A convex programming with nested constraints on the amount of resources

Abstract

This paper investigates the resource allocation problem of maximizing a strictly concave objective function under nested constraints on a weighted amount of resources. The resources are distributed in a multi-dimensional space and the distribution gives a planner some reward. The constraints on the resources are set in such a way that a weighted amount of resources is limited in each level of subspace as well as in the whole space. We call such constraints nested. The purpose of this paper is to devise efficient methods for an optimal distribution of resources under the constraints. For the concave maximization problem, first we derive necessary and sufficient conditions for an optimal solution and, secondly, propose two methods to solve the problem. Both methods manipulate the so-called Lagrangian multipliers and make a sequence of feasible solutions so as to satisfy all the necessary and sufficient conditions. Numerical examination confirms that the proposed methods perform better than some well-known methods of nonlinear programming in terms of computational time.