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PreprintCC BY 4.0arXiv

When Are Nesterov Trajectories Eventually Locally Minimizing?

  • Peyman Morteza
  • Stephen Wright
Open at sourcePDFDOICite

Abstract

We revisit the variational characterization of Nesterov's celebrated method and study when Nesterov trajectories are eventually locally minimizing in a variational sense, a notion that we define precisely. We show that this question is equivalent to the nonnegativity of an associated Schrödinger operator acting on variations of the trajectory. For power objectives of the form f(x)=1/p|x|p (where | ·| denotes the Euclidean norm), we give a complete characterization: Among all exponents p≥ 2, the Nesterov trajectory with non-zero initial position is eventually locally minimizing if and only if p=4. We further interpret this exceptional exponent through the stability theory of semilinear elliptic equations.

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Cite

BibTeXRIS
BibTeX
@misc{morteza2026when,
  title = {{When Are Nesterov Trajectories Eventually Locally Minimizing?}},
  author = {Peyman Morteza and Stephen Wright},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33341},
  eprint = {2610.33341},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33341},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - When Are Nesterov Trajectories Eventually Locally Minimizing?
AU  - Peyman Morteza
AU  - Stephen Wright
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.33341
UR  - https://arxiv.org/abs/2610.33341
AB  - We revisit the variational characterization of Nesterov's celebrated method and study when Nesterov trajectories are eventually locally minimizing in a variational sense, a notion that we define precisely. We show that this question is equivalent to the nonnegativity of an associated Schrödinger operator acting on variations of the trajectory. For power objectives of the form f(x)=1/p|x|^p (where | ·| denotes the Euclidean norm), we give a complete characterization: Among all exponents p≥ 2, the Nesterov trajectory with non-zero initial position is eventually locally minimizing if and only if p=4. We further interpret this exceptional exponent through the stability theory of semilinear elliptic equations.
ER  -