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

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. History: Andrea Lodi, Area Editor for Design & Analysis of Algorithms-Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2022.1259 .

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“…We refer the interested readers to Ahsanullah and Nevzorov (2005), Bertsimas et al (2006), andMehta et al (2020) for details. More recently, Bergman et al (2023) focused on the case with two solutions and used results in order statistics (for two normal random variables) to devise cutting plane algorithms for these problems. Our framework is more versatile and can be extended to handle problems with more than two candidate solutions.…”

“…On this basis, Haugh and Singal (2021) provided an optimization framework to account for an opponent's team selection behavior in daily fantasy sports (DFS). Bergman et al (2023) also applied their methodology, namely maximizing the expectation of the largest order statistics, to select lineups for DFS.…”

“…Using the same moment condition as that of Bergman et al. (2023), we set $$\mu _j=0$$ and $$\sigma _j=1, \forall j\in [n]$$. In this problem setting, similar to problem (), the compact formulation for the case of continuous outcomes can be described by $$$$\begin{equation} \max _{\mathbf {x} \in \Omega _{\mathbf {x}} }\sup _{\mathbb {P}_{\tilde{\mathbf {z}}}\in \mathcal {P}_{\tilde{\mathbf {z}}}} {\mathbb {E}}_{\mathbb {P}_{\tilde{\mathbf {z}}}}{\left[ \max {\left\lbrace \tilde{z}_1{\left(\mathbf {x}_1 \right)},\tilde{z}_2{\left(\mathbf {x}_2 \right)} \right\rbrace} \right]}, \end{equation}$$$$where …”

“…Assuming $$\Omega _{\mathbf {x}}=\lbrace 0,1\rbrace ^{2n}$$, Bergman et al. (2023) have shown that this optimization problem is equivalent to $$$$\begin{eqnarray} &&\max _{\mathbf {x}\in \Omega _{\mathbf {x}}}\;\mathbb {E}{[\tilde{z}_1(\mathbf {x}_1)- \tilde{z}_2(\mathbf {x}_2)]}^2 = \sigma ^2(\tilde{z}_1(\mathbf {x}_1))\nonumber\\ &&\quad +\, \sigma ^2(\tilde{z}_2(\mathbf {x}_2)) - 2 {\rm {Cov}}(\tilde{z}_1(\mathbf {x}_1), \tilde{z}_2(\mathbf {x}_2)), \end{eqnarray}$$$$where $$\sigma ^2(\tilde{z}_i(\mathbf {x}_i))$$ denotes the variance of the payoff of the i th solution and $${\rm {Cov}}(\tilde{z}_1(\mathbf {x}_1), \tilde{z}_2(\mathbf {x}_2))$$…”

“…It is useful to compare the results obtained above to the continuous outcome case. Suppose that the outcome ũj , ∀j ∈ [n], of each game is continuous, which follows the standard normal distribution ũj ∼  (0, 1), then the similar problem is defined to solve max x∈Ω x 𝔼[max{z 1 (x 1 ), z2 (x 2 )}], where zi Bergman et al (2023) have shown that this optimization problem is equivalent to max…”