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Optimizing the Expected Maximum of Two Linear Functions Defined on a Multivariate Gaussian Distribution

Author

Listed:
  • David Bergman

    (Department of Operations and Information Management, University of Connecticut, Storrs, Connecticut 06268)

  • Carlos Cardonha

    (Department of Operations and Information Management, University of Connecticut, Storrs, Connecticut 06268)

  • Jason Imbrogno

    (Department of Finance, Economics, and Data Analytics, University of North Alabama, Florence, Alabama 35632)

  • Leonardo Lozano

    (Department of Operations, Business Analytics, and Information Systems, University of Cincinnati, Cincinnati, Ohio 45221)

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 for a highly nonlinear function, which includes the evaluation of the cumulative distribution function and probability density function 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 suboptimal solutions approaches.

Suggested Citation

  • David Bergman & Carlos Cardonha & Jason Imbrogno & Leonardo Lozano, 2023. "Optimizing the Expected Maximum of Two Linear Functions Defined on a Multivariate Gaussian Distribution," INFORMS Journal on Computing, INFORMS, vol. 35(2), pages 304-317, March.
    • Handle: RePEc:inm:orijoc:v:35:y:2023:i:2:p:304-317
    DOI: 10.1287/ijoc.2022.1259
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    References listed on IDEAS

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