multivariate production systems optimization - Stanford University

direct search method is simple to understand but sometimes challenging to implement.
Furthermore, due to the heuristic nature of direct search methods, no guarantee can be
made of their convergence.
The polytope algorithm (Gill, 1983) is a good example of a function comparison
method. For a problem consisting of n decision variables, a polytope of n+1 points is
created. The objective function is evaluated at each point and then the polytope moves
away from the point with the largest value by replacing it with a new point on the opposite
side of the polytope. This is the reflected point. If the reflected point is a “good” point, the
polytope attempts to expand in this direction. If the reflected point is a “bad” point, the
polytope contracts in size. The polytope moves along, one new function evaluation at a
time, reflecting, expanding, and contracting. At the minimum of the function, it should
contract to a small enough size to satisfy convergence criteria.
For an n-dimensional problem, the polytope consists of n+1 points, x 1 , x 2 , ...,
xn+1 . The objective function is evaluated at each of the points and the function values, F1 ,
F2 , ..., Fn+1 , are ranked such that Fn+1 ≥ Fn ≥ ... ≥ F2 ≥ F1 . The maximum function
value, Fn+1 , and its corresponding point, xn+1 , are removed from the polytope set. The
centroid of the remaining n points is given by
c = 1 n
73
n
∑
j=1
The centroid is used to generate the trial reflection point (see Figure 6.3)
xj
xr = c + α c - xn+1
(6.32)
(6.33)
where α is the reflection coefficient (α ≈ 1). Evaluating the function at xr yields Fr . There
are three possibilities for the reflected function value, Fr , when compared to the existing set
of function values: 1) it is the new low value, 2) it is the new high value, or 3 ) it is
somewhere in between:
1) Fr < F1 . If the reflection function value is the new low value, then we
assume that this is a “good” direction and attempt to expand the polytope
even further along the reflection vector. The expansion point, xe , is
given by
xe = c + β xr - c (6.34)