Lipschitz modulus of linear and convex systems with the Hausdorff metric
Pith reviewed 2026-05-25 09:15 UTC · model grok-4.3
The pith
An explicit formula computes the Lipschitz modulus of the feasible set mapping for linear systems when parameters are closed coefficient sets measured by Hausdorff distance.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
By identifying the parameter space with the collection of closed sets of coefficient vectors in R^{n+1} equipped with the Hausdorff metric, and by means of a suitable indexation, an explicit formula for the Lipschitz modulus of the feasible-set map is obtained. This transfers prior stability results from the Chebyshev setting. The approach also yields Lipschitz modulus estimates for convex systems once their coefficient vectors are allowed to vary.
What carries the argument
An indexation strategy that converts Hausdorff perturbations of coefficient sets into Chebyshev perturbations on an indexed family, thereby transferring prior modulus formulas.
If this is right
- The Lipschitz modulus of feasible sets can be calculated directly from the coefficient data.
- Stability results known for fixed-index Chebyshev perturbations extend to variable-index Hausdorff perturbations.
- Convex feasible sets inherit explicit Lipschitz estimates from their linearized versions.
- Finite and semi-infinite linear systems receive a uniform modulus formula.
Where Pith is reading between the lines
- Numerical evaluation of stability margins for semi-infinite programs becomes feasible once the Hausdorff distance between coefficient sets can be approximated.
- Other set-valued distances on parameters might be related by analogous re-indexing constructions.
- Robust optimization bounds on solution changes could be derived when uncertainty is quantified by set distance rather than componentwise error.
Load-bearing premise
Closed subsets of coefficient vectors in R^{n+1} admit an indexation that preserves the relation between Hausdorff and Chebyshev distances.
What would settle it
A linear system together with a sequence of coefficient-set perturbations whose Hausdorff size tends to zero, yet the ratio of feasible-set distances to perturbation size exceeds the value given by the proposed formula.
read the original abstract
This paper analyzes the Lipschitz behavior of the feasible set in two parametric settings, associated with linear and convex systems in R^n. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of R^(n+1). In this framework, where the Hausdorff distance is used to measure the size of perturbations, an explicit formula for computing the Lipschitz modulus of the feasible set mapping is provided. As direct antecedent, we appeal to its counterpart in the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (pseudo) distance was considered to measure the perturbations. Indeed, the stability (and, particularly, Lipschitz properties) of linear systems in the Chebyshev framework has been widely analyzed in the literature. Here, through an appropriate indexation strategy, we take advantage of previous results to derive the new ones in the Hausdorff setting. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system allows us to provide new contributions on the Lipschitz behavior of convex systems via linearization techniques.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to derive an explicit formula for the Lipschitz modulus of the feasible-set mapping for linear (finite and semi-infinite) systems when the parameter space consists of closed subsets of coefficient vectors in R^{n+1} equipped with the Hausdorff metric; the derivation proceeds by an indexation strategy that reduces the problem to prior results obtained under the Chebyshev metric on a fixed index set T. It further extends the approach to convex systems by linearization.
Significance. If the indexation is shown to preserve distances isometrically, the explicit formula would supply a concrete computational device for stability radii in the Hausdorff setting that is directly applicable to semi-infinite programming; the reduction from an antecedent Chebyshev result is a strength when the transfer is rigorously justified.
major comments (2)
- [Abstract and the section introducing the indexation strategy] The central claim that the Hausdorff modulus equals the transferred Chebyshev modulus rests on the existence of an indexation satisfying d_H(S,S') = d_Cheby(index(S),index(S')) while preserving the feasible-set mapping exactly. The manuscript must supply the explicit construction of this indexation (or prove its existence uniformly for arbitrary closed S) together with a verification that no extra slack is introduced; without this step the formula is not guaranteed to be the true Hausdorff modulus.
- [The part deriving the formula for linear systems] For semi-infinite systems the reduction must hold when the index set T is infinite; the paper should state the precise conditions on the closed coefficient sets under which the indexation remains bijective and distance-preserving, because any dependence of the indexing on the particular S would make the modulus non-uniform.
minor comments (2)
- Notation for the identification of the parameter space with closed subsets of R^{n+1} should be introduced once and used consistently.
- Add a short remark clarifying whether the formula reduces to a known expression when the coefficient set is finite.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and the constructive major comments. We agree that the indexation strategy requires a more explicit and rigorous treatment to justify the transfer of the Lipschitz modulus results. We will revise the paper to supply the requested construction, verifications, and precise conditions. Our point-by-point responses follow.
read point-by-point responses
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Referee: [Abstract and the section introducing the indexation strategy] The central claim that the Hausdorff modulus equals the transferred Chebyshev modulus rests on the existence of an indexation satisfying d_H(S,S') = d_Cheby(index(S),index(S')) while preserving the feasible-set mapping exactly. The manuscript must supply the explicit construction of this indexation (or prove its existence uniformly for arbitrary closed S) together with a verification that no extra slack is introduced; without this step the formula is not guaranteed to be the true Hausdorff modulus.
Authors: We accept the observation. The manuscript introduces the indexation at a conceptual level but does not furnish the explicit map or the isometric verification. In the revised version we will insert a new subsection that constructs the indexation explicitly: each closed set S is mapped to a function on a fixed countable dense subset T of the unit sphere in R^{n+1}, with the value at each t in T given by the supporting hyperplane functional associated with the nearest point in S. We will prove that this map is isometric with respect to the Hausdorff and Chebyshev distances and that the feasible-set mapping is identical, introducing no additional slack. revision: yes
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Referee: [The part deriving the formula for linear systems] For semi-infinite systems the reduction must hold when the index set T is infinite; the paper should state the precise conditions on the closed coefficient sets under which the indexation remains bijective and distance-preserving, because any dependence of the indexing on the particular S would make the modulus non-uniform.
Authors: We agree that the conditions must be stated explicitly. The indexation is built from a single fixed dense countable set T that does not depend on any particular S. In the revision we will add a precise statement: the result holds for every nonempty closed subset S of R^{n+1} (bounded or unbounded), with bijectivity and distance preservation following from the density of T and the closedness of S. This guarantees uniformity of the modulus. We will also note that compactness of S is not required. revision: yes
Circularity Check
No circularity; explicit formula derived by transferring external Chebyshev results via indexation
full rationale
The paper states that it appeals to antecedent results on Lipschitz properties in the Chebyshev framework, which it describes as widely analyzed in the literature, and applies an indexation strategy to obtain the Hausdorff-metric formula. No quoted step shows a self-definitional reduction, a fitted parameter renamed as a prediction, or a load-bearing claim justified solely by overlapping-author citations whose content itself reduces to the target result. The central derivation is presented as building on independent external benchmarks rather than re-expressing its own inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Parameter space of linear systems identified with closed subsets of coefficient vectors in R^{n+1}
- domain assumption Appropriate indexation strategy exists that allows transfer of Chebyshev results to Hausdorff setting
Reference graph
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discussion (0)
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