Ellipsoid Separation in Data-Dependent Time via Infimal Displacement Vectors
Abstract
The infimal displacement vector of the nonexpansive operator underlying the primal-dual hybrid gradient algorithm (PDHG) is known to encode useful information for infeasible optimization problems. In particular, for the ellipsoid separation problem (ESP), where the goal is to find a separating hyperplane between two collections of ellipsoids, the infimal displacement vector corresponding to an infeasible second-order cone program (SOCP) encodes a separating hyperplane. We show FISTA computes a sequence of approximations to the dual component of the infimal displacement vector and establish that a solution to the ESP can be obtained in πͺ(Ξ΄πβ2) iterations. Here Ξ΄π denotes the distance between the convex hulls of the two ellipsoid collections.
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BibTeX
@misc{hough2026ellipsoid,
title = {{Ellipsoid Separation in Data-Dependent Time via Infimal Displacement Vectors}},
author = {Matthew Hough and Stephen A. Vavasis},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33902},
eprint = {2610.33902},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33902},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Ellipsoid Separation in Data-Dependent Time via Infimal Displacement Vectors
AU - Matthew Hough
AU - Stephen A. Vavasis
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33902
UR - https://arxiv.org/abs/2610.33902
AB - The infimal displacement vector of the nonexpansive operator underlying the primal-dual hybrid gradient algorithm (PDHG) is known to encode useful information for infeasible optimization problems. In particular, for the ellipsoid separation problem (ESP), where the goal is to find a separating hyperplane between two collections of ellipsoids, the infimal displacement vector corresponding to an infeasible second-order cone program (SOCP) encodes a separating hyperplane. We show FISTA computes a sequence of approximations to the dual component of the infimal displacement vector and establish that a solution to the ESP can be obtained in πͺ(Ξ΄_π^β2) iterations. Here Ξ΄_π denotes the distance between the convex hulls of the two ellipsoid collections.
ER -