Optimizing Representation in Redistricting: Dual Bounds for Partitioning Problems with Non-Convex Objectives

We investigate optimization models for the purpose of computational redistricting. Our focus is on nonconvex objectives for estimating expected Black Representatives and Political Representation. The objectives are a composition of a ratio of variables and a normal distribution's cumulative distribution function (or ``probit curve"). We extend the work of Validi et al.~\cite{validi2022imposing}, which presented a robust implementation of contiguity constraints. By developing mixed integer linear programming models that closely approximate the parent nonlinear model, our approaches yield tight bounds on these optimization problems. We exhibit the effectiveness of these approaches on county-level data.