IDEAS home Printed from https://ideas.repec.org/a/inm/orijoc/v35y2023i2p304-317.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/ijoc.2022.1259
    Download Restriction: no
    File URL: https://libkey.io/10.1287/ijoc.2022.1259?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Keith C. Brown & Deborah J. Brown, 1986. "Using Order Statistics to Estimate Real Estate Bid Distributions," Management Science, INFORMS, vol. 32(3), pages 289-297, March.
    2. Sujin Kim & Raghu Pasupathy & Shane G. Henderson, 2015. "A Guide to Sample Average Approximation," International Series in Operations Research & Management Science, in: Michael C Fu (ed.), Handbook of Simulation Optimization, edition 127, chapter 0, pages 207-243, Springer.
    3. Charles E. Clark, 1961. "The Greatest of a Finite Set of Random Variables," Operations Research, INFORMS, vol. 9(2), pages 145-162, April.
    4. Diane L. Evans & Lawrence M. Leemis & John H. Drew, 2006. "The Distribution of Order Statistics for Discrete Random Variables with Applications to Bootstrapping," INFORMS Journal on Computing, INFORMS, vol. 18(1), pages 19-30, February.
    5. Bryan Clair & David Letscher, 2007. "Optimal Strategies for Sports Betting Pools," Operations Research, INFORMS, vol. 55(6), pages 1163-1177, December.
    6. Edward H. Kaplan & Stanley J. Garstka, 2001. "March Madness and the Office Pool," Management Science, INFORMS, vol. 47(3), pages 369-382, March.
    7. Dimitrina S. Dimitrova & Zvetan G. Ignatov & Vladimir K. Kaishev, 2019. "Ruin and Deficit Under Claim Arrivals with the Order Statistics Property," Methodology and Computing in Applied Probability, Springer, vol. 21(2), pages 511-530, June.
    8. Martin B. Haugh & Raghav Singal, 2021. "How to Play Fantasy Sports Strategically (and Win)," Management Science, INFORMS, vol. 67(1), pages 72-92, January.
    9. Vasileios M. Koutras & Markos V. Koutras, 2020. "Exact Distribution of Random Order Statistics and Applications in Risk Management," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1539-1558, December.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Stekler Herman O. & Klein Andrew, 2012. "Predicting the Outcomes of NCAA Basketball Championship Games," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 8(1), pages 1-10, March.
    2. Martin B. Haugh & Raghav Singal, 2021. "How to Play Fantasy Sports Strategically (and Win)," Management Science, INFORMS, vol. 67(1), pages 72-92, January.
    3. Martin B. Haugh & Chun Wang, 2022. "Play Like the Pros? Solving the Game of Darts as a Dynamic Zero-Sum Game," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2540-2551, September.
    4. David Bergman & Jason Imbrogno, 2017. "Surviving a National Football League Survivor Pool," Operations Research, INFORMS, vol. 65(5), pages 1343-1354, October.
    5. Irawan, Chandra Ade & Jones, Dylan & Hofman, Peter S. & Zhang, Lina, 2023. "Integrated strategic energy mix and energy generation planning with multiple sustainability criteria and hierarchical stakeholders," European Journal of Operational Research, Elsevier, vol. 308(2), pages 864-883.
    6. Adrian Lee & Sheldon Jacobson, 2011. "Sequential stochastic assignment under uncertainty: estimation and convergence," Statistical Inference for Stochastic Processes, Springer, vol. 14(1), pages 21-46, February.
    7. Bolduc, Denis & Kaci, Mustapha, 1993. "Estimation des modèles probit polytomiques : un survol des techniques," L'Actualité Economique, Société Canadienne de Science Economique, vol. 69(3), pages 161-191, septembre.
    8. Ludden Ian G. & Jacobson Sheldon H. & Khatibi Arash & King Douglas M., 2020. "Models for generating NCAA men’s basketball tournament bracket pools," Journal of Quantitative Analysis in Sports, De Gruyter, vol. 16(1), pages 1-15, March.
    9. Lof, Matthijs & van Bommel, Jos, 2023. "Asymmetric information and the distribution of trading volume," Journal of Corporate Finance, Elsevier, vol. 82(C).
    10. Bayliss, Christopher & Currie, Christine S.M. & Bennell, Julia A. & Martinez-Sykora, Antonio, 2021. "Queue-constrained packing: A vehicle ferry case study," European Journal of Operational Research, Elsevier, vol. 289(2), pages 727-741.
    11. Elmaghraby, Salah E., 2000. "On criticality and sensitivity in activity networks," European Journal of Operational Research, Elsevier, vol. 127(2), pages 220-238, December.
    12. Elmaghraby, S. E. & Fathi, Y. & Taner, M. R., 1999. "On the sensitivity of project variability to activity mean duration," International Journal of Production Economics, Elsevier, vol. 62(3), pages 219-232, September.
    13. Hajivassiliou, Vassilis A. & Ruud, Paul A., 1986. "Classical estimation methods for LDV models using simulation," Handbook of Econometrics, in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 40, pages 2383-2441, Elsevier.
    14. Stekler, H.O. & Sendor, David & Verlander, Richard, 2010. "Issues in sports forecasting," International Journal of Forecasting, Elsevier, vol. 26(3), pages 606-621, July.
      • Herman O. Stekler & David Sendor & Richard Verlander, 2009. "Issues in Sports Forecasting," Working Papers 2009-002, The George Washington University, Department of Economics, H. O. Stekler Research Program on Forecasting.
    15. Qi Fan & Jiaqiao Hu, 2018. "Surrogate-Based Promising Area Search for Lipschitz Continuous Simulation Optimization," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 677-693, November.
    16. Phillip E. Pfeifer, 2016. "The promise of pick-the-winners contests for producing crowd probability forecasts," Theory and Decision, Springer, vol. 81(2), pages 255-278, August.
    17. Dmitry B. Rokhlin, 2021. "Relative utility bounds for empirically optimal portfolios," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(3), pages 437-462, June.
    18. Martinetti, Davide & Geniaux, Ghislain, 2017. "Approximate likelihood estimation of spatial probit models," Regional Science and Urban Economics, Elsevier, vol. 64(C), pages 30-45.
    19. Bhat, Chandra R., 2018. "New matrix-based methods for the analytic evaluation of the multivariate cumulative normal distribution function," Transportation Research Part B: Methodological, Elsevier, vol. 109(C), pages 238-256.
    20. D'Amato, Rebecca M. (Rebecca Marie) & D'Aquila, Richard T. & Wein, Lawrence M., 1998. "Management of antiretroviral therapy for HIV infection : analyzing when to change therapy," Working papers WP 4043-98., Massachusetts Institute of Technology (MIT), Sloan School of Management.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:orijoc:v:35:y:2023:i:2:p:304-317. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.