Birkhoff interpolation models for optimization with some available derivatives
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, as well as a stellarator design problem in which only some derivatives are available.
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BibTeX
@misc{larson2026birkhoff,
title = {{Birkhoff interpolation models for optimization with some available derivatives}},
author = {Jeffrey Larson and Matt Menickelly and Evan Toler},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.28503},
eprint = {2605.28503},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.28503},
}
RIS
TY - UNPB
TI - Birkhoff interpolation models for optimization with some available derivatives
AU - Jeffrey Larson
AU - Matt Menickelly
AU - Evan Toler
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2605.28503
UR - https://arxiv.org/abs/2605.28503
AB - 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, as well as a stellarator design problem in which only some derivatives are available.
ER -