Ramification of Weighted Central Paths and Super-linear Interior-Point Methods for Semi-definite Optimization
Abstract
The local convergence of primal-dual interior-point methods for semi-definite optimization is closely tied to the regularity of the central path at its limiting point. Without strict complementarity, the central path can exhibit fractional power asymptotics and need not be differentiable in the ordinary barrier parameter. We show that this obstruction can be removed simultaneously for a family of weighted \AHO~central paths. More precisely, there exists a common ramification exponent N such that all weighted paths in a fixed neighborhood of the identity become analytic at the limiting point after the reparametrization μ↦μN. Using higher-order derivatives of these reparametrized paths, we construct a feasible pth-order predictor-corrector interior-point method. Under a uniform predictor regularity condition, and without assuming strict complementarity, we prove that the normalized complementarity gaps converge with Q-order at least (p+1)/ρ, where ρ is the reparametrization exponent used by the method. We also establish super-linear convergence of the simultaneously vanishing primal-dual blocks, derive an explicit quantitative reduction bound throughout the predictor regularity region, and obtain a doubly logarithmic outer iteration bound once the iterates enter the local super-linear regime. Finally, we illustrate the predicted higher-order behavior on a collection of degenerate and non-degenerate semi-definite optimization problems.
Related
- Nested Clustered Optimization Is One End of a Schur Bridge, and the Interior Is Sometimes Provably Better
- Interior-point proximal methods for nonsmooth optimization in Hilbert spaces with cone-ordered constraints
- Optimal study interior partitioning (OSIP): An optimization approach to observational study design under fine balance constraints
- Trade-off invariance for weighted scalarizations in multi-objective optimization
- Reliability-Weighted Direct Preference Optimization: Penalized Weights for Robust Preference Learning
BibTeX
@misc{mohammadnezhad2026ramification,
title = {{Ramification of Weighted Central Paths and Super-linear Interior-Point Methods for Semi-definite Optimization}},
author = {Ali Mohammad-Nezhad and Pouya Sampourmahani and Tamas Terlaky},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.07080},
eprint = {2611.07080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.07080},
}
RIS
TY - UNPB
TI - Ramification of Weighted Central Paths and Super-linear Interior-Point Methods for Semi-definite Optimization
AU - Ali Mohammad-Nezhad
AU - Pouya Sampourmahani
AU - Tamas Terlaky
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.07080
UR - https://arxiv.org/abs/2611.07080
AB - The local convergence of primal-dual interior-point methods for semi-definite optimization is closely tied to the regularity of the central path at its limiting point. Without strict complementarity, the central path can exhibit fractional power asymptotics and need not be differentiable in the ordinary barrier parameter. We show that this obstruction can be removed simultaneously for a family of weighted \AHO~central paths. More precisely, there exists a common ramification exponent N such that all weighted paths in a fixed neighborhood of the identity become analytic at the limiting point after the reparametrization μ↦μ^N. Using higher-order derivatives of these reparametrized paths, we construct a feasible pth-order predictor-corrector interior-point method. Under a uniform predictor regularity condition, and without assuming strict complementarity, we prove that the normalized complementarity gaps converge with Q-order at least (p+1)/ρ, where ρ is the reparametrization exponent used by the method. We also establish super-linear convergence of the simultaneously vanishing primal-dual blocks, derive an explicit quantitative reduction bound throughout the predictor regularity region, and obtain a doubly logarithmic outer iteration bound once the iterates enter the local super-linear regime. Finally, we illustrate the predicted higher-order behavior on a collection of degenerate and non-degenerate semi-definite optimization problems.
ER -