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Quantile Inverse Optimization: Improving Stability in Inverse Linear Programming

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Quantile inverse optimization infers LP objective functions that stay stable under noisy, outlier-laden data, with guarantees that kept observations remain within a prescribed distance of the forward-optimal solution.

desk verdict A genuinely new quantile-based inverse LP framework with clever biclique algorithms, but Proposition 4's forward-stability guarantee is false as stated and the numerical support is thin. read the letter →

arxiv 1908.02376 v2 pith:CYRCXVGP submitted 2019-08-06 math.OC

Zahed Shahmoradi , Taewoo Lee This is my paper · ORCID
classification math.OC MSC 90C0590C1190C27
keywords inverseoptimizationlinearprogrammingquantileregressionmixed-integerbicliquestabilitymeasuresonlinelearningrobustestimation
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

Inverse linear programming infers the objective function of an optimization problem from observed decisions, but the cost vectors produced by standard methods can swing wildly when the observations are noisy or contain outliers. This paper introduces quantile inverse optimization (QIO), which keeps only the fraction θ of observations whose optimality error is no larger than a threshold τ and infers cost vectors from that subset. The paper defines formal inverse-stability and forward-stability measures, shows QIO is at least as stable as previous inverse-LP models, and proposes a mixed-integer formulation (MQIO) that maximizes the number of polytope facets spanned by the inferred cost vector. That maximum is a tractable proxy for forward stability: when n facets are selected, the resulting cost vector has a unique forward optimum within distance τ of every kept data point. The authors also connect MQIO to biclique problems, give an exact algorithm and heuristics, and demonstrate the method on diet recommendation and transshipment problems.

What carries the argument

The central machinery is the pair of binary variables (v for facets, u for data points) together with the quantile threshold τ and fraction θ. Constraint (6b) forces any kept observation to lie on all selected facets, and (6c) bounds its perturbation by τ; maximizing Σ v_i then selects the widest facet set consistent with the data. The biclique view—rows are observations, columns are facet normals, and feasible solutions are all-one submatrices with at least ⌈θK⌉ rows—is what lets the authors split the large MIP into per-observation feasibility checks and a smaller clique MIP, and it is also the route to the NP-hardness proof.

What would settle it

Take a two-dimensional forward polytope and place all observations near a single facet, with τ small enough that no other facet can be activated; then the MQIO optimum selects fewer than n facets, Proposition 4's guaranteed forward-stability bound does not apply, and the inferred cost vector can produce a forward solution far from the data. Similarly, reorder any dataset so that an outlier is first; if the exact algorithm then reports u*_1 = 0, its completeness guarantee fails unless a different seeding order is used.

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Extended reading notes

Core claim

The central discovery is a characterization of all inverse-feasible cost vectors for a quantile threshold: a cost vector c makes at least ⌈θK⌉ observations within τ of optimality if and only if it lies in the conic hull of the facet normals ai whose indicator variables are 1 in a feasible solution to the mixed-integer feasibility problem (6), with the data-subset indicators u marking the kept observations (Theorem 1). Building on this, Proposition 4 shows that if a feasible solution activates exactly n facets, then any strict conic combination c of those normals makes the forward LP have a unique optimal solution, and that solution is within τ of every kept data point. The paper's working strategy is therefore to maximize the number of activated facets (the MQIO objective), which Proposition 5 shows cannot hurt either forward or inverse stability, and which numerical experiments confirm brings forward solutions close to the trusted data.

Load-bearing premise

The stability and exactness guarantees depend on the MQIO solution selecting exactly n facets that bound a unique point, and on the first data point being one of the kept points; on degenerate or highly noisy datasets those conditions may not hold.

Editorial extensions

If this is right

  • If MQIO's optimal solution activates all n facets, every cost vector in the returned conic set yields a unique forward solution within distance τ of all retained data points—the forward-stability guarantee is exact, not heuristic.
  • Lowering θ—allowing the model to discard more outliers—monotonically increases inverse stability, so the user can trade fit against tolerance by tuning one parameter.
  • The QIO feasible set contains the cost vectors of the standard inverse-LP model (2) for a suitable τ, so the new method is a strict generalization rather than a replacement.
  • The online-learning extension converges to a cost-vector set that includes the true cost vector in the transshipment experiments, suggesting the method can track drifting preferences.
  • Because maximizing the number of selected facets improves both stability measures, the MQIO objective is a principled surrogate for the hard-to-compute forward-stability measure.

Reading between the lines

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

  • The same biclique machinery could be adapted to inverse problems over other polyhedral sets—e.g., inverse conic or mixed-integer programs—by replacing facet normals with the appropriate extreme rays, though the uniqueness guarantee would need rederivation.
  • The exact algorithm's condition that the first data point be kept (u*_1 = 1) is only checked after the run; a natural robustness upgrade would be to run the seed selection on several candidate points in parallel and take the best, which the paper's own cut-based enumeration already hints at.
  • The forward-stability bound is stated for the kept subset; a practitioner could treat the unkept fraction as outliers and validate the chosen τ on held-out data, giving a cross-validated way to set τ.
  • Since the inverse-stability lower bound grows as θ shrinks, one could combine the method with a breakdown-point argument to choose θ adaptively—matching the paper's suggested interpretation of θ as an outlier fraction.
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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

1 major / 5 minor

Summary. The paper introduces a quantile-based inverse linear programming (QIO) method and its mixed-integer reformulation (MQIO) to address instability of inverse LP under data noise and outliers. It formalizes inverse and forward stability measures, characterizes the set of inverse-feasible cost vectors through conic combinations of constraint normals (Theorem 1), establishes a connection to biclique problems, and proposes an exact algorithm and two heuristics. Numerical experiments are reported on randomly generated LPs, a diet recommendation problem, and a transshipment problem, with comparisons against a previous inverse LP model.

Significance. The quantile idea is a reasonable adaptation of robust regression to inverse LP, and the biclique connection provides a useful algorithmic perspective. The paper contains several formal propositions and a reproducible algorithmic framework, which are strengths. However, the central forward-stability guarantee (Proposition 4) is false as stated, which undermines the paper's main claim of providing forward-stable cost vectors. The inverse-stability monotonicity (Proposition 1) is largely a direct consequence of the model's definition rather than an independent property, and the numerical evaluation is entirely synthetic. If the theoretical gaps are corrected, the framework could be a valuable contribution to robust inverse optimization.

major comments (1)
  1. [Section 3.1, Proposition 1] The inverse-stability monotonicity result is a direct consequence of the fact that decreasing θ enlarges the feasible set of cost vectors, as the proof in Appendix C demonstrates via set inclusion. While mathematically correct, this does not by itself establish that the MQIO model is 'more stable' than the previous inverse LP model in a substantive sense; the reported improvement in inverse stability is partially built into the model definition. The numerical experiments in Section 5.2.2 are the appropriate evidence, and the paper should present Proposition 1 accordingly.
minor comments (5)
  1. [Example 4 (page 19)] The cost vector is written as 0.5a1 + 0.5a1; this appears to be a typo and should read 0.5a1 + 0.5a2.
  2. [Section 2.4] The symbol X is used both for the feasible region of the forward problem and for the shifted dataset in the definitions of inverse and forward stability (e.g., in equations (3) and (4)). This creates notational confusion; a different symbol for the dataset, such as \mathcal{X}, would help.
  3. [Appendix C, proof of Proposition 3] The use of an infinitesimal constant σ in \bar{τ}=τ+σ is informal; a limiting argument with σ followed by σ↓0 would be more rigorous.
  4. [Table 1] Several rows appear to have a missing objective value for the ˜D-Alg-Heuristic column (for example, the row for (n,m,K)=(50,300,500) with τ=3.0). Please check the table entries for completeness and alignment.
  5. [Section 5.2.1] The forward-stability experiment samples 50 cost vectors from each cone C_h rather than computing the worst-case distance over the entire cone. The claim that the distance is 'eventually within the threshold' should be framed as an empirical observation, not a worst-case guarantee.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QIO/MQIO derivation chain is self-contained, with user-chosen thresholds and no fitted parameter renamed as a prediction.

full rationale

The paper's central model is a feasibility/MIP formulation in which tau and theta are user-specified inputs; no parameter is fitted to the stability outcome and then relabeled as a prediction. Proposition 1 is a direct set-inclusion consequence of lowering theta, but it is not a hidden equivalence: the inverse-stability measure (3) is defined independently of the model, and the paper explicitly states that decreasing theta expands the feasible region before deriving the monotonicity. Proposition 4's forward-stability bound is conditional on the model selecting n facets and follows from the model's own constraints (6b)-(6c) together with a uniqueness claim; whether the required vertex intersection actually holds in degenerate polytopes is a correctness and nondegeneracy concern, not a circular reduction. The uniqueness citation to Mangasarian is standard external support, and the coauthored Tavaslioglu et al. citation is paired with that independent source rather than being the sole load-bearing justification. The exactness of Algorithm 2 is honestly conditional on the post-hoc check u*_1 = 1, and the numerical comparisons are run against the previous model on generated data rather than renaming a fitted quantity as an independent result. No derivation in the paper reduces by construction to its own inputs, so there is no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model depends on user-selected theta, tau, and norms rather than fitted constants. The main mathematical assumptions are standard LP duality and the fixed-polyhedron, noisy-data setting. There are no invented physical entities.

free parameters (3)
  • theta (quantile fraction)
    User-specified fraction of observations that must be within the error threshold; experiments use 0.75, 0.85, and 1. It directly controls the size of the inverse-feasible set and the inverse-stability claims.
  • tau (error threshold)
    User-specified distance tolerance for kept observations; experiments use values such as 3, 3.5, and 4, or derive a minimal value via problem (10). Forward and inverse stability bounds depend on it.
  • norm parameters l and p
    Distance and normalization norms chosen by the user, typically infinity in the numerical examples; they define the geometry of Xi distances and the cost normalization.
assumptions (4)
  • domain assumption X is bounded, full-dimensional, and free of redundant constraints; rows a_i are normalized
    Stated at the start of Section 2.1 and used throughout the facet-based analysis and in the proof of Proposition 4.
  • domain assumption The forward LP is the same for all observations; only b or A may vary in extensions
    Section 1 states the paper focuses on fixed constraint matrix A with varying observed decisions; extensions in Appendix A.5 relax this.
  • domain assumption Data noise is modeled as pointwise shifts of observed decisions
    Assumed in Section 3.1.2 and Appendix A.3 to define the inverse stability measure and lower bound.
  • ad hoc to paper Selected n facets define a unique optimal point (nondegenerate vertex)
    Implicit in the proof of Proposition 4, which asserts X*(c) is a singleton whenever |I| = n; not stated as an assumption and not guaranteed for degenerate polytopes.

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Cite this review

Pith. "Pith review of Quantile Inverse Optimization: Improving Stability in Inverse Linear Programming." pith.science (2026). https://pith.science/paper/CYRCXVGP

@misc{pith2026190802376,
  author       = {Pith},
  title        = {Pith review of: Quantile Inverse Optimization: Improving Stability in Inverse Linear Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYRCXVGP}},
  note         = {Machine review of arXiv:1908.02376}
}
read the original abstract

Inverse linear programming (LP) has received increasing attention due to its potential to generate efficient optimization formulations that can closely replicate the behavior of a complex system. However, inversely inferred parameters and corresponding forward solutions from the existing inverse LP method can be highly sensitive to noise, errors, and uncertainty in the input data, limiting its applicability in data-driven settings. We introduce the notion of inverse and forward stability in inverse LP and propose a novel inverse LP method that determines a set of objective functions that are stable under data imperfection and generate solutions close to the relevant subset of the data. We formulate the inverse model as a mixed-integer program and elucidate its connection to bi-clique problems, which we exploit to develop efficient heuristics. We also show how this method can be used for online learning. We numerically evaluate the stability of the proposed method and demonstrate its practical use in the diet recommendation and transshipment applications.

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