Maximizing a Sum of Sigmoids

The problem of maximizing a sum of sigmoidal functions over a convex constraint set arises in many application areas. This objective captures the idea of decreasing marginal returns to investment, and has applications in mathematical marketing, network bandwidth allocation, revenue optimization, optimal bidding, and lottery design. We define the sigmoidal programming problem (SP) and show how it arises in each of these application areas. We show that the general problem is NP-hard, and propose an approximation algorithm (using a branch and bound method) to find a globally optimal approximate solution to the problem. We show that this algorithm finds approximate solutions very quickly on problems of interest: in fact, for problems with few constraints, it frequently finds a solution within an acceptable error margin by solving a single convex optimization problem. To illustrate the power of this approach, we compute the positions which might have allowed the candidates in the 2008 United States presidential election to maximize their vote shares.