6.16 Test Functions 363
Iteration 2 (i= 2 )
The next search direction is determined as
S 2 = −[B 2 ]∇f 2 = −
[ 1
2 −
1
2
−^1252
]{
− 1
− 1
}
=
{
0
2
}
To find the minimizing step lengthλ∗ 2 alongS 2 , we minimize
f(X 2 +λ 2 S 2 )=f
({
− 1
1
}
+λ 2
{
0
2
)}
=f (− 1 , 1 + 2 λ 2 )= 4 λ^22 − 2 λ 2 − 1
with respect toλ 2. Since df/dλ 2 = at 0 λ∗ 2 =^14 , we obtain
X 3 =X 2 +λ∗ 2 S 2 =
{
− 1
1
}
+
1
4
{
0
2
}
=
{
− 1
3
2
}
This point can be identified to be optimum since
∇f 3 =
{
0
0
}
and ||∇f 3 || = 0 <ε
6.16 Test Functions
The efficiency of an optimization algorithm is studied using a set of standard func-
tions. Several functions, involving different number of variables, representing a variety
of complexities have been used as test functions. Almost all the test functions pre-
sented in the literature are nonlinear least squares; that is, each function can be
represented as
f (x 1 , x 2 ,... , xn)=
∑m
i= 1
fi(x 1 , x 2 ,... , xn)^2 (6.139)
wherendenotes the number of variables andmindicates the number of functions(fi)
that define the least-squares problem. The purpose of testing the functions is to show
how well the algorithm works compared to other algorithms. Usually, each test function
is minimized from a standard starting point. The total number of function evaluations
required to find the optimum solution is usually taken as a measure of the efficiency of
the algorithm. References [6.29] to [6.32] present a comparative study of the various
unconstrained optimization techniques. Some of the commonly used test functions are
given below.
1.Rosenbrock’s parabolic valley [6.8]:
f (x 1 , x 2 )= 100 (x 2 −x 12 )^2 +( 1 −x 1 )^2 (6.140)
X 1 =
{
− 1. 2
1. 0
}
, X∗=
{
1
1
}
f 1 = 42. 0 , f∗= 0. 0
