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REVIEW 4 major objections 6 minor 24 references

Data-Driven Adjustable Robust Optimization

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes a two-stage data-driven method that shrinks the uncertainty set until a robustly feasible decision exists, then optimizes over the reduced set.

desk verdict The non-stochastic half is a clean, modest contribution; the stochastic half optimizes the wrong objective, so the advertised distributionally robust guarantees do not hold as written. read the letter →

arxiv 2505.22075 v1 pith:HSMCGJF2 submitted 2025-05-28 math.OC

Xiaoxing Ren , Alessio Moreschini , Zhongda Chu , Yulong Gao , Thomas Parisini This is my paper
classification math.OC MSC 90C4790C1590C90
keywords data-drivenrobustoptimizationadjustableuncertaintysetWassersteinambiguitydistributionallyfeasibility-firstoptimalpowerflowfinitereformulationquantification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a two-stage, data-driven approach to adjustable robust optimization. First it selects a smaller uncertainty set $\hat{S}\subseteq S$, built from observed samples of the uncertain parameter, such that the robust constraint holds for every $\xi\in\hat{S}$; second it solves the cost-minimization problem over that selected set. For non-stochastic uncertainty, the selection is formulated as a scaling problem over common set families, and the penalty term keeps as many samples as possible inside the set. For stochastic uncertainty with unknown distribution, the paper computes the probability that $\xi$ lies in $\hat{S}$ under the worst (or best) distribution inside a Wasserstein ambiguity set around the empirical distribution, and for polytopic sets the selection becomes a finite program. The approach is demonstrated on a numerical example and on an optimal power flow problem with uncertain inertia.

What carries the argument

The central object is the scaled adjustable uncertainty set $\hat{S}(\alpha,v)=\{\xi: V\xi \le \alpha\mathbf{1}+(1-\alpha)Vv,\ Vv\le \mathbf{1},\ \alpha\in[0,1]\}$, where $\alpha$ shrinks the base polytope $\{V\xi\le\mathbf{1}\}$ toward a vertex $v$; the same parametric shrink is used for norm balls, cones, and semidefinite-representable sets. The second load-bearing mechanism is the Wasserstein ambiguity set $B_\epsilon(P_N)$, the ball of distributions within transportation distance $\epsilon$ of the empirical distribution of the samples. Lemma 1 uses the dual formulation of Wasserstein chance constraints to evaluate $\max_{Q\in B_\epsilon(P_N)}Q(\hat{\xi}\notin\hat{S})$ and $\max_{Q\in B_\epsilon(P_N)}Q(\hat{\xi}\in\hat{S})$ as finite optimizations, and Proposition 1 embeds those evaluations into the set-selection problem so that stage one is a finite program for polytopic uncertainty.

What would settle it

Re-run the numerical example of Section IV.A.2 and the power-flow study with the stage-one objective replaced by the worst-case unsafe probability $\min \sup_{Q\in B_\epsilon(P_N)}Q(\xi\notin\hat{S})$; if the chosen uncertainty sets are materially smaller or no feasible set exists for the same $N$ and $\epsilon$, the reported feasibility-preserving behavior depends on the optimistic objective and does not extend to a worst-case safety guarantee.

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Extended reading notes

Core claim

The paper's claim is that feasibility should be treated as a design variable: when the original robust problem is infeasible under the full uncertainty set $S$, the decision-maker should choose the largest subset $\hat{S}\subseteq S$ for which a robustly feasible decision exists, using data to decide where to cut. The objective of this first stage is $\max_{x,\hat{S}}\mu(\hat{S})+\gamma\sum_{i=1}^{N}\mathbb{I}(\hat{\xi}_i\in\hat{S})$, where $\mu$ ranks sets and the penalty rewards covering the observed samples. In the stochastic case the same logic is applied with $\mu(\hat{S})$ replaced by an uncertainty-quantified probability over a Wasserstein ball; Lemma 1 converts the worst-case unsafe probability and the best-case safe probability into finite dual programs when $\hat{S}$ is a polytope, and Proposition 1 builds the stage-one reformulation for scaled polytopic sets. The paper asserts that this construction enlarges the feasible state set at the expense of a possible constraint violation, and reports concrete adjusted uncertainty sets and solutions for the power flow case.

Load-bearing premise

The stochastic stage chooses the uncertainty set by maximizing the best-case safe probability $\sup_{Q\in B_\epsilon(P_N)}Q(\xi\in\hat{S})$; because this is an optimistic upper bound, the claimed safety performance rests on using that particular objective rather than the worst-case probability of violation.

Editorial extensions

If this is right

  • A robust problem that is infeasible under the original $S$ can still admit a decision once $\hat{S}$ is chosen as the largest feasible data-supported subset.
  • The same scaling trick applies across the standard uncertainty-set families, so the method is not tied to a single geometry.
  • For polytopic uncertainty and Wasserstein ambiguity, the first-stage set selection is a finite optimization problem rather than an infinite-dimensional one.
  • In the optimal power flow case, the uncertainty in inertia can be narrowed to an interval that preserves the frequency constraints and yields a feasible dispatch.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: because the stochastic stage maximizes the best-case safe probability, the reported uncertainty sets are optimistic; replacing that objective with the worst-case unsafe probability would likely produce smaller sets for the same samples and radius.
  • Beyond the paper: iterating the two stages, rather than stopping after one shrinkage, could converge to a minimal uncertainty set consistent with a required safety level; the paper does not analyze such a fixed point.
  • Beyond the paper: a natural testable extension is to use the closed-polytope form of Lemma 1 to enforce a lower bound on the worst-case safe probability, turning the method into a calibrated chance-constrained design rather than an optimism-based one.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a two-stage data-driven adjustable robust optimization method. In the first stage, the decision maker selects a subset of a nominal uncertainty set so that a robustly feasible decision exists while the subset is as large as possible and covers as many training samples as possible; in the second stage, the original objective is minimized over that selected subset. For non-stochastic uncertainty, the paper derives reformulations for norm-balls, ellipsoids, polyhedral sets, cone-based sets, and semi-definite representable sets. For stochastic uncertainty with unknown distribution, the paper uses a Wasserstein ambiguity set around the empirical distribution and invokes results from Mohajerin Esfahani and Kuhn to reformulate the stage-one problem, claiming a finite program for polytopic uncertainty sets. The proposed approach is illustrated on a simple numerical example and on an optimal power flow problem.

Significance. If the stochastic formulation were corrected, the paper would make a useful contribution: the non-stochastic part is a coherent and practically motivated way to trade off feasibility against sample coverage, and the power-system case study is a relevant application. The paper also draws a clear connection between adjustable robust optimization and data-driven set construction for several common uncertainty representations. However, the stochastic half currently uses an optimistic safety objective, and the finite-program reformulation contains inconsistencies, so the claimed distributionally robust guarantees are not established as written. These are correctable but load-bearing issues.

major comments (4)
  1. [§III, Eq. (13)-(15) and §IV.A.2, §IV.B.2] The stochastic stage uses the best-case safe probability max_{Q in B_epsilon(P_N)} Q(xi in S_hat) as the objective in the numerical experiments. This quantity is an optimistic upper bound: it can be large even if most distributions in the ambiguity ball make the selected set unsafe, and maximizing it does not control the violation probability under the true distribution or under an adversarial Q. The appropriate distributionally robust objective is to minimize the worst-case unsafe probability (9), whose finite reformulation is (14). As the experiments in Section IV.A.2 and IV.B.2 maximize the best-case branch, the reported uncertainty sets do not support the claimed performance guarantees for a robust scheme. The authors should rerun the stochastic experiments with the worst-case objective, or clearly reframe the contribution as a best-case analysis without DRO guarantees.
  2. [Proposition 1, Eq. (14)] The finite-program reformulation (14) is not well defined as written. The first constraint contains [1 + (1-alpha)V v]_l, but the proof in Eq. (16) and the substitution b = alpha 1 + (1-alpha)V v require [alpha 1 + (1-alpha)V v]_l. In addition, the dual-norm constraint uses a_l and C, which are not defined in the proposition; after substitution they should be V_l and V, respectively. The same applies to (15), where the matrix A in the dual-norm constraint is undefined. Until these substitutions are corrected, the proposition does not provide a valid finite reformulation.
  3. [Section III and Proposition 1, 'finite program' claim] The abstract and introduction state that for polytope uncertainty sets the problem can be reformulated as a finite program. However, the reformulations (14)-(15) only give a finite representation of the probability term; the constraint f(x, xi) <= 0 for all xi in S_hat remains a semi-infinite constraint. No general robust-counterpart reformulation for this constraint is provided in the proposition. To substantiate the finite-program claim, the authors need to impose and state additional structure on f (such as concavity in xi, as hinted in Remark 2) and give the corresponding tractable reformulation, or qualify the claim accordingly.
  4. [Lemma 1 and Proposition 1] The application of Lemma 1 to Proposition 1 is not consistent with the set assumptions. Lemma 1 states that formula (12) applies to a closed safety set S_hat = {xi : A xi <= b}, while Proposition 1 defines S_hat(alpha, v) = {xi : V xi < alpha 1 + (1-alpha)V v} with strict inequalities, i.e., an open set. The paper uses (12) in (15) for this open set without comment. The difference between open and closed polytopes matters for Wasserstein ambiguity set probabilities and for whether the reformulation is valid; the authors should either use the appropriate open-set formula or justify the substitution.
minor comments (6)
  1. [Eq. (8)] The text preceding (8) uses the symbol rho in the reformulation S = {xi : rho I - B(xi) >= 0}, but the constraint in (8) uses alpha I - B(xi) >= 0. Please unify the notation.
  2. [Eq. (6) and Prop. 1] The dimensions of V are inconsistent across the paper: in (6) V is described as an m by n matrix and 1 is an n-vector, while in Proposition 1 (14) the same V appears with inequalities V xi <= alpha 1 + (1-alpha)V v and V v <= 1, and the number of rows is later identified with L. Please align the notation and state the dimensions consistently.
  3. [Remark 2] The statement that adding the indicator function will not influence the solution under the assumption that a larger alpha gives a larger set is valid for the families (5), (7), and (8), but it is not generally true for the polyhedral family (6), where the set depends on both alpha and v and monotonicity in alpha is not guaranteed. Please qualify the remark.
  4. [Section IV.A.1] The solution is reported as v* = 10, but v is a vector in R^2 for the two-dimensional example. This appears to be a typo; please report the full vector.
  5. [Table I] The table reports 'new uncertainty set' intervals for different N and epsilon, but it does not state which stage-one objective (best-case or worst-case) was used. Since the text says the experiments use the best-case probability, this should be stated in the caption so the reader can interpret the numbers.
  6. [Section IV.B] The case study is described as a modified IEEE 5-bus system, but reference [23] points to a 14-bus test case archive. Please provide a specific citation for the 5-bus data or clarify the modification sources.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the two-stage programs are self-contained optimizations, and the Wasserstein duality is imported from an external theorem rather than from the paper's own assumptions.

full rationale

The derivation chain is self-contained and no load-bearing step reduces to its own inputs. The first-stage program (3) maximizes μ(Ŝ)+γΣ_i I(ξ̂_i∈Ŝ) subject to f(x,ξ)≤0 for all ξ∈Ŝ; this is a well-defined optimization over (x,Ŝ), and the second stage (4) solves a standard robust problem on the optimizer Ŝ. The non-stochastic specializations (5)-(8) are direct scalings of norm-ball, polyhedral, cone, and SDP-representable sets; the sample penalty is an extra term, not a hidden fit. The stochastic stage imports the Wasserstein duality formulas (11)-(12) from the external reference [21, Cor. 5.3] (Esfahani and Kuhn), and Proposition 1 substitutes the parametric polytope Ŝ(α,v) into those formulas. That substitution is derivativeness, not circularity: the reformulated finite program is obtained by epigraph manipulation of an external duality theorem, and the optimal α and v are the outcome of the program, not pre-imposed. The self-references [16]-[18] are related-work comparisons and are not used to justify the main result. Two non-circular technical concerns should be weighed separately: Eq. (14) appears to contain a typo, writing [1+(1−α)Vv]_l where Lemma 1 requires [α1+(1−α)Vv]_l, and A and C are not explicitly defined in Proposition 1; moreover, the stochastic experiments maximize the best-case safe probability sup_Q Q(ξ∈Ŝ) (Eqs. (21) and IV.B.2), which is an optimistic quantity and does not by itself certify a low worst-case violation probability. These are correctness or modeling risks, not circular reductions, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The framework is not built on fitted constants; the set parameters alpha and v are decision variables. The hand-picked tuning parameters are gamma and epsilon. The stochastic safety objective is not a fitted quantity, but it is mis-specified as an upper bound rather than a guarantee. No new physical or mathematical entities are introduced.

free parameters (2)
  • gamma (sample-coverage penalty weight) = not reported
    The objective (3), (20), and (23) mixes set size alpha with sample count, and gamma sets the trade-off. Its value is never stated or varied in the experiments.
  • Wasserstein radius epsilon = 0.05 and 0.1 in experiments
    Chosen by hand for Section IV; no calibration rule, cross-validation, or sensitivity analysis is provided.
assumptions (3)
  • domain assumption The uncertainty samples are i.i.d. from an unknown distribution P when the uncertainty is stochastic, and the empirical distribution with a Wasserstein ball is a valid model for P.
    Used in Section III to justify B_epsilon(P_N) and the uncertainty quantification formulas (9)-(12).
  • domain assumption Lemma 1's conditions hold for the optimized sets: the open or closed polytope safety set has nonempty intersection with S, and each defining half-space has nonempty intersection with S.
    Invoked in Proposition 1 without proof. If these conditions fail for the parameterized sets S_hat(alpha,v), formulas (11) and (12) are not valid.
  • domain assumption The function f(x,.) is concave for each x and f(.,xi) is convex for each xi, so the robust counterpart can be reformulated tractably.
    Remark 2 relies on this to make Stage 2 tractable. The paper's experiments use linear constraints, but Proposition 1 is stated for general f without showing the robust dual.

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Cite this review

Pith. "Pith review of Data-Driven Adjustable Robust Optimization." pith.science (2026). https://pith.science/paper/HSMCGJF2

@misc{pith2026250522075,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Adjustable Robust Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSMCGJF2}},
  note         = {Machine review of arXiv:2505.22075}
}
read the original abstract

In this paper, we develop a two-stage data-driven approach to address the adjustable robust optimization problem, where the uncertainty set is adjustable to manage infeasibility caused by significant or poorly quantified uncertainties. In the first stage, we synthesize an uncertainty set to ensure the feasibility of the problem as much as possible using the collected uncertainty samples. In the second stage, we find the optimal solution while ensuring that the constraints are satisfied under the new uncertainty set. This approach enlarges the feasible state set, at the expense of the risk of possible constraint violation. We analyze two scenarios: one where the uncertainty is non-stochastic, and another where the uncertainty is stochastic but with unknown probability distribution, leading to a distributionally robust optimization problem. In the first case, we scale the uncertainty set and find the best subset that fits the uncertainty samples. In the second case, we employ the Wasserstein metric to quantify uncertainty based on training data, and for polytope uncertainty sets, we further provide a finite program reformulation of the problem. The effectiveness of the proposed methods is demonstrated through an optimal power flow problem.

Figures

Figures reproduced from arXiv: 2505.22075 by the authors.

Figure 1
Figure 1. Illustration of the feasible and unfeasible regions of the problem [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Motivating example for (3). On the left-hand side, we depict the problem with its original uncertainty set and the randomly generated uncertainty samples. On the right-hand side, we show the new uncertainty set that contains the most uncertainty samples while ensuring feasibility. III. DATA-DRIVEN DISTRIBUTIONALLY ADJUSTABLE ROBUST OPTIMIZATION In this section, we consider the case where the uncertainty ξ is stochas… view at source ↗
Figure 3
Figure 3. IEEE-5-bus-system where the 500 uncertainty samples ˆξi=1,...,500 are randomly generated from S = {ξ ∈ R 2 | Bξ ≤ d}. The solution of (20) is α ∗ = 3 4 , v ∗ = 10, the new uncertainty set is Sˆ = {ξ ∈ R 2 | Bξ ≤ [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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