We consider optimal decision-making problems in an uncertain environment. In
particular, we consider the case in which the distribution of the input is
unknown, yet there is abundant historical data drawn from the distribution. In
this paper, we propose a new type of distributionally robust optimization model
called the likelihood robust optimization (LRO) model for this class of
problems. In contrast to previous work on distributionally robust optimization
that focuses on certain parameters (e.g., mean, variance, etc.) of the input
distribution, we exploit the historical data and define the accessible
distribution set to contain only those distributions that make the observed
data achieve a certain level of likelihood. Then we formulate the targeting
problem as one of optimizing the expected value of the objective function under
the worst-case distribution in that set. Our model avoids the
over-conservativeness of some prior robust approaches by ruling out unrealistic
distributions while maintaining robustness of the solution for any
statistically likely outcomes. We present statistical analyses of our model
using Bayesian statistics and empirical likelihood theory. Specifically, we
prove the asymptotic behavior of our distribution set and establish the
relationship between our model and other distributionally robust models. To
test the performance of our model, we apply it to the newsvendor problem and
the portfolio selection problem. The test results show that the solutions of
our model indeed have desirable performance.

本文提出了一种新型的分布鲁棒优化模型 —— 似然鲁棒优化模型，以历史数据为依据，用置信区间代替概率分布，解决了当环境不确定且输入分布未知的决策问题，避免了以往过于谨慎的方法对真实分布的偏离和对输出的限制。