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arxiv: 2604.20506 · v1 · submitted 2026-04-22 · 🧮 math.OC

A unified framework for inexact adaptive stepsizes in the gradient methods, the conjugate gradient methods and the quasi-Newton methods for strictly convex quadratic optimization

Zexian Liu This is my paper

Pith reviewed 2026-05-10 00:03 UTC · model grok-4.3

classification 🧮 math.OC
keywords approximately optimal stepsizeinexact adaptive stepsizegradient methodconjugate gradient methodquasi-Newton methodBarzilai-Borwein stepsizestrictly convex quadratic optimizationglobal convergence
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The pith

The paper introduces the approximately optimal stepsize as a unified framework for inexact adaptive stepsizes in gradient, conjugate gradient, and quasi-Newton methods on strictly convex quadratic problems.

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

This paper aims to supply a single rule for choosing inexact but adaptive step sizes that applies equally to the gradient method, the conjugate gradient method, and quasi-Newton methods when the objective function is strictly convex and quadratic. The author defines the approximately optimal stepsize and proves global convergence plus a convergence rate for the gradient method by linking the new choice to the Barzilai-Borwein stepsizes. A sympathetic reader would care because the framework removes the need for exact line searches while still promising reliable progress, and the paper reports numerical gains when the same rule is used across all three methods. Several open questions about extending the full theory to conjugate gradient and quasi-Newton cases are left for later work.

Core claim

The central claim is that a single construction called the approximately optimal stepsize provides a unified framework for inexact adaptive stepsizes that can be used in the gradient method, the conjugate gradient method, and the quasi-Newton method for strictly convex quadratic optimization. For the gradient method the global convergence and the convergence rate follow from the relation between this stepsize and the Barzilai-Borwein stepsizes. Numerical experiments confirm practical advantages when the framework is applied to any of the three methods.

What carries the argument

The approximately optimal stepsize, an inexact but controlled approximation to the exact optimal stepsize that permits a common rule across the three methods and preserves the relation to Barzilai-Borwein stepsizes needed for convergence proofs.

Load-bearing premise

The approximately optimal stepsize can be defined and implemented for the conjugate gradient and quasi-Newton methods so that the convergence properties and the Barzilai-Borwein relation established for the gradient method extend without hidden restrictions on the approximation or on problem conditioning.

What would settle it

A concrete quadratic test problem together with an explicit implementation of the approximately optimal stepsize for the conjugate gradient method in which the sequence of iterates diverges or converges slower than predicted by the gradient-method theory would falsify the unified framework.

read the original abstract

The inexact adaptive stepsizes for the conjugate gradient method and the quasi-Newton method are very rare. The exact stepsizes in the gradient method, the conjugate gradient method and the quasi-Newton method for strictly convex quadratic optimization have a unified framework, while the unified framework for inexact adaptive stepsizes in the gradient method, the conjugate gradient method and the quasi-Newton method for strictly convex quadratic optimization still remains unknown. Based on the above observations, we propose a unified framework for inexact adaptive stepsizes in the gradient method, the conjugate gradient method and the quasi-Newton method for strictly convex quadratic optimization, which is called approximately optimal stepsize. The global convergence and the convergence rate of the gradient method with the approximately optimal stepsize are established by exploring the relation between the approximately optimal stepsize and the famous Barzilai-Borwein (BB) stepsizes. Some numerical results are presented, which confirm the remarkable numerical advantage of the gradient method, the conjugate gradient method and the quasi-Newton method with the unified framework for inexact adaptive stepsizes. Some open problems about the gradient method, the conjugate gradient method and the quasi-Newton method with approximately optimal stepsize are raised.

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

2 major / 1 minor

Summary. The manuscript proposes a unified framework called the 'approximately optimal stepsize' for inexact adaptive stepsizes in the gradient method, conjugate gradient method, and quasi-Newton methods for strictly convex quadratic optimization. It establishes global convergence and a convergence rate specifically for the gradient method by relating the new stepsize to Barzilai-Borwein stepsizes, presents numerical results confirming performance advantages across the methods, and raises open problems.

Significance. If the approximately optimal stepsize extends rigorously beyond the gradient method while preserving convergence, the unified framework would fill a noted gap in adaptive inexact stepsizes for CG and QN on quadratics and offer a practical contribution grounded in the established BB theory. The explicit relation to BB stepsizes for the gradient case is a clear strength, as is the provision of numerical confirmation. However, the current theoretical scope limits the unification claim to numerics for CG and QN.

major comments (2)
  1. [Abstract] Abstract: The central claim is a single unified framework with convergence properties for gradient, CG, and QN methods, yet global convergence and rate are established only for the gradient method via the BB relation. No re-derivation or error bound is indicated for non-steepest-descent directions, which is load-bearing for the unification claim given that CG conjugacy and QN Hessian approximations alter the search direction properties.
  2. [Abstract] Abstract: The definition of 'approximately optimal stepsize' and its implementation for CG and QN is not specified, preventing assessment of whether the BB relation transfers without hidden tolerances, conditioning restrictions, or post-hoc adjustments that would undermine the 'unified' and 'inexact adaptive' assertions.
minor comments (1)
  1. The abstract refers to 'some numerical results' without specifying problem dimensions, test functions, comparison baselines, or performance metrics, which reduces clarity on the claimed 'remarkable numerical advantage'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, clarifying the scope of the theoretical results and the definition of the approximately optimal stepsize. We will revise the abstract and introduction for greater precision.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim is a single unified framework with convergence properties for gradient, CG, and QN methods, yet global convergence and rate are established only for the gradient method via the BB relation. No re-derivation or error bound is indicated for non-steepest-descent directions, which is load-bearing for the unification claim given that CG conjugacy and QN Hessian approximations alter the search direction properties.

    Authors: We agree that global convergence and the convergence rate are established only for the gradient method, by relating the approximately optimal stepsize to the Barzilai-Borwein stepsizes. The unified framework consists of a single definition of the stepsize that applies to any search direction, including those arising in CG and QN methods. For CG and QN we provide only numerical evidence of effectiveness and explicitly raise open problems on extending the convergence analysis, precisely because the conjugacy conditions and Hessian approximations change the direction properties and prevent a direct transfer of the gradient-method proof. We will revise the abstract to state more clearly that the convergence theory applies to the gradient method while the framework and numerics cover all three methods. revision: yes

  2. Referee: [Abstract] Abstract: The definition of 'approximately optimal stepsize' and its implementation for CG and QN is not specified, preventing assessment of whether the BB relation transfers without hidden tolerances, conditioning restrictions, or post-hoc adjustments that would undermine the 'unified' and 'inexact adaptive' assertions.

    Authors: The definition of the approximately optimal stepsize is given in the main text as a general expression that approximates the exact line-search stepsize for strictly convex quadratics and is independent of the particular search direction. Its implementation for CG uses the CG direction inside the same formula; for QN it incorporates the current Hessian approximation. The BB relation is invoked solely in the gradient-method convergence proof and is not asserted for CG or QN. No additional tolerances, conditioning restrictions, or post-hoc adjustments are introduced. To improve accessibility we will add a short explicit statement of the CG and QN implementations in the abstract and introduction. revision: yes

Circularity Check

0 steps flagged

No circularity: convergence established via external relation to Barzilai-Borwein stepsizes

full rationale

The paper defines a new 'approximately optimal stepsize' as a unified inexact adaptive choice and proves global convergence plus rate for the gradient method specifically by relating it to the established Barzilai-Borwein (BB) stepsizes from independent 1988 literature. This link supplies external grounding rather than reducing the result to the paper's own inputs or definitions. Numerical results support the framework on CG and QN methods, while open problems are explicitly raised, confirming that the theoretical claims do not collapse into self-definition, fitted inputs renamed as predictions, or self-citation chains. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the new stepsize definition and its relation to Barzilai-Borwein stepsizes. No explicit free parameters are named in the abstract, but the 'approximately' qualifier implies at least one tolerance or approximation parameter. The setting assumes strictly convex quadratic problems.

axioms (2)
  • domain assumption The optimization problem is strictly convex quadratic.
    Repeated in title and abstract as the problem class for which the framework applies.
  • ad hoc to paper A relation exists between the approximately optimal stepsize and Barzilai-Borwein stepsizes sufficient to prove global convergence and rate.
    Invoked to establish the convergence result for the gradient method.
invented entities (1)
  • approximately optimal stepsize no independent evidence
    purpose: To serve as a unified inexact adaptive stepsize for gradient, conjugate gradient, and quasi-Newton methods.
    Newly introduced framework whose precise mathematical definition is not given in the abstract.

pith-pipeline@v0.9.0 · 5513 in / 1730 out tokens · 59577 ms · 2026-05-10T00:03:08.521136+00:00 · methodology

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