Allocation of limited resources under quadratic constraints

Abstract

The proper allocation/distribution of limited resources is a traditional problem with various applications. The mathematical formulation of such problems usually includes constraints describing the set of feasible solutions (feasible set), from which the (nearly) optimal or equilibrium solution should be selected. Often the feasible set is more difficult to determine than to find the optimal or equilibrium solution. Alternatively, the already known feasible set often makes it easier to select the optimal or equilibrium solution. In some other cases, any feasible solutions are the same satisfactory, additional optimization is needless. Accordingly, the main or only task in many cases is to determine the feasible set itself. In the paper, a new theorem is proved for the explicit expression of properly assigned (dependent) variables by means of the other (independent) variables in a system of inequality and quadratic equality constraints. The sum of the (nonnegative) variables can be either prefixed or not. The constraints may describe the feasible set in various resource allocation tasks (possibly in optimization or game-theoretical contexts) or in other problems. Two new lemmas are proved for supporting the proof of the above mentioned theorem, nevertheless, they can also be considered independent results, which may help future mathematical derivations. Supported by a further new lemma, a practical algorithm is derived for assigning in a feasible way the independent variables, to which (possibly limited) arbitrary nonnegative values can be prescribed. Various practical examples are provided to facilitate utilizing the results.