Source-linked AI summary
Likelihood Robust Optimization for Data-driven Problems
Zizhuo Wang, Peter Glynn, Yinyu Ye
TL;DR
The paper addresses decision-making when input distributions are unknown despite abundant historical data. It introduces likelihood robust optimization, which optimizes worst-case expected objectives over distributions making the observed data sufficiently likely, and analyzes its statistical meaning and applications. The reported results show tractable optimization and desirable performance in newsvendor and portfolio selection problems while avoiding over-conservatism.
Problem
Unknown input distributions and the limitations of moment-based robust sets motivate using abundant historical data more fully in distributionally robust decision-making.
Method
LRO defines the robust distribution set using distributions under which the observed historical data achieve at least a specified likelihood, then optimizes the worst-case expected objective over that set.
Results
The model is easily solvable, has statistical interpretations linked to Bayesian statistics and empirical likelihood theory, and shows desirable performance in newsvendor and portfolio selection applications.
Takeaways & Limitations
LRO uses full data information while retaining robustness and is reported to avoid the over-conservatism of other robust models.
Takeaways & Limitations
Portfolio experiments illustrate decent performance and desired features rather than establishing that LRO is the best portfolio-construction approach; the support set Ξ also materially affects the distribution set and solution.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper proposes likelihood robust optimization (LRO), which uses historical-data likelihood to define a tractable distributionally robust model. Statistical analyses connect LRO to Bayesian statistics and empirical likelihood theory, while applications report favorable performance in newsvendor and portfolio selection problems.
- Motivation and approach: Mean-variance DRO can discard distributional information and select unrealistic worst-case distributions, potentially producing overly conservative decisions.The paper contrasts exponential and normal distributions with similar moments but different properties, and notes two-point worst-case distributions in the newsvendor setting.
- Motivation and approach: LRO defines the accessible distribution set as distributions that assign the observed historical data at least a specified likelihood level.This replaces moment-only distribution sets with a likelihood-based construction using the full observed data.
- Model properties: LRO is highly tractable because duality reformulates its robust counterpart as a single convex optimization problem.The distribution set can also incorporate arbitrary convex constraints, including moment constraints, while preserving tractability.
- Statistical analysis: The paper links LRO to Bayesian statistics and empirical likelihood theory, interpreting its distribution set as a confidence region for distributions given observed data.It also discusses selecting the likelihood parameter for a specified confidence level and connects LRO with mean-variance DRO.
- Applications: In the newsvendor problem, LRO gives results similar to mean-variance DRO for symmetric distributions but performs much better for asymmetric distributions.The comparison targets the differing behavior of the two approaches under distributional asymmetry.
- Applications: In portfolio selection using real historical data, LRO achieves decent returns, naturally diversifies portfolios, and produces relatively small return fluctuations.The paper evaluates the framework through newsvendor and portfolio selection applications.
2 Likelihood Robust Optimization Model
The LRO model selects worst-case distributions that give observed historical data sufficient likelihood, yielding a data-driven robust optimization framework. The section establishes tractability, statistical interpretations, and links to empirical-likelihood-based robust optimization.
- Model formulation: LRO defines the accessible distribution set as distributions under which observed data achieve at least a specified likelihood.The parameter γ controls the likelihood threshold.
- Tractability: When h(x, ξ) is concave in x, LRO can be solved through an equivalent convex optimization problem.The corresponding optimal solution also identifies the worst-case distribution.
- Flexible distribution sets: LRO can incorporate additional convex information, including mean, variance, and linear constraints on distribution probabilities, while preserving tractability.Linear constraints Ap ≥ b include moment constraints as a special case.
- Statistical interpretation: Bayesian and empirical-likelihood analyses interpret the likelihood distribution set as a confidence region for distributions supported by the observed data.The paper also studies how to select γ for a desired confidence level.
- Asymptotic behavior: As the data set grows, the likelihood-robust distribution set requires allowable distributions to be very close to the empirical distribution.This distinguishes LRO from some moment-based robust sets that can retain distributions far from the empirical distribution even with large samples.
- Relation to other DRO models: Empirical likelihood theory connects LRO with other distributionally robust models and can produce asymptotic confidence regions for uncertain means.For mean uncertainty, thresholds can be found by bisection to construct an interval with asymptotic probability 1 − α under the Dirichlet measure.
3 Continuous State Space Case
The paper extends likelihood-based robust optimization from discrete uncertainty to continuous scalar state spaces. It also uses CDF bands, including Kolmogorov-Smirnov bands, to obtain statistically meaningful robust formulations.
- Continuous formulation: The continuous-state extension considers a scalar uncertain parameter and represents the CDF through its probability density function.The resulting formulation is initially a semi-infinite program.
- CDF-based alternative: A direct density-based likelihood extension is ineffective with finite data because constraints imposed only at observed points do not meaningfully restrict the distribution.The alternative CDF-band approach retains the empirical-likelihood foundation while providing a statistically meaningful robust counterpart.
- Statistical bands: Kolmogorov-Smirnov bands bound the empirical CDF at observed points and cover the true CDF at least 1 − α of the time when calibrated by the test statistic.The bands can be modified with different weights at different points.
- Finite reformulation: If h(x, ξ) is concave in ξ, the semi-infinite constraints can be reduced to finite constraints.The reduction uses intervals between ordered observations and explicitly defined boundary points.
- Tractability: If h(x, ξ) is also concave in x, the robust counterpart becomes a finite-dimensional convex program.The paper states that this formulation can therefore be solved easily.
4 Applications and Numerical Results
The paper applies LRO to newsvendor and portfolio selection problems, combining likelihood-based distribution sets with numerical tests. Results indicate improved robustness for asymmetric demand and decent, implicitly diversified portfolio performance.
- Newsvendor problem: LRO defines the newsvendor model over historical demand counts, with a likelihood constraint selecting worst-case distributions and a convex reformulation preserving tractability.The outer problem chooses stocking quantity, while the inner problem selects a distribution meeting the likelihood threshold.
- Newsvendor problem: LRO’s worst-case distributions are closer to empirical data and more plausible than Scarf’s two-point worst-case distribution.With mean and variance constraints, LRO produces the worst-case distribution closest to the empirical data among the three approaches.
- Newsvendor problem: Compared with Scarf’s mean-variance approach, LRO produces similar solutions for symmetric normal demand but performs much better for asymmetric exponential demand.The exponential-demand result is attributed to LRO adapting to data asymmetry rather than using only mean and variance.
- Newsvendor problem: Adding mean and variance constraints to LRO usually improves performance by concentrating the distribution set around shapes resembling the empirical distribution.The empirical distribution performs well under the true distribution but is less robust when the underlying distribution may deviate within a 95% confidence range.
- Portfolio selection problem: In portfolio selection, LRO outperforms all tested methods in return frequency and average overall return across 100 experiments, while maintaining a decent return standard deviation.Its standard deviation is much smaller than SS and comparable to DRO; the study uses 721 days of returns.
- Portfolio selection problem: LRO implicitly diversifies portfolios: it selects one, two, three, or all four stocks in 52%, 39%, 8%, and 1% of experiments, respectively.The authors identify diversification as a generally desired feature of portfolio selection.
5 Conclusion
The paper introduces likelihood robust optimization (LRO), which selects worst-case distributions consistent with a specified likelihood of observed data. It reports that LRO is solvable, statistically meaningful, less conservative than other robust models, and promising in two applications.
- LRO optimizes the worst-case expected objective over distributions that give the observed data a specified likelihood.
- The model is easily solvable and has strong statistical meanings.
- LRO avoids the over-conservatism of other robust models while protecting decisions from reasonable deviations from empirical data.
- Applications to newsvendor and portfolio selection problems produced numerical results that may make LRO appealing in several applications.