A concave maximization problem with double layers of constraints on the total amount of resources

This paper investigates a kind of resource allocation problem which maximizes a strictly concave objective function with double layers of constraints on the total amount of resources. Resources are distributed on a two-dimensional space, say, a geographical space with time flow, and are doubly constrained in the sense that the total amount is limited on the whole space and the subtotal amount is constrained at each time too. We derive necessary and sufficient conditions for an optimal solution and propose two methods of solving it. Both methods manipulate Lagrange multipliers and make a sequence of feasible solutions that ultimately satisfy necessary and sufficient conditions for optimality. It is shown by numerical computation that the proposed methods are faster than other well-known methods.