386 Nonlinear Programming III: Constrained Optimization Techniques
4.If at any stage, the reflected pointXr (found in step 3) violates any of the
constraints [Eqs. (7.7b)], it is moved halfway in toward the centroid until it
becomes feasible, that is,
(Xr)new=^12 (X 0 +Xr) (7.12)This method will progress toward the optimum point as long as the complex
has not collapsed into its centroid.
5.Each time the worst pointXh of the current complex is replaced by a new
point,the complex gets modified and we have to test for the convergence of the
process. We assume convergence of the process whenever the following two
conditions are satisfied:
(a) The complex shrinks to a specified small size (i.e., the distance between
any two vertices amongX 1 ,X 2 ,... ,Xkbecomes smaller than a prescribed
small quantity,ε 1.
(b) The standard deviation of the function value becomes sufficiently small
(i.e., when
1
k∑kj= 1[ (f X)−f (Xj)]^2
1 / 2≤ε 2 (7.13)whereXis the centroid of all thekvertices of the current complex, and
ε 2 > 0 is a specified small number).Discussion. This method does not require the derivatives off (X)andgj( X)to find
the minimum point, and hence it is computationally very simple. The method is very
simple from programming point of view and does not require a large computer storage.
1.A value of 1.3 for the initial value ofαin Eq. (7.10) has been found to be
satisfactory by Box.
2.Box recommended a value ofk≃ 2 n(although a lesser value can be chosen
ifnis greater than, say, 5). Ifkis not sufficiently large, the complex tends to
collapse and flatten along the first constraint boundary encountered.
3.From the procedure above, it can be observed that the complex rolls over and
over, normally expanding. However, if a boundary is encountered, the complex
contracts and flattens itself. It can then roll along this constraint boundary and
leave it if the contours change. The complex can also accommodate more than
one boundary and can turn corners.
4.If the feasible region is nonconvex, there is no guarantee that the centroid of all
feasible points is also feasible. If the centroid is not feasible, we cannot apply
the procedure above to find the new pointsXr.
5 .The method becomes inefficient rapidly as the number of variables increases.
6.It cannot be used to solve problems having equality constraints.
7.This method requires an initial pointX 1 that is feasible. This is not a major
restriction. If an initial feasible point is not readily available, the method
described in Section 7.13 can be used to find a feasible pointX 1.