $x$ variable lies between 0 and 5.

So, $a=0,b=5$.

Now,

$\begin{array}{l}r=\frac{5^{1/2}-1}{2}\\ \text{ =0}\text{.618}\end{array}$

Determine the new lower and upper limits as follows,

$\begin{array}{l}{x}_1=b-r(b-a)\\ \text{ =}1.9098\\ x_2=a+r(b-a)\\ \text{ =}3.0902\end{array}$

Now, $f$ is unimodal function of $x$with $f(x)=3x-x^2$.

So, for these new limits, we can obtain the function,

$\begin{array}{l}f(x_1)=3\times 1.9098-{(}^{1.9098}\\ \text{ =}2.0820\\ f(x_2)=3\times 3.0901-{(}^{3.0901}\\ \text{ =}-.027864\end{array}$

Since, $f(x_1)>f(x_2)$, the new interval of uncertainty will be,

$\begin{array}{l}{I}_1=[a,x_2)\\ \text{ =[0,3}\end{array}$