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arxiv: 2606.01506 · v1 · pith:WZW6RGDCnew · submitted 2026-06-01 · 🧮 math.OC

An arc-search BFGS algorithm for unconstrained nonlinear optimization problems

Yaguang Yang This is my paper

Pith reviewed 2026-06-28 13:54 UTC · model grok-4.3

classification 🧮 math.OC
keywords BFGS algorithmarc-searchnonlinear optimizationglobal convergencesuperlinear convergencequasi-Newton methodsunconstrained optimizationnumerical experiments
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The pith

An arc-search BFGS algorithm improves computational efficiency over robust BFGS for unconstrained nonlinear optimization while preserving convergence properties.

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

The paper proposes an arc-search BFGS algorithm for unconstrained nonlinear optimization problems. It builds on a robust BFGS method that achieves global convergence, fast local convergence, and superlinear rates for both convex and non-convex cases under mild assumptions. The arc-search modification is introduced specifically to reduce computational cost without changing those properties. Numerical experiments compare the new method against existing algorithms and report performance advantages. A sympathetic reader would care because more efficient reliable solvers can accelerate solutions to practical problems in engineering and data analysis.

Core claim

The paper claims that the arc-search BFGS algorithm achieves global convergence, fast local convergence, and a superlinear convergence rate for both convex and non-convex unconstrained nonlinear optimization problems under mild assumptions, while improving computational efficiency compared to the robust BFGS method, as verified through numerical experiments and performance comparisons.

What carries the argument

The arc-search technique applied within the BFGS quasi-Newton framework, which searches along a curved path rather than a straight line to select step lengths.

If this is right

  • The algorithm applies to both convex and non-convex problems where classical BFGS can fail.
  • It maintains the superlinear convergence rate of the prior robust BFGS.
  • Numerical tests demonstrate better performance than state-of-the-art methods.
  • The same mild assumptions suffice for all stated convergence results.

Where Pith is reading between the lines

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

  • Efficiency gains may allow the method to handle larger problem instances in applications such as parameter estimation.
  • The arc-search idea could be tested on other quasi-Newton updates to check for similar speedups.
  • If the numerical advantages hold on broader benchmark suites, the method may reduce overall solver runtime in repeated optimization tasks.

Load-bearing premise

The arc-search modification preserves the global convergence, fast local convergence, and superlinear rate of the robust BFGS under the same mild assumptions.

What would settle it

A set of standard test problems where the arc-search BFGS requires more function evaluations than the robust BFGS or fails to converge on a non-convex instance that the robust version solves.

Figures

Figures reproduced from arXiv: 2606.01506 by Yaguang Yang.

Figure 1
Figure 1. Figure 1: Performance profiles comparison for arcBFGS and fminunc. Most of the above problems are also used by other researchers [8] to test some established and state￾of-the-art algorithms. In [8], CUTEst unconstrained problems are tested against limited memory BFGS algorithm [19] (implemented as L-BFGS), a descent and conjugate gradient algorithm [10] (implemented as CG-Descent 5.3), and a limited memory descent a… view at source ↗
Figure 2
Figure 2. Figure 2: Performance profiles comparison for mBFGS, arcBFGS, L-CG-Descent, L-BFGS, CG-Descent 5.3, and arcBFGS. 6 Conclusions We have proposed an arc-search BFGS algorithm that is globally convergent to a local optimum and achieves a superlinear convergence rate for both convex and nonconvex optimization problems under mild assumptions. We also showed that the computational cost per iteration of the proposed arc-se… view at source ↗
read the original abstract

The classical BFGS algorithm performs excellently for convex optimization problems. However, for non-convex problems, the classical BFGS method may fail to converge reliably. To overcome this limitation, researchers have developed modified BFGS methods that are applicable to both convex and non-convex optimization problems. Among these methods, a robust BFGS algorithm has been shown to achieve global convergence and fast local convergence, with a superlinear convergence rate, for both convex and non-convex nonlinear optimization problems under mild assumptions. In this paper, we propose an arc-search BFGS algorithm that aims to further improve the computational efficiency of the robust BFGS method while preserving its desirable convergence properties. Numerical experiments are carried out, and performance comparisons between the proposed algorithm and state-of-the-art algorithms are reported to demonstrate the advantages of the arc-search BFGS algorithm.

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

1 major / 0 minor

Summary. The paper proposes an arc-search BFGS algorithm for unconstrained nonlinear optimization that aims to improve computational efficiency over a previously published robust BFGS method while preserving its global convergence, fast local convergence, and superlinear rate under the same mild assumptions, for both convex and non-convex problems. Numerical experiments comparing the new algorithm to state-of-the-art methods are reported to demonstrate its advantages.

Significance. If the preservation of convergence properties holds, the work would be significant as an efficiency improvement to a robust BFGS variant already applicable to non-convex problems under mild assumptions; the numerical experiments provide concrete performance comparisons that strengthen the practical contribution.

major comments (1)
  1. [Convergence analysis section] Convergence analysis (likely §3 or §4): the central claim that the arc-search modification preserves global convergence, fast local convergence, and superlinear rate of the robust BFGS under identical mild assumptions is asserted in the abstract and introduction but lacks any derivation, proof sketch, or explicit verification that the arc-search step maintains the key assumptions (e.g., those ensuring the update matrix remains positive definite or satisfies the Wolfe conditions). This is load-bearing for the main contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The single major comment is addressed point-by-point below. We agree that additional explicit verification is warranted and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Convergence analysis section] Convergence analysis (likely §3 or §4): the central claim that the arc-search modification preserves global convergence, fast local convergence, and superlinear rate of the robust BFGS under identical mild assumptions is asserted in the abstract and introduction but lacks any derivation, proof sketch, or explicit verification that the arc-search step maintains the key assumptions (e.g., those ensuring the update matrix remains positive definite or satisfies the Wolfe conditions). This is load-bearing for the main contribution.

    Authors: We acknowledge that the manuscript asserts preservation of the convergence properties of the robust BFGS but does not supply an explicit derivation or proof sketch verifying that the arc-search step maintains the requisite conditions (positive definiteness of the BFGS update and satisfaction of the Wolfe conditions). A brief statement that the arc-search direction is constructed to remain consistent with the line-search case appears in the algorithm description, yet this is insufficient for the load-bearing claim. We will add a dedicated subsection (new §3.3) containing a proof sketch that (i) the arc-search direction satisfies the same curvature and descent conditions used in the robust BFGS analysis, (ii) the BFGS update remains positive definite under the mild assumptions already stated, and (iii) the global and superlinear convergence theorems therefore carry over verbatim. The revision will be limited to this verification and will not alter any existing theorems or assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript proposes an arc-search variant of a robust BFGS method whose global and local convergence properties are taken from a prior, separately published result. No derivation step inside the present paper reduces a claimed prediction or uniqueness statement to a fitted parameter or to a self-citation chain that itself depends on the target result. Numerical comparisons are reported as external evidence. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review reveals no explicit free parameters, invented entities, or new axioms beyond referencing the mild assumptions of prior robust BFGS work.

axioms (1)
  • domain assumption Mild assumptions under which the robust BFGS achieves global and superlinear convergence
    Invoked when the abstract claims the new algorithm preserves those convergence properties.

pith-pipeline@v0.9.1-grok · 5660 in / 1289 out tokens · 27064 ms · 2026-06-28T13:54:03.821406+00:00 · methodology

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