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arxiv: 2503.17740 · v2 · submitted 2025-03-22 · 🧮 math.OC

Directional differentiability for solution operators of sweeping processes with convex polyhedral admissible sets

Martin Brokate , Constantin Christof This is my paper

Pith reviewed 2026-05-22 23:26 UTC · model grok-4.3

classification 🧮 math.OC
keywords directional differentiabilitysweeping processesvector playvector stoppolyhedral setsHadamard differentiabilityrate-independent evolutionoptimal control
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The pith

The vector play and stop operators are Hadamard directionally differentiable pointwise if and only if the admissible set is non-obtuse.

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

The paper establishes that the solution operators for sweeping processes, specifically the vector play and vector stop, possess Hadamard directional derivatives in a pointwise sense precisely when the convex polyhedral admissible set satisfies the non-obtuse condition. This characterization is obtained when the operators act on the space of continuous functions of bounded variation. The result matters because these operators appear in models of rate-independent systems such as elastoplasticity and friction, and directional differentiability supplies the stationarity conditions needed for optimal control of such systems. When the derivatives exist they are shown to satisfy a closed system of projection identities and variational inequalities. The paper further demonstrates that the non-obtuse condition is necessary, since differentiability fails for obtuse sets even after restricting the domain to Lipschitz continuous functions.

Core claim

When the domain is the space of continuous functions of bounded variation, the vector play and the vector stop are Hadamard directionally differentiable in a pointwise manner if and only if the convex polyhedral admissible set is non-obtuse; in the cases where the derivatives exist they are uniquely characterized by a system of projection identities and variational inequalities, while no such differentiability holds for obtuse sets even on the smaller space of Lipschitz functions.

What carries the argument

The non-obtuse condition on a convex polyhedral set, which ensures that the vector play and stop operators admit pointwise Hadamard directional derivatives on the space of continuous functions of bounded variation.

If this is right

  • Bouligand stationarity conditions become available for optimal control problems that involve sweeping processes with non-obtuse polyhedral sets.
  • The directional derivatives of the play and stop are uniquely determined by a system of projection identities and variational inequalities.
  • Directional differentiability is lost for any obtuse admissible set, even when the input is restricted to Lipschitz functions.
  • The necessity of the non-obtuse condition supplies a sharp geometric criterion for when these rate-independent operators can be differentiated.

Where Pith is reading between the lines

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

  • The same non-obtuse criterion may govern directional differentiability for other solution operators of sweeping processes beyond the play and stop.
  • Optimal control problems with sweeping processes could be discretized while preserving the stationarity conditions once the admissible set is verified to be non-obtuse.
  • The result suggests examining whether analogous differentiability statements hold when the admissible set is replaced by a smooth convex body rather than a polyhedron.

Load-bearing premise

The domain of definition for the solution operators is the space of continuous functions of bounded variation.

What would settle it

An explicit obtuse polyhedral set together with a bounded-variation input function at which the directional derivative of the vector play operator exists.

read the original abstract

We study directional differentiability properties of solution operators of rate-independent evolution variational inequalities with full-dimensional convex polyhedral admissible sets. It is shown that, if the space of continuous functions of bounded variation is used as the domain of definition, then the most prototypical examples of such solution operators - the vector play and stop - are Hadamard directionally differentiable in a pointwise manner if and only if the admissible set is non-obtuse. We further prove that, in those cases where they exist, the directional derivatives of the vector play and stop are uniquely characterized by a system of projection identities and variational inequalities and that directional differentiability cannot be expected in the obtuse case even if the solution operator is restricted to the space of Lipschitz continuous functions. Our results can be used, for example, to formulate Bouligand stationarity conditions for optimal control problems involving sweeping processes.

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

0 major / 2 minor

Summary. The manuscript studies directional differentiability of solution operators for sweeping processes with convex polyhedral admissible sets. It proves that the vector play and stop operators are Hadamard directionally differentiable pointwise on the space of continuous functions of bounded variation if and only if the admissible set is non-obtuse. The derivatives are characterized by a system of projection identities and variational inequalities. It also shows that differentiability fails for obtuse sets even when restricted to Lipschitz continuous functions. The results are positioned for use in formulating Bouligand stationarity conditions for optimal control problems involving sweeping processes.

Significance. If the results hold, this provides a precise if-and-only-if characterization of directional differentiability for key operators in rate-independent systems, along with explicit derivative characterizations via projections and variational inequalities. The sharpness via the CBV domain restriction and the explicit failure on Lipschitz functions for obtuse sets are notable strengths that support applications to stationarity conditions in optimal control.

minor comments (2)
  1. [Abstract] The abstract states the if-and-only-if result clearly but does not indicate the precise definition of 'non-obtuse' used in the theorems; adding a brief parenthetical or reference to the relevant definition in §2 would improve readability.
  2. [Main theorems] In the characterization of the directional derivatives (likely in the main theorems), the system of projection identities and variational inequalities is presented; confirming that the uniqueness proof does not rely on additional regularity beyond the stated assumptions would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the if-and-only-if characterization, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes an if-and-only-if theorem on Hadamard directional differentiability of the vector play and stop operators (pointwise) when the domain is restricted to continuous functions of bounded variation and the admissible set is non-obtuse. This is framed as a direct consequence of properties of convex polyhedral sets and variational inequalities, with the derivatives characterized by projection identities plus variational inequalities. No load-bearing step reduces by the paper's own equations to a fitted input, self-definition, or self-citation chain; the central claim remains an independent mathematical characterization rather than a renaming or tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard background results from convex analysis and the theory of rate-independent evolution variational inequalities; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Standard properties of convex polyhedral sets, metric projections, and rate-independent variational inequalities in Hilbert spaces.
    These are foundational, previously established results invoked to define the operators and state the differentiability claims.

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Reference graph

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