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PreprintCC BY 4.0HAL

Uniform Quadratic Growth and Second-Order Variational Curvature in Hilbert Spaces

  • Samir Adly
  • Ralph Tyrrell Rockafellar
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Abstract

In second-order variational analysis, a fundamental property of interest is the level of quadratic growth of a function at a point relative to sublinearization for a subgradient at that point. Uniformity in a primal-dual sense is captured by a modulus which, in finite dimensions, can be expressed by a second-subderivative formula. But what about infinite dimensions? The answer, supplied here, is that the formula can fail even for a very nice function on an infinite-dimensional Hilbert space, reducing only to an inequality. However, equality can be restored under an extra condition of directional compactness that is brought to light. This condition is automatic in finite dimensions. For an L 1 -penalized Dirichlet energy on L 2 (Ω), it holds at every point of the subgradient graph. If the multiplier is globally saturated and has the same sign as the state, the modulus coincides with the lowest eigenvalue of the associated elliptic operator.

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Cite

BibTeXRIS
BibTeX
@misc{adly2026uniform,
  title = {{Uniform Quadratic Growth and Second-Order Variational Curvature in Hilbert Spaces}},
  author = {Samir Adly and Ralph Tyrrell Rockafellar},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  url = {https://cnrs.hal.science/hal-05768304v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniform Quadratic Growth and Second-Order Variational Curvature in Hilbert Spaces
AU  - Samir Adly
AU  - Ralph Tyrrell Rockafellar
PY  - 2026
DA  - 2026/09/28
PB  - HAL
UR  - https://cnrs.hal.science/hal-05768304v1
AB  - In second-order variational analysis, a fundamental property of interest is the level of quadratic growth of a function at a point relative to sublinearization for a subgradient at that point. Uniformity in a primal-dual sense is captured by a modulus which, in finite dimensions, can be expressed by a second-subderivative formula. But what about infinite dimensions? The answer, supplied here, is that the formula can fail even for a very nice function on an infinite-dimensional Hilbert space, reducing only to an inequality. However, equality can be restored under an extra condition of directional compactness that is brought to light. This condition is automatic in finite dimensions. For an L 1 -penalized Dirichlet energy on L 2 (Ω), it holds at every point of the subgradient graph. If the multiplier is globally saturated and has the same sign as the state, the modulus coincides with the lowest eigenvalue of the associated elliptic operator.
ER  -