A Family of Convex Models to Achieve Fairness through Dispersion Control

TL;DR

This paper introduces a family of convex models that control decision-variable dispersion via a parameterized coefficient of variation, enabling fairness and load-balancing in optimization problems.

Contribution

It develops a novel family of convex models parameterized by a single variable to regulate dispersion, bridging trivial and equality constraints.

Findings

Coefficient of variation bounds are monotonic with the parameter.

The models relate feasible solution spaces across different parameter values.

Solution quality varies with dispersion control in assignment problems.

Abstract

Controlling the dispersion of a subset of decision variables in an optimization problem is crucial for enforcing fairness or load-balancing across a wide range of applications. Building on the well-known equivalence of finite-dimensional norms, the article develops a family of parameterized convex models that regulate the dispersion of a vector of decision-variable values through its coefficient of variation. Each model has a single parameter taking values in the interval $[0,1]$. When the parameter is set to zero, the model imposes only a trivial constraint on the optimization problem; when set to one, it enforces equality of all the decision variables. As the parameter varies, the coefficient of variation is provably bounded above by a monotonic function of that parameter. The article also presents theoretical results relating the space of feasible solutions across all models. Finally, it compares the models' solution quality on a variant of the assignment problem that regulates the dispersion in the assignment costs.