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Linear convergence of relocated fixed-point iterations

T0 review · 1 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Relocated fixed-point iterations converge linearly under a uniform linear error bound, and this yields linear convergence for variable-stepsize Douglas–Rachford and resolvent-splitting iterations.

desk verdict Main theorem is solid and the rate results are new, but the headline Douglas–Rachford application leans on a delegated proof that should be supplied before publication. read the letter →

arxiv 2512.12954 v2 pith:G6TCUP24 submitted 2025-12-15 math.OC

Felipe Atenas , Farhana Ahmed Simi , Matthew K Tam This is my paper · ORCID
classification math.OC MSC 47H0447H0547H0965K10
keywords fixed-pointiterationserrorboundlinearconvergenceresolventsplittingDouglas–RachfordmonotoneoperatorsR-linearrelocated
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that relocated fixed-point iterations—where the algorithmic operator depends on a step-size parameter γ that changes at every step—converge R-linearly, meaning the error contracts by a fixed factor per iteration, provided the family of operators satisfies a uniform linear error bound (bounded linear regularity). This closes a gap: earlier work showed only weak convergence for such iterations and could not say anything about the rate. The significance is that many practical algorithms, including Douglas–Rachford with variable step sizes, fall outside the classical common-fixed-point analysis because the fixed-point set of the operator changes with γ; the new framework handles this without requiring Fejér monotonicity. The theorem is then applied to derive explicit linear rates for the relocated Douglas–Rachford algorithm under strong-monotonicity-type conditions, and for a multi-operator resolvent splitting algorithm.

What carries the argument

The key machinery is the notion of uniform bounded linear regularity of the family (Tγ): the distance from any point in a bounded set to the fixed-point set of Tγ is controlled by the displacement ∥x − Tγx∥, with a constant independent of γ. This turns the averaged nonexpansiveness of Tγ into a per-step contraction on the squared distance to Fix Tγn. The second component is the fixed-point relocator Qδ←γ, which maps the fixed-point set of Tγ into that of Tδ, together with Assumption 3.4 that these relocators are Lipschitz in the parameter; this allows the iteration to be compared to a 'shadow' sequence anchored at a fixed point of the limit operator, yielding a Cauchy estimate that forces R-

What would settle it

Numerically monitor the relocated Douglas–Rachford iteration and plot the ratio dist(xn, Fix Tγn)/∥xn − Tγn xn∥ along the iterate path, with γn converging geometrically to γ∗. If this ratio is unbounded, uniform linear regularity fails and Theorem 3.6's conclusion need not hold; if the ratio stays bounded and yet the errors do not decay geometrically, the theorem would be contradicted. A direct construction would be a family of averaged nonexpansive operators with fixed-point sets that drift linearly in γ while the displacement ∥x − Tγx∥ stays small on a bounded set far from Fix Tγ.

Watch

Extended reading notes

Core claim

The central result, Theorem 3.6, states the following. Let (Tγ)γ∈Γ be a family of α-averaged nonexpansive operators with nonempty fixed-point sets, equipped with fixed-point relocators Qδ←γ satisfying a Lipschitz condition in both the point and the parameter (Assumption 3.4), and suppose (Tγ) is uniformly boundedly linearly regular: for every bounded set S there is a single κ with dist(x, Fix Tγ) ≤ κ∥x − Tγx∥ for all x ∈ S and all γ ∈ Γ. If the step-size sequence γn converges R-linearly to γ∗ and the relocator deviations are summable, then the relocated iteration xn+1 = Qγn+1←γn Tγn xn produces sequences (xn) and (Tγn xn) that converge R-linearly to a common point in Fix Tγ∗ . The proof is a

Load-bearing premise

The load-bearing premise is that the family of operators is uniformly boundedly linearly regular: within every bounded region, the distance to each operator's fixed-point set is bounded by a single constant times the displacement produced by that operator, with the same constant for all parameter values; if no such uniform error bound exists, the contraction argument collapses.

Editorial extensions

If this is right

  • Variable-stepsize Douglas–Rachford converges R-linearly when one of the operators is strongly monotone relative to the primal solution set and one inverse is strongly monotone relative to the dual solution set (Corollary 4.3).
  • The same holds when one operator is Lipschitz continuous and the other strongly monotone, in which case the Douglas–Rachford operators themselves are contractions (Corollary 4.5).
  • Multi-operator resolvent splitting (the relocated version of the minimal-lifting scheme) converges R-linearly under Lipschitz and strong-monotonicity assumptions (Corollary 5.5).
  • For any family of contractions, uniform linear regularity holds automatically with constant 1/(1−β), so relocated fixed-point iterations of contractions always converge R-linearly provided the parameter sequence does (Corollary 3.8).
  • If the step-size sequence converges only sublinearly, R-linear convergence of the iterates can fail even for the simplest contraction family, so R-linear parameter convergence is necessary in general (Remark 3.9).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The uniform bounded linear regularity assumption is essentially a parameter-uniform metric subregularity of Id − Tγ; any family that satisfies such an error bound—not necessarily monotone or convex—should inherit the same R-linear rate argument, so the framework may extend to nonconvex problems where error bounds are known.
  • Because the proof never uses Fejér monotonicity, it suggests that the same technique could yield rates for other non-stationary methods, such as linesearch-based proximal gradient or adaptive ADMM, as long as the varying operators admit relocators with the prescribed Lipschitz behavior.
  • The necessity of R-linear parameter convergence (Remark 3.9) implies that adaptive step-size rules designed for fast asymptotic convergence of γn to γ∗ may not improve—and could destroy—the linear rate; a practical rule might instead aim for exactly geometric decay.
  • One might test the tightness of the constant: in the Douglas–Rachford case, the rate factor is determined by κγ = 4(1 + max{1/(γμ), γ/ρ}); whether the factor 4 is sharp is not addressed, and a refined error bound could improve the rate estimate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper studies the rate of convergence of relocated fixed-point iterations, a framework introduced by the same group in [3] for variable-stepsize operator splitting when the underlying operators do not share a common fixed point. The main result, Theorem 3.6, states that if the family (T_γ) is α-averaged nonexpansive, uniformly boundedly linearly regular, the relocators satisfy a Lipschitz-type assumption (Assumption 3.4), and the stepsize sequence (γ_n) converges R-linearly, then both dist(x_n, Fix T_{γ_n}) and the iterates (x_n), (T_{γ_n}x_n) converge R-linearly to a fixed point of T_{γ*}. Corollary 3.8 specializes to contractions. The paper then applies the framework to the relocated Douglas--Rachford algorithm (Section 4) and the relocated Malitsky--Tam resolvent splitting algorithm (Section 5), claiming R-linear convergence under strong monotonicity and Lipschitz continuity assumptions. The proof of the general theorem is detailed and appears correct; the principal gap is in the Douglas--Rachford application, where the uniform linear regularity of the family is asserted via a proof delegated to an external convex-analysis theorem.

Significance. If the Douglas--Rachford application is fully substantiated, the paper makes a genuinely useful contribution: it gives the first R-linear convergence rates for relocated/non-stationary fixed-point iterations without Fejér monotonicity, and it covers variable-stepsize Douglas--Rachford and multioperator resolvent splitting under natural strong-monotonicity and Lipschitz conditions. The proof of Theorem 3.6 is a highlight: the contraction-type estimate (18), the boundedness argument for the projections p_n, and the Cauchy estimate (22) that upgrades R-linear decay of the distance-to-fixed-point sets to R-linear convergence of the iterates are all internally sound and checkable. The contraction case (Corollary 3.8) is a clean consequence. However, the application to Douglas--Rachford rests on Lemma 4.2, whose proof is not given; until that lemma is proved or replaced by a genuinely applicable external theorem, the announced Douglas--Rachford rates are conditional rather than established.

major comments (1)
  1. [Section 4, Lemma 4.2] The proof of Lemma 4.2 is not supplied. The text states only that the proof is 'analogous to [25, Theorem 2]' with Proposition 2.1 used in place of [25, Proposition 1]. This is the exact step that provides the uniform linear regularity of the Douglas--Rachford operators (T_γ), which is a hypothesis of Theorem 3.6 and therefore load-bearing for Corollary 4.3. The cited [25, Theorem 2] is stated for subdifferentials of convex functions, and its proof may rely on Fenchel conjugacy or subgradient calculus that does not automatically extend to general paramonotone maximally monotone operators. Moreover, [25] is an arXiv preprint, making verification difficult. Please provide a complete proof of Lemma 4.2, or cite a theorem that explicitly covers paramonotone operators. Without this, the Douglas--Rachford application is not proved.
minor comments (5)
  1. [Corollary 4.5, proof] The first sentence of the proof says 'By Lemma 3.3, there exists β∈[0,1) such that (T_γ) is a family of β-contractions.' This should cite Lemma 4.4; Lemma 3.3 only states a consequence for a given family of contractions and does not establish existence of β.
  2. [Section 4 and 5, hypotheses on Γ] Several statements say 'Γ⊆R++ be a nonempty closed interval,' but the proofs use compactness of Γ: Lemma 4.1 invokes 'Γ is compact in R' and Lemma 4.2 uses that κ_γ attains its maximum on the compact set Γ. If Γ is allowed to be unbounded (e.g., [a,∞)), the uniformity claims can fail. Please state explicitly that Γ is a compact (equivalently, bounded closed) interval, or adjust the proofs and statements accordingly.
  3. [Equation (38)] The identity in (38) is missing the square on the last term: it should be ∥αu+(1−α)v∥² = α∥u∥² + (1−α)∥v∥² − α(1−α)∥u−v∥².
  4. [Lemma 3.5, proof] In the induction step, the expression 'c_{n+1} = Q_{γ_{n+1}←γ_n} c_n ⊆ Q_{γ_{n+1}←γ_n}(Fix T_{γ_n})' uses set notation; since Q is single-valued, this should be '∈' rather than '⊆'.
  5. [Corollary 4.3(iii)] The phrase 'in view of the second step in (25)' appears inaccurate; the identity w_n − y_n = x_n − z_n follows from the first line of (25) (or from the displayed update in Step 2). Please correct the cross-reference.

Circularity Check

0 steps flagged · score 2.0 of 10

No constructive circularity: the linear-rate theorem derives its conclusion from the assumed uniform linear regularity; the self-cited relocated-fixed-point framework supplies prior support, not the target result.

full rationale

The central result, Theorem 3.6, is not circular. Its key hypothesis, uniform bounded linear regularity (Definition 3.1, Eq. (11)), is an input about the operator family and is not derived from or equivalent to the R-linear convergence conclusion; the proof combines the α-averaged nonexpansive inequality with the Lipschitz relocator bound to obtain the contraction estimate (18), then uses the R-linear stepsize sequence to upgrade the distance decay to convergence of (x_n) and (T_{γ_n}x_n). The use of Theorem 2.8 from the authors' own prior framework [3] supplies only weak convergence/boundedness and relocator properties; it does not assume the rate result, so the self-citation is foundational but not a circular reduction. Lemma 3.3 independently proves linear regularity for contractions. In the DR application, Lemma 4.2 delegates the linear-regularity verification to the external [25, Theorem 2] with no proof supplied for the paramonotone case; this is a missing-support/correctness gap, not a circular step, since [25] is not by the present authors and its constant κγ enters Theorem 3.6 as an input rather than as the desired convergence conclusion. The compactness of Γ is used in Lemmas 4.1 and 4.2 without explicitly assuming Γ is bounded; that is likewise a correctness concern, not circularity. Overall, no equation is equivalent by construction to its own output.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. Central claim rests on external results: the relocated-framework machinery [3] (same authors), linear regularity transfer [25], and contraction estimates [23].

assumptions (6)
  • domain assumption FixTγ = P + γD for paramonotone A1,A2 (Proposition 2.1, via [5, Corollary 5.5(iii)])
    Used in Lemma 4.2 to establish uniform linear regularity; proof is cited, not reproduced.
  • domain assumption Theorem 2.8 (weak convergence of relocated iterations) from [3] by the same authors
    Gives boundedness of (x_n), needed to apply bounded linear regularity in Theorem 3.6; the proof is in the companion paper [3].
  • domain assumption Lipschitz continuity of (x,γ) ↦ J_{γA}x on compact subsets of H×R++ (Remark 2.2, [3, Prop 3.4])
    Used in Corollaries 4.3 and 5.5 to control parameter changes for resolvents and relocators.
  • domain assumption Lemma 4.2's linear regularity of the DR operator: proof 'analogous to [25, Theorem 2]' is not given
    This is the load-bearing verification of uniform linear regularity for the DR application; the actual argument lives in an external preprint.
  • domain assumption Contraction constant βγ for DR when A1 Lipschitz and A2 strongly monotone is quoted from [23, Theorem 4.3]
    Used in Lemma 4.4/Corollary 4.5; the formula is imported without derivation.
  • standard math Standard Hilbert-space monotone operator theory (resolvent averagedness, projection nonexpansiveness, etc.)
    Background throughout; cited to [6].

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Cite this review

Pith. "Pith review of Linear convergence of relocated fixed-point iterations." pith.science (2026). https://pith.science/paper/G6TCUP24

@misc{pith2026251212954,
  author       = {Pith},
  title        = {Pith review of: Linear convergence of relocated fixed-point iterations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G6TCUP24}},
  note         = {Machine review of arXiv:2512.12954}
}
read the original abstract

We establish linear convergence of relocated fixed-point iterations as introduced by Atenas et al. (2026) DOI: 10.1137/25M1776810 assuming the algorithmic operator satisfies a linear error bound. In particular, this framework applies to the setting where the algorithmic operator is a contraction. As a key application of our framework, we obtain linear convergence of the relocated Douglas--Rachford algorithm for finding a zero in the sum of two monotone operators in a setting with Lipschitz continuity and strong monotonicity assumptions. We also apply the framework to deduce linear convergence of variable stepsize resolvent splitting algorithms for multioperator monotone inclusions.

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