Optimizing the expected maximum of two linear functions defined on a multivariate Gaussian distribution

TL;DR

This paper addresses the complex problem of optimizing the expected maximum of two linear functions on a multivariate Gaussian, introducing a cutting-plane algorithm that handles nonlinearity and demonstrates practical applications in sports analytics and manufacturing.

Contribution

It develops a novel cutting-plane algorithm for a NP-hard stochastic optimization problem involving correlated Gaussian variables, extending existing methods to nonlinear functions with real-world applications.

Findings

Algorithm outperforms sub-optimal approaches in computational tests.

Effective discretization and upper bounding improve computational efficiency.

Application to fantasy sports and manufacturing demonstrates practical relevance.

Abstract

We study stochastic optimization problems with objective function given by the expectation of the maximum of two linear functions defined on the component random variables of a multivariate Gaussian distribution. We consider random variables that are arbitrarily correlated, and we show that the problem is NP-hard even if the space of feasible solutions is unconstrained. We exploit a closed-form expression for the objective function from the literature to construct a cutting-plane algorithm that can be seen as an extension of the integer L-shaped method for a highly nonlinear function, which includes the evaluation of the c.d.f and p.d.f of a standard normal random variable with decision variables as part of the arguments. To exhibit the model's applicability, we consider two featured applications. The first is daily fantasy sports, where the algorithm identifies entries with positive returns during the 2018-2019 National Football League season. The second is a special case of makespan minimization for two parallel machines and jobs with uncertain processing times; for the special case where the jobs are uncorrelated, we prove the equivalence between its deterministic and stochastic versions and show that our algorithm can deliver a constant-factor approximation guarantee for the problem. The results of our computational evaluation involving synthetic and real-world data suggest that our discretization and upper bounding techniques lead to significant computational improvements and that the proposed algorithm outperforms sub-optimal solutions approaches.