Conditions 3 and 4 ensure that the differential equations that characterize the optimal sharing rules have concave functions
on both sides of the equal sign.
Therefore, this technique is not comparable with genetic algorithm, since it could be applied for only one field of problems and is not applicable for concave functions
, although genetic algorithm has no limits.
The three are variants of Jensen's inequality, which is concerned with convex and concave functions.
An example of a concave function is the logarithmic function:
Another application of a concave function is the utility of wealth, or of consumption, where the utility function takes the following two forms:
The algorithm was developed as a particular case of the simplified algorithm for minimizing separable concave functions over linear polyhedra (see Falk and Soland).
ij]), the first term in the total production cost is concave because multiplication by non-negative number preserves the concavity and because the sum of concave functions is concave.
2]] are non-positive, and R ([center dot]) and [Alpha] ([center dot]) are concave functions, then the optimal number of inspections in a production run is one.
If L[double prime] are non-positive, and R ([center dot]) and a ([center dot]) are concave functions, then the optimal number of inspections in a production run is one.
Concave functions have a unique maximal point whereas convex functions have a unique minimal point.
This is because a concave function can have only one local maximum whereas in the computational experimentation section, many local maxima of the function T exist for an example problem.
As the product of real number and concave function
and the sum of several concave functions
are both concave, it is easy to deduce that the objective function R([lambda], t) is concave too.