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arxiv: 2505.19724 · v2 · submitted 2025-05-26 · 🧮 math.OC

Local near-quadratic convergence of Riemannian interior point methods

Mitsuaki Obara , Takayuki Okuno , Akiko Takeda This is my paper

Pith reviewed 2026-05-19 14:27 UTC · model grok-4.3

classification 🧮 math.OC
keywords Riemannian optimizationinterior point methodslocal convergencesuperlinear convergencenear-quadratic convergenceconstrained optimizationsecond-order stationarity
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The pith

Riemannian interior point methods achieve local near-quadratic convergence by solving one linear system per outer iteration with a specific barrier update.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper analyzes a class of Riemannian interior point methods for optimization problems with equality and inequality constraints on manifolds. It shows that under standard assumptions the outer iteration requires only a single linear system solve to reach local superlinear convergence, with no further inner-iteration work needed. A tailored update rule for the barrier parameter then lifts the rate to local near-quadratic convergence. The same analysis is applied to an existing algorithm to establish both its local rates and its global convergence to second-order stationary points.

Core claim

Under standard assumptions the algorithm achieves local superlinear convergence by solving a linear system at each outer iteration, removing the need for further computations in the inner iterations. A specific update for the barrier parameter achieves local near-quadratic convergence of the algorithm. When applied to the method of Obara, Okuno, and Takeda (2026) the analysis also confirms second-order stationarity in the global convergence result.

What carries the argument

The outer-inner iteration structure of the Riemannian interior point method together with a barrier-parameter update rule that controls the local convergence rate.

If this is right

  • Only one linear solve per outer iteration is required for local superlinear convergence.
  • The barrier-parameter update produces local near-quadratic convergence near a solution.
  • The analysis supplies both local rates and global convergence to a second-order stationary point for the algorithm to which it is applied.
  • The same framework covers problems with both equality and inequality constraints on Riemannian manifolds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The single-solve outer iteration may be reusable as a local accelerator inside other Riemannian constrained solvers.
  • The barrier update strategy could be tested on higher-dimensional manifold problems to check whether the predicted rates survive floating-point effects.
  • Similar local-rate results might be derived for Riemannian variants of other classical interior-point families such as primal-dual or penalty methods.

Load-bearing premise

The standard assumptions on constraint qualifications, second-order stationarity, and the properties of the Riemannian metric and barrier function hold in a neighborhood of the solution.

What would settle it

Numerical runs on a low-dimensional constrained Riemannian problem that show the error sequence remains only linearly convergent even after the proposed barrier update is applied would refute the near-quadratic claim.

read the original abstract

We consider Riemannian optimization problems with inequality and equality constraints and analyze a class of Riemannian interior point methods for solving them. The algorithm of interest consists of outer and inner iterations. We show that, under standard assumptions, the algorithm achieves local superlinear convergence by solving a linear system at each outer iteration, removing the need for further computations in the inner iterations. We also provide a specific update for the barrier parameter that achieves local near-quadratic convergence of the algorithm. We apply our results to the method proposed by Obara, Okuno, and Takeda (2026) and show its local superlinear and near-quadratic convergence with an analysis of the second-order stationarity. To our knowledge, this is the first algorithm for constrained optimization on Riemannian manifolds that achieves both local convergence and global convergence to a second-order stationary point. Numerical results support the theoretical analyses of the proposed methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript analyzes a class of Riemannian interior point methods for constrained optimization problems on Riemannian manifolds with equality and inequality constraints. Under standard assumptions (constraint qualifications, second-order stationarity, and suitable smoothness/convexity properties of the barrier function with respect to the Riemannian metric), it establishes that the algorithm achieves local superlinear convergence because each outer iteration reduces to solving a single linear system whose solution yields the step, eliminating further inner-iteration computations. A specific barrier-parameter update is proposed that yields local near-quadratic convergence. The analysis is applied to the concrete method of Obara, Okuno, and Takeda (2026), proving both local rates and second-order stationarity; global convergence to a second-order stationary point is shown separately via a potential-function argument. Numerical results support the theory, and the work claims to be the first Riemannian constrained optimizer with both local convergence and global convergence to second-order points.

Significance. If the results hold, the manuscript makes a solid contribution to Riemannian optimization by supplying the first local superlinear and near-quadratic convergence analysis for interior-point methods on manifolds, together with global convergence to second-order stationary points. The explicit verification that the listed assumptions are inherited by the Obara et al. method, the clean separation between the global potential-function argument and the local error-recursion analysis, and the retraction-based feasibility arguments are strengths that enhance applicability and credibility. The work provides a reusable framework that can be instantiated on other Riemannian IPMs and addresses a genuine gap in the literature on constrained manifold optimization.

minor comments (3)
  1. [Local-analysis section] § Local-analysis section: the precise definition of the 'near-quadratic' rate (how the barrier update modifies the standard quadratic error recursion) would benefit from an explicit side-by-side comparison with the Euclidean IPM quadratic rate to clarify the distinction.
  2. [Local-analysis section] The statement of the standard assumptions is given in the local-analysis section but would be easier to reference if collected into a single numbered assumption block or table.
  3. [Numerical results] Numerical results section: adding log-log plots of the error versus iteration count would make the observed superlinear/near-quadratic behavior visually immediate and strengthen the empirical support.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and thorough evaluation of our manuscript on Riemannian interior point methods. We appreciate the recognition of the local superlinear and near-quadratic convergence results, the global convergence to second-order stationary points, and the reusable framework for other Riemannian IPMs. The recommendation for minor revision is noted, and we will make appropriate improvements to enhance the presentation and clarity.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives local superlinear and near-quadratic convergence rates for a class of Riemannian interior-point methods by showing that, under explicitly enumerated standard assumptions (constraint qualifications, second-order stationarity, smoothness of the barrier with respect to the Riemannian metric), each outer iteration reduces to a single linear-system solve whose step, combined with the proposed barrier-parameter update, produces the stated rates via standard error-recursion arguments. The global convergence to a second-order stationary point is established separately via a potential-function argument that does not rely on the local-rate machinery. Application of the general results to the concrete method of Obara et al. (2026) consists of verifying that the listed assumptions are inherited by that method; this verification step is independent of the rate derivations themselves. No equation reduces by construction to a fitted parameter, no central premise rests solely on a self-citation whose content is unverified, and the analysis remains self-contained against external benchmarks once the stated assumptions are granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on unspecified 'standard assumptions' typical of Riemannian optimization literature; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Standard assumptions for local convergence of Riemannian interior-point methods (constraint qualifications, second-order stationarity, smoothness of the barrier function)
    Invoked to guarantee the local superlinear and near-quadratic rates; location referenced in abstract as 'under standard assumptions'.

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