The impact of sampling methods on bias and variance in stochastic linear programs

Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the problem based on this subset, an approach known as sample path optimization. The value of the optimal solution to the sampled problem provides an estimate of the true objective function value. This estimator is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative sampling methods, antithetic variates and Latin Hypercube sampling, affect both the bias and variance, and thus the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. Whether the bias reduction or variance reduction plays a larger role in MSE reduction is problem and parameter specific.