6.4 Univariate Method 315
variables (n=10), the number of grid points will be 3^10 = 95 ,049 withpi= and 3
410 = 1 , 048 ,576 withpi=. However, for problems with a small number of design 4
variables, the grid method can be used conveniently to find an approximate minimum.
Also, the grid method can be used to find a good starting point for one of the more
efficient methods.
6.4 Univariate Method
In this method we change only one variable at a time and seek to produce a sequence
of improved approximations to the minimum point. By starting at a base pointXiin the
ithiteration, we fix the values ofn−1 variables and vary the remaining variable. Since
only one variable is changed, the problem becomes a one-dimensional minimization
problem and any of the methods discussed in Chapter 5 can be used to produce a new
base pointXi+ 1. The search is now continued in a new direction. This new direc tion
is obtained by changing any one of then−1 variables that were fixed in the previous
iteration. In fact, the search procedure is continued by taking each coordinate direction
in turn. After all thendirections are searched sequentially, the first cycle is complete
and hence we repeat the entire process of sequential minimization. The procedure is
continued until no further improvement is possible in the objective function in any of
thendirections of a cycle. The univariate method can be summarized as follows:
1.Choose an arbitrary staring pointX 1 and seti= 1.
2.Find the search directionSias
STi =
( 1 , 0 , 0 ,... , 0 ) for i= 1 , n+ 1 , 2 n+ 1 ,...
( 1 , 0 , 0 ,... , 0 ) for i= 2 , n+ 2 , 2 n+ 2 ,...
( 0 , 0 , 1 ,... , 0 ) for i= 3 , n+ 3 , 2 n+ 3 ,...
..
.
( 0 , 0 , 0 ,... , 1 ) for i=n, 2 n, 3 n,...
(6.22)
3.Determine whetherλishould be positive or negative. For the current direction
Si, this means find whether the function value decreases in the positive or
negative direction. For this we take a small probe length (ε) and evaluatefi=
f(Xi),f+= f(Xi+εSi) and, f−= f(Xi−εSi) If. f+< fi,Siwill be the
correct direction for decreasing the value off and iff−< fi,−Siwill be the
correct one. If bothf+andf−are greater thanfi, we takeXias the minimum
along the directionSi.
4 .Find the optimal step lengthλ∗isuch that
f(Xi±λ∗iSi) =min
λi
(Xi±λiSi) (6.23)
where+or−sign has to be used depending upon whetherSior−Siis the
direction for decreasing the function value.
5.SetXi+ 1 =Xi±λ∗iSi depending on the direction for decreasing the function
value, andfi+ 1 = f(Xi+ 1 ).
6 .Set the new value ofi=i+1 and go to step 2. Continue this procedure until
no significant change is achieved in the value of the objective function.
