Optimal Allocations of Continuous Resources to Several Activities with a Concave Return Function—Some Theoretical Results

The problem of allocating J continuous resources to K competing activities each with a concave return function is considered. The following results pertaining to the space of optimal solution to this problem are proved:

—under the most unrestricted conditions an optimal solution is almost always unique;

—there exists at least one optimal solution with J + K − 1 or less allocations;

—there exists at least one optimal solution for which the number of activities having multiple allocations is at most J − 1;

Also, sufficient input conditions for a unique optimal solution are established and it is shown how the dimension and extent of the optimal solution space can be derived directly from the output matrix.