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arxiv: 2605.28503 · v2 · pith:RQ3QDFPUnew · submitted 2026-05-27 · 🧮 math.OC

Birkhoff interpolation models for optimization with some available derivatives

Jeffrey Larson , Matt Menickelly , Evan Toler This is my paper

Pith reviewed 2026-06-29 10:49 UTC · model grok-4.3

classification 🧮 math.OC
keywords Birkhoff interpolationderivative-free optimizationpartial derivativestrust-region methodspolynomial modelspoised setsmodel accuracyinterpolation-based optimization
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The pith

Birkhoff interpolation enables construction of local polynomial models from arbitrary mixtures of function values and partial derivatives.

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

The paper introduces a Birkhoff interpolation framework that builds local polynomial models for optimization when only some derivatives are available. This setup handles any pattern of function evaluations and partial derivatives without demanding complete derivative sets at every point. Conditions are given under which the resulting systems stay poised, yielding accuracy bounds for fully quadratic models. A trust-region algorithm is developed that keeps the interpolation sets poised while selectively adding the available derivative information. The framework generalizes standard interpolation-based derivative-free methods and connects them to settings with partial derivative access.

Core claim

A Birkhoff interpolation framework is introduced that permits arbitrary patterns of derivative availability and enables the construction of local polynomial models using mixtures of function values and partial derivative information. In contrast to Hermite interpolation approaches, the proposed framework does not require all available derivatives to be queried at every interpolation point. Conditions are developed under which the resulting interpolation systems are poised and corresponding model-accuracy bounds are established for fully quadratic interpolation models. A trust-region framework maintains poised interpolation sets while selectively incorporating derivative information, generali

What carries the argument

Birkhoff interpolation framework that permits arbitrary patterns of derivative availability without requiring complete derivatives at each point.

If this is right

  • Arbitrary patterns of available derivatives can be used without forcing full derivative sets at all interpolation points.
  • Poisedness conditions guarantee model accuracy bounds for fully quadratic models.
  • The trust-region method maintains poised sets while incorporating whatever derivative information is present.
  • The approach generalizes existing interpolation-based derivative-free optimization algorithms.
  • It bridges derivative-free and derivative-based optimization by handling partial derivative availability.

Where Pith is reading between the lines

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

  • The method could reduce the cost of derivative computations in applications like adjoint-enabled simulations by using only the cheapest available partials.
  • Higher-order models beyond quadratic could be explored if similar poisedness conditions can be derived for other patterns.
  • Real derivative-availability patterns from legacy code or structured models might be tested to check performance beyond the synthetic CUTEst cases.
  • The framework might combine with existing poised-set maintenance techniques from derivative-free optimization literature to handle dynamic availability.

Load-bearing premise

The interpolation systems must remain poised for the chosen derivative-availability patterns so that the model-accuracy bounds hold.

What would settle it

A concrete derivative-availability pattern for which the Birkhoff interpolation matrix for a quadratic model becomes singular, violating poisedness and falsifying the accuracy bounds.

Figures

Figures reproduced from arXiv: 2605.28503 by Evan Toler, Jeffrey Larson, Matt Menickelly.

Figure 1
Figure 1. Figure 1: Heatmaps of the Birkhoff poisedness constant Λ for two interpolation configurations in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Performance profiles on 89 CUTEst problems comparing our Birkhoff approach and Hermite [ [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
read the original abstract

We consider interpolation-based derivative-free optimization in settings where only some derivatives are available. Such situations arise in scientific computing applications involving simulations, adjoint-enabled components, legacy software, or partially differentiable models. We introduce a Birkhoff interpolation framework that permits arbitrary patterns of derivative availability and enables the construction of local polynomial models using mixtures of function values and partial derivative information. In contrast to Hermite interpolation approaches, the proposed framework does not require all available derivatives to be queried at every interpolation point. We develop conditions under which the resulting interpolation systems are poised and establish corresponding model-accuracy bounds for fully quadratic interpolation models. We develop a trust-region framework that maintains poised interpolation sets while selectively incorporating derivative information. The method generalizes an established class of interpolation-based derivative-free optimization algorithms and naturally bridges derivative-free and derivative-based settings. We evaluate our approach on a collection of CUTEst test problems with synthetically generated derivative-availability patterns.

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 / 3 minor

Summary. The paper introduces a Birkhoff interpolation framework for derivative-free optimization that accommodates arbitrary patterns of available partial derivatives at interpolation points, without requiring full Hermite data at every point. It develops explicit conditions ensuring the resulting linear systems are poised and derives corresponding model-accuracy bounds for fully quadratic models. A trust-region algorithm is proposed that maintains poised interpolation sets while selectively incorporating available derivative information, generalizing existing interpolation-based DFO methods. The approach is evaluated on CUTEst test problems using synthetically generated derivative-availability patterns.

Significance. If the poisedness conditions and accuracy bounds hold as claimed, the work provides a principled way to exploit partial derivative information in DFO settings common in simulation-based and adjoint-enabled applications. It bridges derivative-free and derivative-based regimes within a single interpolation model class and supplies the theoretical guarantees (poisedness, error bounds) that many mixed-information methods have left implicit. The trust-region framework and CUTEst experiments offer a concrete algorithmic realization and initial empirical evidence of practicality.

minor comments (3)
  1. Abstract and §1: the description of the trust-region framework states that it 'maintains poised interpolation sets,' but the precise mechanism (e.g., which points are replaced when a new derivative pattern is encountered) is not summarized; a one-sentence outline would help readers assess the algorithmic novelty.
  2. Evaluation section: the synthetic derivative-availability patterns are described only as 'generated'; explicit rules or pseudocode for their construction (e.g., probability per derivative order, correlation across points) would allow reproduction and assessment of whether the reported gains are robust to different pattern distributions.
  3. Notation: the distinction between the Birkhoff interpolation matrix and the standard Hermite matrix is introduced but the precise block structure for a mixed pattern is not illustrated with a small example (e.g., 2-D quadratic with f at one point and ∇f at another); adding such an example would clarify the framework for readers unfamiliar with Birkhoff theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of the manuscript, the assessment of its significance in bridging derivative-free and derivative-based regimes, and the recommendation for minor revision. No specific major comments appear in the provided report, so we have no individual points requiring point-by-point rebuttal or revision at this stage. We remain available to address any minor clarifications or editorial suggestions in a revised version.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces a Birkhoff interpolation framework that extends established DFO interpolation methods to handle arbitrary patterns of partial derivative availability. It explicitly develops poisedness conditions for the interpolation systems and derives corresponding model-accuracy bounds for quadratic models, rather than assuming them or reducing them to prior self-citations. The trust-region framework maintains poised sets while incorporating selective derivatives, generalizing existing algorithms without any load-bearing step that reduces by construction to fitted inputs, self-definitions, or renamed known results. The derivation chain is self-contained against external benchmarks in the DFO literature.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Framework rests on standard interpolation theory and DFO poised-set concepts; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Standard smoothness assumptions on the objective function sufficient for quadratic model error bounds
    Invoked to establish model-accuracy bounds for fully quadratic models

pith-pipeline@v0.9.1-grok · 5681 in / 1125 out tokens · 46112 ms · 2026-06-29T10:49:38.073995+00:00 · methodology

discussion (0)

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Reference graph

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