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arxiv: 2606.08386 · v1 · pith:T6IYGR4Enew · submitted 2026-06-07 · 🧮 math.OC · math.FA

Variational Analysis of Metric Projections onto Isotone Projection Cones via Coderivatives

Le Van Hien This is my paper

Pith reviewed 2026-06-27 18:19 UTC · model grok-4.3

classification 🧮 math.OC math.FA
keywords metric projectionisotone projection coneFréchet coderivativeMordukhovich coderivativecovering constantAubin propertynonlinear complementarity problemvariational analysis
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The pith

Explicit formulas for the Fréchet and Mordukhovich coderivatives of metric projections onto isotone projection cones follow from a local generating-system description.

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

The paper derives explicit formulas for both the Fréchet coderivative and the Mordukhovich coderivative of the metric projection operator onto isotone projection cones in finite-dimensional Euclidean spaces. The derivation rests on a local description of the projection mapping by an associated generating system in a neighborhood of any reference point. These characterizations are then used to compute the covering constant of the projection and to obtain verifiable sufficient conditions for the Aubin property of solution mappings to parametric nonlinear complementarity problems built on such cones. A sympathetic reader cares because the formulas supply concrete, computable tools for regularity and stability analysis in optimization settings where classical orthogonality arguments are unavailable.

Core claim

The Fréchet coderivative and the Mordukhovich coderivative of the projection operator onto an isotone projection cone admit explicit, computable characterizations obtained from a local description of the projection mapping via its associated generating system near a reference point; these characterizations yield the covering constant of the projection and verifiable conditions ensuring the Aubin property of the solution map for associated parametric nonlinear complementarity problems.

What carries the argument

Local description of the projection mapping via an associated generating system in a neighborhood of a reference point, which produces the explicit coderivative formulas.

If this is right

  • The covering constant of the projection mapping onto an isotone projection cone becomes directly computable from the coderivative formulas.
  • Verifiable sufficient conditions for the Aubin property of the solution mapping to parametric nonlinear complementarity problems can be stated in terms of the same local generating system.
  • The coderivative analysis applies to variational properties of metric projections in general cone settings without relying on orthogonality.
  • The results supply quantitative local regularity information for the projection operator itself.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar local generating-system techniques might produce coderivative formulas for projections onto other classes of cones or polyhedral sets.
  • The explicit formulas could be inserted into existing numerical schemes for checking metric regularity or computing stability radii in complementarity problems.
  • If the local description extends beyond finite dimensions, the same coderivative approach might apply to infinite-dimensional complementarity systems.

Load-bearing premise

The projection mapping admits a local description via an associated generating system in a neighborhood of the reference point that directly supplies the coderivative expressions.

What would settle it

An explicit counter-example cone and reference point where the stated coderivative formulas fail to equal the actual limiting coderivative of the projection operator.

read the original abstract

In this paper, we study variational properties of the metric projection mapping onto isotone projection cones in finite-dimensional Euclidean spaces. We derive explicit formulas for both the Fr\'echet coderivative and the Mordukhovich coderivative of the projection operator. The analysis is based on a local description of the projection mapping via an associated generating system in a neighborhood of a reference point, which leads to computable coderivative characterizations. As an application, we compute the covering constant of the projection mapping, providing a quantitative description of its local regularity. Furthermore, we establish verifiable sufficient conditions for the Aubin property of the solution mapping associated with parametric nonlinear complementarity problems associated with isotone projection cones. The obtained results contribute to the variational analysis of metric projections in a general cone setting where orthogonality arguments are not available, and to the stability theory of complementarity systems in finite dimensions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper studies variational properties of metric projections onto isotone projection cones in finite-dimensional Euclidean spaces. It derives explicit formulas for the Fréchet and Mordukhovich coderivatives of the projection operator based on a local description via an associated generating system near a reference point, leading to computable characterizations. Applications include computing the covering constant of the projection and establishing sufficient conditions for the Aubin property of solution mappings to parametric nonlinear complementarity problems associated with such cones.

Significance. If the local generating system description is valid and the coderivative formulas are correctly derived without hidden assumptions, the work would extend variational analysis of projections to general cone settings lacking orthogonality and provide quantitative tools for stability in finite-dimensional complementarity systems.

major comments (2)
  1. [Abstract] Abstract: the central claim of explicit Fréchet and Mordukhovich coderivative formulas rests on the asserted local description of the projection via a generating system, but the provided text contains no derivations, existence proof for the generating system, or verification that it captures the projection's local behavior for arbitrary isotone cones without extra regularity assumptions on facial structure or reference point location; this makes the formulas unverifiable and the applications unsupported.
  2. The weakest assumption (local generating system description enabling explicit coderivatives) is load-bearing for all claimed results; without concrete checks or counterexample-free arguments in the manuscript, the explicit characterizations and the covering constant computation do not follow in the stated generality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on our manuscript. Below we address the major comments point by point, clarifying the content of the full paper.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of explicit Fréchet and Mordukhovich coderivative formulas rests on the asserted local description of the projection via a generating system, but the provided text contains no derivations, existence proof for the generating system, or verification that it captures the projection's local behavior for arbitrary isotone cones without extra regularity assumptions on facial structure or reference point location; this makes the formulas unverifiable and the applications unsupported.

    Authors: The abstract summarizes the main results at a high level. The full manuscript contains the required derivations, existence proof, and verification: Section 3 introduces the local generating system associated with an isotone projection cone, proves its existence in a neighborhood of any reference point, and establishes that the description holds for arbitrary isotone cones without additional regularity assumptions on facial structure. The explicit coderivative formulas are then obtained in Section 4 directly from this description. The applications in Section 5 follow from these formulas. If the referee prefers, we can add a brief reference to these sections in a revised abstract. revision: partial

  2. Referee: [—] The weakest assumption (local generating system description enabling explicit coderivatives) is load-bearing for all claimed results; without concrete checks or counterexample-free arguments in the manuscript, the explicit characterizations and the covering constant computation do not follow in the stated generality.

    Authors: Section 3 provides the general proof that the local generating system description is valid for any isotone projection cone in finite dimensions and at any reference point. Concrete illustrative examples are given in Sections 2 and 5 to verify the description and the resulting coderivative formulas. The arguments contain no hidden assumptions and are counterexample-free under the stated hypotheses; the covering constant computation in Theorem 5.1 follows immediately from the coderivative formula via the standard variational-analytic relation between coderivatives and covering constants. revision: no

Circularity Check

0 steps flagged

No circularity; derivation rests on asserted local generating system description

full rationale

The abstract states that explicit Fréchet and Mordukhovich coderivative formulas are derived from a local description of the projection mapping via an associated generating system. No self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation is visible in the provided text. The local description is presented as an input assumption enabling the characterizations, with no indication that it is constructed from the target coderivative formulas themselves. This matches the default case of a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no access to specific free parameters, axioms, or invented entities used in the derivations.

pith-pipeline@v0.9.1-grok · 5674 in / 1034 out tokens · 21184 ms · 2026-06-27T18:19:30.044006+00:00 · methodology

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