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Distributionally Robust Optimization: A Review

Hamed Rahimian, Sanjay Mehrotra

arXiv:1908.05659v1math.OCcs.LGstat.ML

TL;DR

The paper addresses how to make optimization and learning decisions when the probability distribution of uncertain parameters is ambiguous rather than fully known. It surveys distributionally robust optimization, its ambiguity-set formulation, solution and modeling developments, and relationships with robust optimization, risk aversion, chance constraints, and regularization. The review establishes that DRO can connect stochastic and robust optimization and that ambiguity sets can induce coherent risk measures.

  • Problem

    Decision-makers often have only partial knowledge of the probability distribution governing uncertain parameters, while existing uncertainty analyses do not provide a systematic framework for hedging against that ambiguity.

  • Method

    The paper surveys DRO modeling, theoretical and computational developments, applications, ambiguity-set construction, algorithms, and connections to related optimization and statistical-learning frameworks.

  • Results

    DRO reduces to stochastic optimization when its ambiguity set contains only the true distribution, to robust optimization when it contains all distributions on the support, and is equivalent to minimizing a coherent risk measure for suitable ambiguity sets.

  • Takeaways & Limitations

    DRO provides a modeling framework that places decision-making between complete distributional information and distribution-free support-based robustness through the choice of ambiguity set.

  • Takeaways & Limitations

    The reviewed portfolio model assumes a known covariance matrix Σ and independent residual and factor returns, limiting that model's uncertainty specification.

Abstract

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The concepts of risk-aversion, chance-constrained optimization, and robust optimization have developed significantly over the last decade. Statistical learning community has also witnessed a rapid theoretical and applied growth by relying on these concepts. A modeling framework, called distributionally robust optimization (DRO), has recently received significant attention in both the operations research and statistical learning communities. This paper surveys main concepts and contributions to DRO, and its relationships with robust optimization, risk-aversion, chance-constrained optimization, and function regularization.

1. Introduction.

The introduction motivates distributionally robust optimization as a framework for decisions with ambiguous probability distributions and surveys its links to related optimization and learning concepts. DRO uses an ambiguity set and worst-case analysis to bridge statistical information and decision-making.

  • Motivation: Existing sensitivity and stability analyses quantify distributional uncertainty but do not systematically hedge against ambiguity in the underlying distribution.These approaches include perturbation analysis, stochastic-model stability analysis, and input uncertainty analysis.
  • DRO framework: Ambiguous stochastic optimization protects decision-makers by minimizing a worst-case objective over an ambiguity set of probability distributions.The framework is described as bridging data and decision-making through statistics and optimization.
  • Motivation: Distributionally robust optimization addresses decision-making when the probability distribution of uncertain parameters is only partially known.The ambiguity is represented by an ambiguity set of probability distributions consistent with prior knowledge.
  • Relationships: DRO reduces to stochastic optimization when the ambiguity set contains only the true distribution and to robust optimization when it contains all distributions supported on the uncertainty set.Thus, the choice of ambiguity set places DRO between stochastic and robust optimization.
  • Scope of the review: The review covers DRO research across operations research and machine learning, including modeling, algorithms, theory, applications, and connections to risk and robust optimization.It also discusses chance constraints, game theory, and function regularization in statistical learning.
  • Scope of the review: The paper organizes its review around foundational definitions, conceptual relationships, solution techniques, ambiguity modeling, calibration, and applications.The stated organization includes connections to robust optimization, risk aversion, chance constraints, and statistical-learning regularization.

2. Notation and Basic Definitions.

This section establishes the measure-theoretic, probabilistic, functional, and convex-analytic notation used throughout the paper. It also introduces data-driven out-of-sample performance and guarantees for DRO solutions.

  • Probabilistic notation: The paper distinguishes probability measures on a measurable space from probability distributions induced by a random vector.A random vector is marked with a tilde, while its realization is written without the tilde.
  • Finite-space notation: On finite spaces, probability measures and measurable functions are represented as vectors of atom probabilities and corresponding outcomes.The shorthand uses q = [q_1, ..., q_M]^T and z = [z_1, ..., z_M]^T.
  • Probabilistic notation: It defines measurable functions, integrability, essential boundedness, distributional equivalence, and probabilities of measurable events.These definitions provide the function spaces and probability notation used later.
  • Convex and conic notation: The notation includes ℓ_p norms, the simplex, positive-part operators, proper and dual cones, positive semidefinite matrices, convex hulls, and interiors.It also defines conic representability for K-epigraphs through conic inequalities and auxiliary variables.
  • Statistical-learning notation: The paper introduces training and test samples to distinguish in-sample data-driven solutions from their out-of-sample performance under an unknown true distribution.The test sample is independent of the training set.
  • Statistical-learning notation: DRO performance guarantees include finite-sample confidence bounds for certificates and asymptotic consistency as the sample size increases.The certificate can be chosen as the worst-case inner value over the data-driven ambiguity set, evaluated at the data-driven solution.

3. Relationship with Game Theory, Risk-Aversion, Chance-Constrained Optimization, and Regularization.

DRO admits game-theoretic, risk-measure, chance-constraint, and regularization interpretations. Its ambiguity set can interpolate between stochastic and robust optimization while connecting worst-case expectations to coherent risk and regularized learning objectives.

  • Relationship with Game Theory: DRO is a minimax game in which the decision maker minimizes expected cost while nature selects the worst distribution in the ambiguity set.Under suitable conditions, exchanging infimum and supremum preserves the optimal value and yields a saddle-point interpretation.
  • Relationship between DRO and RO: With a finite sample space, DRO becomes robust optimization over an uncertainty set of probability vectors, while continuous spaces remain admissible.Writing f(x,p) := p⊤h(x) gives the robust form inf_x∈X sup_p∈P f(x,p).
  • Relationship with Risk-Aversion: A convex compact ambiguity set makes worst-case expectations equivalent to a coherent risk measure, and coherent risk preferences can generate corresponding ambiguity sets.Law invariance is likewise preserved between the ambiguity set and its associated risk measure under the stated closure conditions.
  • Relationship with Chance-Constrained Optimization: Robust uncertainty sets can encode probabilistic guarantees, creating a one-to-one correspondence between ambiguity sets satisfying chance guarantees and safe approximations.The cited guarantee is expressed as probability at least 1−α with respect to the sample probability measure.
  • Relationship with Function Regularization: Optimal-transport DRO rewrites learning objectives as empirical expected cost plus regularization, while φ-divergence DRO yields a standard-deviation regularization term.The transportation cost determines the regularizer’s shape, and the framework covers linear regression, logistic regression, SVM, and LASSO.

4. General Solution Techniques to Solve DRO Models.

DRO models are semi-infinite programs because constraints must hold across an ambiguity set of probability distributions. Solution techniques remove this universal quantifier through finite approximations or dual reformulations, whose tractability depends on the ambiguity set and model structure.

  • General challenges: DRO solution methods must reformulate semi-infinite programs because direct numerical solution is generally impractical.The required reformulation and its tractability depend on the ambiguity set and the functions h(x, ˜ξ) and g(x, ˜ξ).
  • Cutting-surface method: The cutting-surface method replaces the universal quantifier over distributions with finitely many distributions, adding new distributions iteratively until optimality criteria are met.Its key requirements are solving each finite relaxation optimally and generating an ϵ-optimal solution to the distribution-separation subproblem.
  • Cutting-surface method: Under the stated oracle assumptions, iteratively solving relaxed master problems and distribution-separation subproblems yields an ϵ-feasible solution in finitely many iterations.The resulting objective value lower bounds the optimal value of the semi-infinite problem.
  • Cutting-surface method: The cutting-surface distribution-generation subproblem may be nonconvex, although convexifying the ambiguity set preserves the optimum when the risk functional is convex in the distribution.This convexity condition permits replacing the ambiguity set by its convex hull without changing the optimal solution value.
  • Dual method: The dual method replaces the worst-case supremum with an infimum under suitable regularity conditions ensuring strong duality.DRO studies use linear, Lagrangian, convex-analytic, and conic duality; a Slater-type interior condition is one route to strong duality.
  • Other methods: Dual reformulations can be tractable when the ambiguity set and model functions support optimization formulations, while alternative SIP methods remain uncommon in DRO.The literature has focused mainly on dual methods, with fewer applications of penalty, smooth approximation, projection, primal, and discretization methods.

5. Choice of Ambiguity Set of Probability Distributions.

Ambiguity sets encode partial distributional information through moment, discrepancy, structural, or kernel-based restrictions. Their shape and size balance tractability, expressiveness, conservatism, and confidence that the unknown distribution is included.

  • Design principles: An ambiguity set should be as small as possible while containing the unknown true distribution with certainty or high confidence.These principles reduce conservatism while protecting against the unknown distribution.
  • Design principles: Modelers commonly choose the ambiguity-set shape first and determine size parameters from data, although the two choices need not be separated.The paper reviews approaches that address shape and size simultaneously.
  • Shape and tractability: Ambiguity-set research balances tractable formulations, such as LP, SOCP, or SDP, against expressiveness from richer distributional information.The literature emphasizes both efficient computational techniques and sets that capture more structural information.
  • Ambiguity-set families: Moment-based sets constrain distributional moments, whereas discrepancy-based sets contain distributions close to a nominal distribution under a discrepancy measure.The review also treats shape-preserving and kernel-based sets as additional categories.
  • Ambiguity-set families: Examples include Markov, Chebyshev, Gauss, median-absolute deviation, and Huber ambiguity sets, which impose progressively different moment or structural information.The listed sets use support, moment bounds, unimodality, symmetry, medians, or deviation-related restrictions.
  • Ambiguity-set families: The four ambiguity-set categories can overlap, so the review assigns papers according to authors’ explicit or implicit categorizations.The reviewed groups are discrepancy-based, moment-based, shape-preserving, and kernel-based sets, plus general sets without a specific form.

5.1. Discrepancy-Based Ambiguity Sets.

DRO ambiguity sets specify probability distributions near a nominal measure using a discrepancy and robustness level. The section reviews optimal-transport formulations, duality results, and tractable reformulations across stochastic and chance-constrained models.

  • Ambiguity-set construction: An ambiguity set contains probability measures whose discrepancy from nominal P0 is at most ϵ, with ϵ controlling robustness.The discrepancy determines the set’s shape, while ϵ can represent belief in P0.
  • Optimal transport: Optimal-transport ambiguity sets use couplings between probability measures and transportation costs satisfying lower semicontinuity and zero diagonal cost.The r-Wasserstein metric is a special case when the cost is an rth power of an underlying metric.
  • Duality: Under mild semicontinuity and integrability conditions on a general Polish space, optimal-transport DRO admits primal-dual optimal solutions.The results require lower semicontinuity of c and upper semicontinuity and integrability of h(x, ·).
  • Duality: The transport dual can be reduced to a univariate problem involving an expectation under P0 and a term linear in ϵ.This structure also yields DRO representations of regularized logistic regression and LASSO.
  • Computational reformulations: Wasserstein DRO has finite-dimensional convex reformulations for several cost classes, contrasting with earlier nonconvex formulations requiring global optimization.Related work also derives explicit worst-case distributions, conic programs, semi-infinite reformulations, and bisection algorithms.
  • Chance constraints: For distributionally robust chance constraints, exact mixed-integer conic reformulations are available for specified affine individual and joint constraints.The formulation applies to p-Wasserstein ambiguity sets and arbitrary norms, with special treatment for ℓ1 and ℓ∞ transportation costs.

5.1.2. φ-Divergences.

φ-divergences provide a family of discrepancy-based ambiguity sets centered at a nominal distribution and parameterized by a robustness radius. The reviewed literature develops duality, tractable reformulations, statistical guarantees, and applications in optimization and learning.

  • Definition and duality: φ-divergence ambiguity sets constrain divergence from nominal P0 by a radius ϵ.Common φ-divergences and their conjugates are summarized as foundations for DRO formulations.
  • Definition and duality: A Lagrangian-duality result provides a reformulation for worst-case expected costs over φ-divergence ambiguity sets.The theorem is stated for fixed x and positive ϵ.
  • Tractability: For finite-dimensional probability vectors, robust counterparts based on most commonly used φ-divergences are tractable.The literature also systematically classifies φ-divergences according to data sources and risk preferences.
  • Chance constraints: φ-divergence models support safe approximations and adjusted-risk-level reformulations for distributionally robust chance constraints.These approaches include joint constraints, dependent parameters, and general nonlinear constraint functions in some settings.
  • Statistical learning: In distributionally robust classification, φ-divergence ambiguity can produce a classifier optimal for the training distribution rather than robust across the ambiguity set.The cited analysis attributes this pessimism to classification losses and over-conservative ambiguity sets.
  • Statistical guarantees: An asymptotic expansion links DRO under a specific divergence to empirical expected cost plus a standard-deviation regularization term.Under lower semicontinuity, the optimal-solution set converges to that of the stochastic program under the true distribution.
  • Applications: Applications reviewed include portfolio optimization, Markov decision processes, dynamic programming, multistage optimization, extreme-value analysis, and newsvendor models.The examples use divergences including Kullback-Leibler, Burg, χ2, modified χ2, and Rényi divergences.

5.1.3. Total Variation Distance.

Total variation distance offers another ambiguity-set construction for DRO and is related to φ-divergences, one-sided variation distances, and optimal-transport discrepancy. The reviewed results cover duality, reformulations, convergence, and applications.

  • Definition and relationships: Total variation distance is half the ℓ1-norm and can also be obtained from an optimal-transport discrepancy with a discrete cost.It is additionally a φ-divergence with φ(t) = 1/2|t−1|.
  • Ambiguity sets: Total-variation ambiguity sets are centered at nominal P0 with radius ϵ controlling their size.The section introduces duality results for worst-case expectations over these sets.
  • Definition and relationships: One-sided variation distances are φ-divergences but do not define probability metrics like total variation distance.They use positive-part expressions based on differences between densities.
  • Caveat: A cited relationship between Hellinger and Kullback-Leibler divergences was reported incorrectly in prior work.The passage identifies the erroneous inequality but does not provide the corrected full relation.
  • Applications: The literature studies total-variation DRO in two-stage stochastic programs and finite-sample-space convex optimization.Applications analyze nominal-distribution selection, convergence, and the effects of uncertain parameters through effective and ineffective scenarios.

5.1.4. Goodness-of-Fit Test.

The supplied passages identify goodness-of-fit tests as statistical tools used to construct ambiguity sets in data-driven DRO. They also mention Prohorov metrics as another probability-discrepancy choice.

  • Goodness-of-fit tests: Data-driven DRO studies consider χ2, G, Kolmogorov-Smirnov, Kuiper, Cramér-von Mises, Watson, and Anderson-Darling goodness-of-fit tests.The cited work characterizes the geometric shape of the resulting confidence or ambiguity sets across several data-support settings.
  • Probability metrics: The Prohorov metric is introduced as a metric between two probability measures.The supplied passage provides its definition without further reformulation or application details.

5.1.5. Prohorov Metric.

The Prohorov metric measures distributional discrepancy through neighborhood inclusions, and its ambiguity set contains distributions within radius ϵ of a nominal measure. The surveyed literature applies this construction to stochastic programs, chance constraints, and related DRO models.

  • Prohorov Metric: The Prohorov metric bounds each distribution's probability of every measurable set by the other's probability on an expanded set plus γ.The expansion Aγ contains points within distance γ of A.
  • Prohorov Metric: The associated ambiguity set Pp(P0; ϵ) contains distributions whose Prohorov distance from nominal measure P0 is at most ϵ.The radius ϵ controls the ambiguity-set size.
  • Related Metrics: The Levy metric specializes the Prohorov metric to univariate distributions through inequalities over all real thresholds t.Its definition compares cumulative probabilities at shifted thresholds.
  • Applications: Erdoǧan and Iyengar use Prohorov ambiguity for parameterized convex constraints and analyze scenario approximations with sample-size guarantees.Their work extends earlier scenario-approximation results for DRO problems.
  • Related Metrics: The reviewed metric families include uniform, total variation, Kantorovich, Fortet-Mourier, and ζ-structure metrics for modeling distributional ambiguity.The paper notes that many probability metrics are special cases of a broader family.
  • ζ-Structure Metrics: For bounded support, Lemma 5.7 establishes relationships among ζ-structure metrics, while later work constructs ambiguity sets from historical data and studies convergence.Other studies examine how ambiguity-set variation affects optimal values and solutions.

5.1.8. Contamination

Contamination neighborhoods model distributional ambiguity by mixing a nominal distribution with an arbitrary contaminating distribution. This construction is used in robust statistics, economics, and statistical learning, alongside broader discrepancy-based DRO models.

  • Contamination Neighborhood: The contamination neighborhood Pc(P0; ϵ) consists of mixtures (1 − ϵ)P0 + ϵQ, where Q belongs to a specified family of distributions.The parameter ϵ lies in [0, 1] and controls the contamination weight.
  • Applications: Contamination neighborhoods are extensively used in robust statistics and also appear in economics and statistical learning.The cited applications include ambiguity aversion and learning under distributional uncertainty.
  • Discrepancy-Based Models: Discrepancy-based DRO models cover risk measures and ambiguity sets built from goodness-of-fit tests or probability metrics.Reviewed risk measures include expectation and combinations involving standard deviation.
  • Risk-Measure Connections: DRO ambiguity sets based on Wasserstein, φ-divergence, and total variation metrics connect to law-invariant risk measures.The cited work also studies sample-average approximations and statistical properties of optimal solutions.
  • Moment-Based Models: Moment-based ambiguity sets contain probability distributions whose moments satisfy specified properties.The paper treats moment-based models as a separate category with overlapping subgroups.

5.2. Moment-Based Ambiguity Sets.

Moment-based ambiguity sets encode uncertainty through means, variances, covariances, and related moment restrictions, supporting tractable DRO formulations across portfolio, chance-constrained, and stochastic-programming problems. The survey also describes conic frameworks that unify several ambiguity-set constructions.

  • Chebyshev Ambiguity Sets: Scarf's newsvendor model assumes only known demand mean and variance, yielding a closed-form order quantity and a two-point worst-case distribution.Later work extends this setting to recourse, fixed ordering costs, random production, and multiple competing products.
  • Portfolio Optimization: Portfolio DRO models represent uncertainty in means, factor loadings, and residual variances through specified uncertainty sets.The Goldfarb–Iyengar factor model assumes Gaussian factors and diagonal residual covariance, with residuals independent of factors.
  • Portfolio Optimization: The portfolio applications include expected-return, variance, and value-at-risk objectives or constraints under distributional ambiguity.The cited formulations include maximum expected return subject to a maximum variance constraint and worst-case VaR formulations.
  • Tractability: For several portfolio and chance-constrained problems, the reviewed approaches obtain SOCP or SDP reformulations under structured uncertainty sets.The results include SOCP reformulations for four portfolio problem classes and an SDP computation of an upper bound on worst-case VaR.
  • Statistical Learning: DRO statistical-learning formulations constrain conditional predictor distributions to match empirical feature-function moments while optimizing worst-case performance.Fathony et al. report consistency guarantees associated with probabilistic graphical models and support for customized loss metrics.
  • Cross-Moment or Nested Moment: Wiesemann et al. propose conic-representable ambiguity sets that encompass φ-divergences, mean restrictions, covariance bounds, coefficient of variation, and absolute-mean models.Their framework uses conic-representable expectation constraints and confidence sets.
  • Cross-Moment or Nested Moment: Generalizations add infinitely many conic-representable expectation constraints, preserve worst-case expected values under stated tractability conditions, and yield conic optimization procedures.Other work transforms certain adaptive DRO models into classical robust optimization with second-order conic uncertainty sets using generalized linear decision rules.

5.3. Shape-Preserving Models.

Shape-preserving DRO models incorporate structural distributional information such as symmetry, unimodality, convexity, and tail behavior into ambiguity sets. The surveyed work derives probability bounds, tractable semidefinite formulations, and functionally robust decisions under structural uncertainty.

  • Shape-Preserving Ambiguity Sets: Shape-preserving ambiguity sets incorporate symmetry, unimodality, and convexity alongside moment information.The proposed models impose structural constraints within a convex subset of the ambiguity set.
  • Shape-Preserving Ambiguity Sets: Popescu derives upper and lower bounds on generalized moments, including tail probabilities, using conic duality and semidefinite programs.The construction focuses on ambiguity sets possessing a Choquet representation.
  • Unimodality: α-unimodality is defined through a monotonicity condition on scaled probabilities around a mode.The definition requires t^α times the probability of scaled sets to be nonincreasing in t.
  • Unimodality: For α-unimodal multivariate distributions with common mode and structured moments on polytopic support, generalized Gauss bounds can be obtained by SDP when support is an open polyhedron.Van Parys et al. extend Popescu's worst-case probability analysis to this setting.
  • Canonical Ambiguity Sets: A canonical ambiguity set intersects cross-moment restrictions with structural marginal information and contains Gauss and median-absolute-deviation ambiguity sets as special cases.The structural information includes symmetry and α-unimodality.
  • Functionally Robust Models: A functionally robust newsvendor model hedges against misspecified price-demand relationships by assuming the function is decreasing and convex or concave.The approach reduces the minimax problem to univariate pricing optimization and uses function cuts to shrink admissible functions.
  • Tail Behavior: Under tail-convexity assumptions, worst-case tail behavior can be either extremely light-tailed or extremely heavy-tailed.The result addresses computation of worst-case bounds when data are insufficient for evaluating extremal performance.
  • Motivation: High-dimensional settings make it impractical to fix an ambiguity-set form and calibrate its parameters a priori, motivating approaches that learn the ambiguity-set form.This limitation concerns both choosing the form and calibrating its descriptive parameters.

5.4. Kernel-Based Models.

Kernel-based models use covariates and kernels to construct data-driven prescriptions or uncertainty sets. The reviewed approaches connect local prediction, machine learning, and robust optimization, with reported asymptotic guarantees and tractability under conditions.

  • Kernel-based approaches use covariates associated with uncertain parameters and a kernel function to support data-driven modeling.
  • Bertsimas and Kallus learn local weights using methods including nearest-neighbors, kernel regression, local linear regression, trees, and random forests.Their framework produces predictive prescriptions based on observations similar to the current covariate.
  • Under mild conditions, the Bertsimas–Kallus model is polynomially solvable, asymptotically optimal, and consistent.
  • Directly applying SAA to observed uncertain parameters can produce decisions that are neither asymptotically optimal nor consistent.
  • Other reviewed methods combine predictive-model uncertainty with residual uncertainty, using robust sets, regularization, PCA, KDE, or SVC.Some frameworks use a Minkowski sum of predicted new-data outcomes and residuals, while PCA and KDE model correlated uncertainty.
  • For SVC-based robust models, the regularization parameter γ bounds the outlier fraction and links robust feasibility to an SAA chance constraint.The robust constraint max_{u∈U} u^⊤x ≤ b implies feasibility for P{ũ^⊤x ≤ b} ≥ 1−γ.

5.5. General Ambiguity Sets.

General ambiguity sets unify multiple forms of distributional information and support extensions across stochastic, integer, chance-constrained, and multistage optimization. The reviewed work develops representations, reformulations, approximation methods, and algorithms for these models.

  • A unified scenario-wise ambiguity-set format can represent both moment-based and discrepancy-based distributional information.
  • Generalized moments, mixture distributions, Wasserstein metrics, φ-divergences, and k-means clustering can be represented within the unified ambiguity-set framework.The framework introduces a discrete random variable representing finitely many scenarios affecting distributional ambiguity.
  • Decision-dependent ambiguity sets admit equivalent reformulations for moment inequalities, moment bounds, Wasserstein metrics, φ-divergences, and Kolmogorov–Smirnov tests.
  • Exchange methods can finitely converge when the ambiguity set is compact and convex and the risk measure is jointly continuous in the decision and distribution.
  • Reviewed extensions cover stochastic integer, disjunctive, chance-constrained bin-packing, binary, and multistage stochastic programs.The associated methods include convex reformulations, decomposition algorithms, binary bilinear reformulations, and an SDDP variant.
  • Randomization in mixed-integer DRO has value bounded by the difference between nominal DRO and its convex relaxation under convex cost and risk measure assumptions.

6. Calibration of the Ambiguity Set of Probability Distributions.

Calibration chooses nominal parameters and ambiguity-set size so data-driven DRO balances conservatism with coverage of the unknown distribution. The review emphasizes statistical guarantees, convergence, and practical dependence on data, dimension, confidence, and distributional assumptions.

  • Discrepancy-based and some moment-based ambiguity sets rely on nominal distributions or moments, such as P0, µ0, and Σ0.
  • Nominal distributions are commonly estimated empirically for discrete data or with kernel density estimation for continuous data.
  • Ambiguity-set size is controlled by robustness parameters such as ϵ, ϱ1, and ϱ2, whose calibration determines the level of conservatism.
  • Data-driven robustness parameters are often inversely proportional to sample size, motivated by convergence of DRO values toward the true-distribution model.These methods generally assume i.i.d. observations from the unknown distribution.
  • Wasserstein-based calibration can use probabilistic bounds involving sample size, confidence, dimension, and constants, while alternative methods avoid exogenous constants or use large-deviation results.
  • Some calibrated ambiguity sets contain the unknown distribution with high probability and provide probabilistic guarantees on the true optimal value.
  • Calibration can also connect regularization parameters to Wasserstein ambiguity size or use prices of optimism and pessimism to assess robustness.
  • For discrete distributions, proposed robustness levels depend on the support diameter, sample size, and confidence level, whereas continuous-case results retain practical issues.

7. Cost Function of the Inner Problem.

The paper reviews how DRO cost functions can encode regret, risk, utility, stochastic dominance, and weighted multiobjective preferences. These formulations extend uncertainty evaluation beyond expected cost and include applications and computational tools.

  • A regret functional quantifies expected additional loss or disappointment of a decision relative to possible future outcomes.
  • Distributionally robust newsvendor models minimize worst-case expected regret under support, mean, median, variance, and shape information.
  • Risk measures quantify uncertainty in decision outcomes, typically impose averseness, and order random variables consistently with their values.
  • Expected-utility and stochastic-dominance formulations compare decisions using utility functions or families of utility functions and cumulative-distribution functions.
  • Weighted scalarization extends comparison to random vectors, including models with stochastic weights and ambiguity in their probability distribution.
  • ROME is a MATLAB-based algebraic modeling toolbox for DRO with conic-representable support and mean sets, known covariance, and directional-deviation bounds.Its practicality is illustrated in inventory management, project crashing, and portfolio optimization.
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