5.13 Practical Considerations 293
Iteration 2
Sincef′(λ 2 ) =+ 0. 0105789 >0, we set newA= 0. 4 , f′(A) =− 0. 103652 , B=λ 2 =
0. 545757 , f′(B)=f′(λ 2 ) =+ 0 .0105789, and compute
λ 3 =A−
f′(A)(B −A)
f′(B)−f′(A)
= 0. 490632
Convergence check:|f′(λ 3 ) = |+| 0.00151235|< ε.
Since the process has converged, the optimum solution is given byλ∗≈λ 3 =
0. 4 90632.
5.13 Practical Considerations
5.13.1 How to Make the Methods Efficient and More Reliable
In some cases, some of the interpolation methods discussed in Sections 5.10 to 5.12
may be very slow to converge, may diverge, or may predict the minimum of the func-
tion,f (λ), outside the initial interval of uncertainty, especially when the interpolating
polynomial is not representative of the variation of the function being minimized. In
such cases we can use the Fibonacci or golden section method to find the minimum. In
some problems it might prove to be more efficient to combine several techniques. For
example, the unrestricted search with an accelerated step size can be used to bracket
the minimum and then the Fibonacci or the golden section method can be used to find
the optimum point. In some cases the Fibonacci or golden section method can be used
in conjunction with an interpolation method.
5.13.2 Implementation in Multivariable Optimization Problems
As stated earlier, the one-dimensional minimization methods are useful in multivariable
optimization problems to find an improved design vectorXi+ 1 from the current design
vectorXiusing the formula
Xi+ 1 =Xi+λ∗iSi (5.76)
whereSi is the known search direction andλ∗i is the optimal step length found by
solving the one-dimensional minimization problem as
λ∗i= inm
λi
[f(Xi+λiSi)] (5.77)
Here the objective functionf is to be evaluated at any trial step lengtht 0 as
f(t 0 ) =f(Xi+t 0 Si) (5.78)
Similarly, the derivative of the functionfwith respect toλcorresponding to the trial
step lengtht 0 is to be found as
df
dλ
∣
∣
∣
∣
λ=t 0
=STi f|λ=t 0 (5.79)
Separate function programs or subroutines can be written conveniently to implement
Eqs. (5.78) and (5.79).
