A theoretical framework for solving the optimal admissions control with sigmoidal utility functions

This paper describes a social welfare maximization oriented admissions control over users who adopt their utility functions in the form of x(t)·f(x(t)) with x(t) being a quantity and f(x(t)) being a sigmoidal function. An optimal admissions control problem is first formulated into a network utility maximization (NUM) problem with the constraints of satisfying the utility targets. Finding a solution to the optimal admission control problem is to find a partition of a divisible resource such that the aggregate utility is maximized while ensuring the admitted users to satisfy the utility targets. Partitioning a divisible resource into appropriate bundle sizes would lead to a vast combinatorial space in which the complexity of finding a solution is NP-complete. In the NUM framework, finding a maximizer to the optimal admission control problem is to equivalent to find an optimal price. This paper describes a linear-time algorithm to search for the optimal price. The process of approaching the optimal price is formulated into a multi-round auction mechanism.