Engineering Optimization: Theory and Practice, Fourth Edition

316 Nonlinear Programming II: Unconstrained Optimization Techniques

The univariate method is very simple and can be implemented easily. However,

it will not converge rapidly to the optimum solution, as it has a tendency to oscil-

late with steadily decreasing progress toward the optimum. Hence it will be better to

stop the computations at some point near to the optimum point rather than trying to

find the precise optimum point. In theory, the univariate method can be applied to find

the minimum of any function that possesses continuous derivatives. However, if the

function has a steep valley, the method may not even converge. For example, consider

the contours of a function of two variables with a valley as shown in Fig. 6.5. If the

univariate search starts at pointP, the function value cannot be decreased either in

the direction±S 1 or in the direction±S 2. Thus the search comes to a halt and one

may be misled to take the pointP, which is certainly not the optimum point, as the

optimum point. This situation arises whenever the value of the probe lengthεneeded

for detecting the proper direction (±S 1 or±S 2 ) happens to be less than the number of

significant figures used in the computations.

Example 6.4 Minimize f (x 1 , x 2 )=x 1 −x 2 + 2 x^21 + 2 x 1 x 2 +x 22 with the starting

point (0, 0).

SOLUTION We will take the probe length (ε) as 0.01 to find the correct direction for

decreasing the function value in step 3. Further, we will use the differential calculus

method to find the optimum step lengthλ∗i along the direction±Siin step 4.

Iterationi= 1

Step 2: Choose the search directionS 1 asS 1 =

{ 1

0

}

.

Figure 6.5 Failure of the univariate method on a steep valley.