Pith. sign in

REVIEW 2 major objections 4 minor 2 cited by

Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Variable stepsizes are now safe in Douglas–Rachford splitting.

desk verdict A clean, genuinely useful fixed-point framework that removes the common-fixed-point assumption for variable-stepsize resolvent splitting; the bounded-variation stepsize condition is explicit and reasonable, and the paper deserves serious refereeing. read the letter →

arxiv 2507.07428 v3 pith:GDWMYPI3 submitted 2025-07-10 math.OC

Felipe Atenas , Heinz H. Bauschke , Minh N. Dao , Matthew K. Tam This is my paper · ORCID
classification math.OC MSC 47H0547H0947N1065K0547H04
keywords fixed-pointiterationsdemiclosednessprinciplenonexpansiveoperatorresolventsplittingDouglas–Rachfordvariablestepsize
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

This paper proves that iterative splitting algorithms like Douglas–Rachford can let the stepsize change from one iteration to the next and still converge to a solution, even though the fixed-point sets of the operators change with the stepsize and may be completely disjoint. The key mechanism is a 'fixed-point relocator': after applying the current operator, a second affine step maps fixed points of the current parameter to fixed points of the next parameter, keeping a Fejér-type distance inequality alive. Convergence follows from a new parametric demiclosedness principle, needing only that the stepsizes converge and that their upward moves have finite total sum. If correct, adaptive and line-search based resolvent splitting methods inherit weak convergence without the common-fixed-point assumption that previously blocked variable stepsize analysis.

What carries the argument

The paper's central object is the fixed-point relocator (Definition 4.1): a family of Lipschitz maps $Q_{\delta\leftarrow\gamma}$ whose restriction to $\operatorname{Fix}T_\gamma$ is a bijection onto $\operatorname{Fix}T_\delta$, with cocycle identity $Q_{\varepsilon\leftarrow\delta}Q_{\delta\leftarrow\gamma}=Q_{\varepsilon\leftarrow\gamma}$ on fixed points and continuity in the parameter. Paired with the parametric demiclosedness principle (Theorem 3.9), which extends Browder's principle to families continuous in a parameter, it lets the proof transfer the classical Opial argument to a moving target: the sequence of 'anchor' fixed points $c_n$ tracks the changing fixed-point set and the inequality $\|x_{n+1}-c_{n+1}\|\le L_{\gamma_{n+1}\leftarrow\gamma_n}\|x_n-c_n\|$ converts the bounded Lipschitz sum into a Robbins–Siegmund recursion.

What would settle it

Take $X=\mathbb{R}$, $A=N_{\{1\}}$, $B=\partial(-\ln)$, so $\operatorname{Fix}T_\gamma=\{1+\gamma\}$ (Remark 4.13), and run Algorithm 1 with $\gamma_n=1+\sum_{k=1}^n k^{-1}$, an increasing sequence with infinite upward variation. If $x_n$ fails to converge or $z_n$ fails to approach $1$, the assumption (17) is necessary; conversely, replacing $\gamma_n$ by $1+\sum k^{-2}$ (finite upward variation) must yield weak convergence of $z_n$ to $1$ for the theorem to hold. A single such counterexample or verification would settle the claim.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.5: for any family of nonexpansive operators $(T_\gamma)$ with fixed points and fixed-point relocators $(Q_{\delta\leftarrow\gamma})$ whose Lipschitz constants satisfy $\sum_n (L_{\gamma_{n+1}\leftarrow\gamma_n}-1)<\infty$, the relocated iteration $x_{n+1}=Q_{\gamma_{n+1}\leftarrow\gamma_n}T_{\gamma_n}x_n$ is Opial with respect to $\operatorname{Fix}T_\gamma$, and under uniform averagedness and joint continuity it converges weakly to a point in $\operatorname{Fix}T_\gamma$. Applied to Douglas–Rachford, the paper exhibits an explicit relocator $Q_{\delta\leftarrow\gamma}=\frac{\delta}{\gamma}\mathrm{Id}+(1-\frac{\delta}{\gamma})J_{\gamma A}$, yielding a variable stepsize Douglas–Rachford method (Algorithm 1) in which the step sequence needs only $\inf_n\gamma_n>0$ and $\sum_n(\gamma_{n+1}-\gamma_n)_+<\infty$. The same scheme extends to graph-based resolvent splitting for sums of $N\ge2$ maximally monotone operators.

Load-bearing premise

The step-size sequence must have a positive lower bound and its upward jumps must be summable, $\sum_n(\gamma_{n+1}-\gamma_n)_+<\infty$; if upward jumps accumulate, the relocated anchor points drift without a limiting distance and the proof's central inequality no longer yields convergence.

Editorial extensions

If this is right

  • Variable stepsize Douglas–Rachford (Algorithm 1) converges weakly without requiring a common fixed point across iterations.
  • The graph-based Douglas–Rachford extension for sums of $N\ge2$ monotone operators admits variable stepsizes (Corollary 5.4).
  • The Malitsky–Tam resolvent splitting method can be run with changing stepsizes at the price of a small modification, keeping one resolvent per operator per iteration (Corollary 5.7).
  • Any stepsize rule with bounded upward increments and positive lower bound, including increasing bounded sequences, is admissible; stepsize rules that can only be shown to have such structure become analysable.

Reading between the lines

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

  • One can view the relocator as a 'moving anchor' mechanism; the same idea may apply to other parametrized operator families, such as three-operator or Davis–Yin splitting, where the fixed-point set depends on a parameter.
  • The condition $\sum_n(\gamma_{n+1}-\gamma_n)_+<\infty$ is an asymmetric bounded-variation condition: upward jumps must be summable, while downward jumps are unrestricted. This asymmetry may be intrinsic to the anchor-drift argument, though the paper does not prove necessity.
  • A natural numerical test would use an increasing, bounded stepsize sequence (admissible under Remark 4.10(ii)) and compare the relocated DR limit point with the expected shadow limit; if upward jumps are chosen adaptively, the weak limit of the shadow sequence should track a solution.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes a convergence framework for iterations of the form x_{n+1}=Q_{γ_{n+1}←γ_n}T_{γ_n}x_n, where (T_γ) is a one-parameter family of nonexpansive operators and (Q_{δ←γ}) are fixed-point relocators mapping Fix T_γ bijectively onto Fix T_δ. The main abstract result, Theorem 4.5, shows that if the relocator Lipschitz constants satisfy sum_n(L_{γ_{n+1}←γ_n}-1)<∞, then the iteration is Opial with respect to Fix T_γ; under uniform averagedness and joint continuity, both (x_n) and (T_{γ_n}x_n) converge weakly to a common point of Fix T_γ. The framework is applied to variable-stepsize Douglas–Rachford splitting (Algorithm 1) and to graph-based Douglas–Rachford and Malitsky–Tam resolvent splitting (Algorithms 2 and 3). The paper also develops a parametric demiclosedness principle (Theorem 3.9) and resolvent identities (Lemma 3.1, Theorem 3.5) used to construct the relocators.

Significance. The abstract framework is a genuine contribution: it removes the common-fixed-point assumption that is standard in non-stationary fixed-point theory, and the proof of Theorem 4.5 is clean and machine-checkable in structure. The parametric demiclosedness principle is a natural and useful extension of Browder's principle, and the two-operator Douglas–Rachford application (Algorithm 1, Corollary 4.11) is convincing and correctly implemented. The Malitsky–Tam application (Algorithm 3, Corollary 5.7) is also well supported. However, the graph-based Algorithm 2, which is the centerpiece of Section 5.1, is not correctly derived from Proposition 5.3: it applies the relocator using the z-sweep for x_n rather than for w_n=T_{γ_n}x_n. This is a load-bearing flaw in the claimed convergence of Algorithm 2, though it is localized and fixable without changing the abstract theory.

major comments (2)
  1. [Section 5.1, Algorithm 2 and Proposition 5.3] Algorithm 2 does not implement the relocated fixed-point iteration associated with the relocator of Proposition 5.3. In Step 1, z_n is computed from x_n by (25), so w_n = T_{γ_n}x_n. The relocator of Proposition 5.3 requires, at the argument w_n, the vector e(w_n) built from the z-sweep at w_n. Algorithm 2 Step 2 instead builds e_n from the already computed z_n, i.e. from e(x_n). These differ in general. A concrete counterexample is N=2 with graph E={(1,2)}, A_1=∂ι_{0}, A_2=0, θ=1, and x_0≠0. For any γ_0, one obtains z_1=0, z_2=-x_0, w_0=x_0-S^*z_0=0, while e(x_0)≠0; hence Algorithm 2 produces x_1=(1-γ_1/γ_0)S^†e(x_0)≠0 when γ_1≠γ_0, whereas the relocated iteration gives x_1=Q_{γ_1←γ_0}(w_0)=0. Consequently, the assertion in Corollary 5.4 that Algorithm 2 satisfies x_{n+1}=Q_{γ_{n+1}←γ_n}T_{γ_n}x_n is incorrect. The algorithm must be modified to recompute the z-sweep at w_n (or to define e_n from z(w_n)), and Corollary 5.4 should be restated for the corrected algorithm.
  2. [Fact 5.1(i) and Algorithm 2 initialization] Fact 5.1(i) states d_1=1 for every connected directed graph satisfying (20), but this is false. For example, the graph with edges (1,2) and (1,3) is connected, satisfies (i,j)∈E ⇒ i<j, and has d_1=2; the paper's own Malitsky–Tam example in Section 5.2 has d_i=2 for all i, including i=1. The identity d_1^+=0 is true and is the only part of Fact 5.1(i) needed in Lemma 5.2. However, the false statement d_1=1 enters Algorithm 2: the initialization z_{0,1}=J_{γ_1A_1}(x_{0,1}) is not the correct first component of (25), which should be J_{γ_1/d_1A_1}((1/d_1)∑_j S_{1j}x_{0,j}). This must be corrected, together with the Fact 5.1(i) statement.
minor comments (4)
  1. [Introduction, paragraph after (4)] There is a duplicated word in 'a variable stepsize version of of DR given by'. Please correct the typo.
  2. [Fact 2.6] The notation in the second sentence, '(J^{γA} T^n_γ x)n∈N', appears garbled; it should presumably read (J_{γA}T_γ^n x)_{n∈N}.
  3. [Proposition 5.3] The phrase 'with is Lipschitz constants' should be 'with Lipschitz constants'.
  4. [Section 5.1, Algorithm 2] The initialization comment is tied to the major comment on Fact 5.1(i); beyond that, the algorithm would benefit from a line stating that z_{0,1} must be the value from (25) for i=1, not an independently chosen resolvent evaluation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified; the relocated fixed-point framework proves its convergence claims from explicit assumptions and constructs relocators from resolvent identities.

full rationale

The central results are not circular. Theorem 4.5 is a conditional derivation: given nonexpansive operators (T_gamma), fixed-point relocators (Q_{delta<-gamma}), and the explicit excess condition sum_n(L_{gamma_{n+1}<-gamma_n}-1)<infinity, it proves Opialness via the anchor sequence c_n and the Robbins-Siegmund lemma; no fitted quantity is renamed as a prediction. The Douglas-Rachford relocator (6) is derived, not assumed: Lemma 3.1 proves the resolvent identity J_{beta A}((beta/alpha)Id + (1-beta/alpha)J_{alpha A}) = J_{alpha A}, and Theorem 3.5 uses it to establish the bijection FixT_alpha -> FixT_beta. The graph-based relocators in Proposition 5.3 are likewise constructed from the fixed-point characterization and the pseudo-inverse solution of (27), with the Lipschitz bound proved by induction. The step-size conditions (17) and (15) are explicit assumptions, and Remark 4.10 shows they are equivalent to the known bounded-variation condition (18), so restricting to them is not a hidden input. Self-citations, such as [3] for Opial's lemma and [37] for the graph Laplacian kernel, concern standard or independently verifiable facts and are not load-bearing for the paper's new convergence claims. The applications are checked against known results (e.g., [23, Theorem 3.2] in Remark 4.12), which further supports that the derivation is self-contained rather than circular.

Assumptions & free parameters 0 free parameters · 8 assumptions · 1 invented entities

The paper relies on standard monotone operator theory (resolvent identities, demiclosedness, Robbins-Siegmund, Opial) plus explicit structural assumptions: Hilbert space setting, maximal monotonicity, nonempty solution set, graph connectivity, and step-size variation summability. There are no data-fitted free parameters.

assumptions (8)
  • domain assumption X is a real Hilbert space and the operators A_i are maximally monotone with a nonempty solution set.
    This is the standing setting for the inclusions (1) and (19), stated in Section 1 and used in Corollaries 4.11, 5.4, and 5.7.
  • standard math Demiclosedness of maximally monotone operators and Browder's demiclosedness principle.
    Facts 2.2 and 3.7 are used in Theorem 3.9 and in the weak-limit passages of Corollaries 4.11 and 5.4.
  • standard math Robbins-Siegmund lemma.
    Fact 2.7 is the summation engine in Theorem 4.5, converting the perturbed Fejer inequality into convergence and summability of residuals.
  • standard math Opial property facts for sequences.
    Fact 2.9 is used in Theorem 4.5(iii) to turn cluster-point information into weak convergence.
  • standard math Resolvent identities, nonexpansiveness, and continuity of the resolvent as a function of the parameter.
    Fact 2.5 and Proposition 3.4 underlie the Douglas-Rachford relocator construction and the continuity assumption Assumption 4.4(ii).
  • domain assumption Step-size condition: inf_n gamma_n > 0 and sum_n (gamma_{n+1}-gamma_n)_+ < infinity.
    Assumption (17) and Lemma 4.9; this is the load-bearing condition that makes the relocator Lipschitz constants summable in (15).
  • domain assumption For graph-based extensions, G is a connected directed graph satisfying (20) and G' is a connected subgraph; zer(sum_i A_i) is nonempty.
    Section 5.1 and Fact 5.1 set the graph topology and solvability conditions for the multi-operator algorithms.
  • standard math The sum of a maximally monotone operator and a skew-symmetric linear operator with full domain is maximally monotone.
    Fact 2.3 and Example 20.35 of the cited monograph are invoked in Corollary 4.11 and Corollary 5.4 to obtain demiclosedness of the lifted operator.
invented entities (1)
  • Fixed-point relocator family (Q_{delta<-gamma}) independent evidence
    purpose: Bijective map carrying fixed points of T_gamma to fixed points of T_delta so that a relocated iteration (5) can be analyzed without a common fixed point.
    The relocator is not an unobserved entity: explicit constructions with inverses and Lipschitz constants are proven for the Douglas-Rachford operator (Lemma 4.7), the graph-based operator (Proposition 5.3), and the Malitsky-Tam operator (Proposition 5.5).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting." pith.science (2026). https://pith.science/paper/GDWMYPI3

@misc{pith2026250707428,
  author       = {Pith},
  title        = {Pith review of: Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GDWMYPI3}},
  note         = {Machine review of arXiv:2507.07428}
}
abstract

In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is followed by a second step which ''relocates'' fixed points of the current operator to the next. As a consequence, our analysis does not require the family of nonexpansive operators to have a common fixed point, as frequently assumed in the literature. Our analysis uses a parametric extension of the demiclosedness principle for nonexpansive operators. As an application of our convergence results, we develop a version of the graph-based extension of the Douglas--Rachford algorithm for finding a zero of the sum of $N\geq 2$ maximally monotone operators, which does not require the resolvent parameter to be constant across iterations.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Convergence Analysis of the Restarted Moving-Anchored Extra-Gradient Method in the Absence of Local Lipschitz Continuity

    math.OC 2026-07 accept novelty 7.0 of 10

    The MAEG-R method achieves convergence for monotone inclusions with merely continuous operators via a moving-anchor restart strategy, while preserving O(1/k) complexity in the Lipschitz case.

  2. Linear convergence of relocated fixed-point iterations

    math.OC 2025-12 conditional novelty 6.0 of 10

    Relocated fixed-point iterations converge R-linearly under uniform bounded linear regularity, yielding linear rates for variable-stepsize Douglas-Rachford and multioperator resolvent splitting.

Reference graph

Works this paper leans on

39 extracted references · 37 canonical work pages · cited by 2 Pith papers

  1. [1]

    Splitting the Forward-Backward Algorithm: A Full Characterization

    A. ˚Akerman, E. Chenchene, P . Giselsson, and E. Naldi. “Splitting the Forward-Backward Algorithm: A Full Characterization”. In:arXiv preprint arXiv:2504.10999(2025)

    arXiv 2025

  2. [2]

    Forward-backward algorithms de- vised by graphs

    F. J. Arag ´on-Artacho, R. Campoy, and C. L ´opez-Pastor. “Forward-backward algorithms de- vised by graphs”. In:arXiv preprint arXiv:2406.03309(2024)

    arXiv 2024

  3. [3]

    On Opial's Lemma

    A. Arakcheev and H. H. Bauschke. “On Opial’s Lemma”. In:arXiv preprint arXiv:2503.22004 (2025)

    work page Pith review arXiv 2025

  4. [4]

    Understanding the Douglas–Rachford splitting method through the lenses of Moreau-type envelopes

    F. Atenas. “Understanding the Douglas–Rachford splitting method through the lenses of Moreau-type envelopes”. In:Computational Optimization and Applications90 (2025), pp. 881– 910

    work page 2025

  5. [5]

    Bauschke and P

    H. Bauschke and P . Combettes.Convex Analysis and Monotone Operator Theory in Hilbert Spaces. CMS Books in Mathematics. Springer International Publishing, 2017. 27

    work page 2017

  6. [6]

    Bhatia.Perturbation Bounds for Matrix Eigenvalues

    R. Bhatia.Perturbation Bounds for Matrix Eigenvalues. Classics in Applied Mathematics. Soci- ety for Industrial and Applied Mathematics, 2007

    work page 2007

  7. [7]

    Degenerate preconditioned proximal point algorithms

    K. Bredies, E. Chenchene, D. A. Lorenz, and E. Naldi. “Degenerate preconditioned proximal point algorithms”. In:SIAM Journal on Optimization32.3 (2022), pp. 2376–2401

    work page 2022

  8. [8]

    Graph and distributed extensions of the Douglas– Rachford method

    K. Bredies, E. Chenchene, and E. Naldi. “Graph and distributed extensions of the Douglas– Rachford method”. In:SIAM Journal on Optimization34.2 (2024), pp. 1569–1594

    work page 2024

Show all 39 references
  1. [9]

    Semicontractive and semiaccretive nonlinear mappings in Banach spaces

    F. E. Browder. “Semicontractive and semiaccretive nonlinear mappings in Banach spaces”. In:Bulletin of the American Mathematical Society74 (1968), pp. 660–665

    1968

  2. [10]

    Regular sequences of quasi-nonexpansive operators and their applications

    A. Cegielski, S. Reich, and R. Zalas. “Regular sequences of quasi-nonexpansive operators and their applications”. In:SIAM Journal on Optimization28.2 (2018), pp. 1508–1532

    2018

  3. [11]

    Quasi-Fej ´erian analysis of some optimization algorithms

    P . L. Combettes. “Quasi-Fej ´erian analysis of some optimization algorithms”. In:Inherently parallel algorithms in feasibility and optimization and their applications (Haifa, 2000). Vol. 8. Stud. Comput. Math. North-Holland, Amsterdam, 2001, pp. 115–152

    2000

  4. [12]

    Adaptive Douglas–Rachford splitting algorithm for the sum of two operators

    M. N. Dao and H. M. Phan. “Adaptive Douglas–Rachford splitting algorithm for the sum of two operators”. In:SIAM Journal on Optimization29.4 (2019), pp. 2697–2724

    2019

  5. [13]

    A general approach to distributed operator split- ting

    M. N. Dao, M. K. Tam, and T. D. Truong. “A general approach to distributed operator split- ting”. In:Journal of Mathematical Analysis and Applications562.2 (2026), p. 130692

    2026

  6. [14]

    A three-operator splitting scheme and its optimization applications

    D. Davis and W. Yin. “A three-operator splitting scheme and its optimization applications”. In:Set-Valued and Variational Analysis25 (2017), pp. 829–858

    2017

  7. [15]

    On the numerical solution of heat conduction problems in two and three space variables

    J. Douglas and H. H. Rachford. “On the numerical solution of heat conduction problems in two and three space variables”. In:Transactions of the American mathematical Society82.2 (1956), pp. 421–439

    1956

  8. [16]

    From perspective maps to epigraphical projections

    M. P . Friedlander, A. Goodwin, and T. Hoheisel. “From perspective maps to epigraphical projections”. In:Mathematics of Operations Research48.3 (2023), pp. 1711–1740

    2023

  9. [17]

    On the convergence of the proximal point algorithm for convex minimization

    O. G ¨uler. “On the convergence of the proximal point algorithm for convex minimization”. In:SIAM Journal on Control and Optimization29.2 (1991), pp. 403–419

    1991

  10. [18]

    Horn and C

    R. Horn and C. Johnson.Matrix Analysis. Cambridge University Press, 2012

    2012

  11. [19]

    Douglas–Rachford splitting for nonconvex optimization with appli- cation to nonconvex feasibility problems

    G. Li and T. K. Pong. “Douglas–Rachford splitting for nonconvex optimization with appli- cation to nonconvex feasibility problems”. In:Mathematical programming159 (2016), pp. 371– 401

    2016

  12. [20]

    Survey: sixty years of Douglas–Rachford

    S. B. Lindstrom and B. Sims. “Survey: sixty years of Douglas–Rachford”. In:Journal of the Australian Mathematical Society110.3 (2021), pp. 333–370

    2021

  13. [21]

    Splitting algorithms for the sum of two nonlinear operators

    P .-L. Lions and B. Mercier. “Splitting algorithms for the sum of two nonlinear operators”. In: SIAM Journal on Numerical Analysis16.6 (1979), pp. 964–979

    1979

  14. [22]

    The degenerate variable metric proximal point algorithm and adaptive stepsizes for primal–dual Douglas–Rachford

    D. A. Lorenz, J. Marquardt, and E. Naldi. “The degenerate variable metric proximal point algorithm and adaptive stepsizes for primal–dual Douglas–Rachford”. In:Optimization74.6 (2025), pp. 1355–1381

    2025

  15. [23]

    Non-stationary Douglas–Rachford and alternating direc- tion method of multipliers: adaptive step-sizes and convergence

    D. A. Lorenz and Q. Tran-Dinh. “Non-stationary Douglas–Rachford and alternating direc- tion method of multipliers: adaptive step-sizes and convergence”. In:Computational Opti- mization and Applications74 (2019), pp. 67–92

    2019

  16. [24]

    A forward-backward splitting method for monotone inclusions without cocoercivity

    Y. Malitsky and M. K. Tam. “A forward-backward splitting method for monotone inclusions without cocoercivity”. In:SIAM Journal on Optimization30.2 (2020), pp. 1451–1472. 28

    2020

  17. [25]

    Resolvent splitting for sums of monotone operators with mini- mal lifting

    Y. Malitsky and M. K. Tam. “Resolvent splitting for sums of monotone operators with mini- mal lifting”. In:Mathematical Programming201.1 (2023), pp. 231–262

    2023

  18. [26]

    Monotone (nonlinear) operators in Hilbert space

    G. J. Minty. “Monotone (nonlinear) operators in Hilbert space”. In:Duke Mathematical Journal 29 (1962), pp. 341–346

    1962

  19. [27]

    A quantitative Robbins-Siegmund theorem

    M. Neri and T. Powell. “A quantitative Robbins-Siegmund theorem”. In:The Annals of Ap- plied Probability36.1 (2026), pp. 636–651

    2026

  20. [28]

    Weak convergence of the sequence of successive approximations for nonexpansive mappings

    Z. Opial. “Weak convergence of the sequence of successive approximations for nonexpansive mappings”. In:Bulletin of the American Mathematical Society73 (1967), pp. 591–597

    1967

  21. [29]

    Adaptive three operator splitting

    F. Pedregosa and G. Gidel. “Adaptive three operator splitting”. In:International Conference on Machine Learning. PMLR. 2018, pp. 4085–4094

    2018

  22. [30]

    Peypouquet.Convex Optimization in Normed Spaces: Theory, Methods and Examples

    J. Peypouquet.Convex Optimization in Normed Spaces: Theory, Methods and Examples. Springer- Briefs in Optimization. Springer International Publishing, 2015

    2015

  23. [31]

    Linear convergence of the Douglas–Rachford method for two closed sets

    H. M. Phan. “Linear convergence of the Douglas–Rachford method for two closed sets”. In: Optimization65.2 (2016), pp. 369–385

    2016

  24. [32]

    Polyak.Introduction to Optimization

    B. Polyak.Introduction to Optimization. Translations series in mathematics and engineering. Optimization Software, Publications Division, 1987

    1987

  25. [33]

    A convergence theorem for non negative almost super- martingales and some applications

    H. Robbins and D. Siegmund. “A convergence theorem for non negative almost super- martingales and some applications”. In:Optimizing Methods in Statistics. Elsevier, 1971, pp. 233–257

    1971

  26. [34]

    On the virtual convexity of the domain and range of a nonlinear maximal monotone operator

    R. Rockafellar. “On the virtual convexity of the domain and range of a nonlinear maximal monotone operator”. In:Mathematische Annalen185.2 (1970), pp. 81–90

    1970

  27. [35]

    Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting

    E. K. Ryu. “Uniqueness of DRS as the 2 operator resolvent-splitting and impossibility of 3 operator resolvent-splitting”. In:Mathematical Programming182.1 (2020), pp. 233–273

    2020

  28. [36]

    On weak convergence of the Douglas–Rachford method

    B. F. Svaiter. “On weak convergence of the Douglas–Rachford method”. In:SIAM Journal on Control and Optimization49.1 (2011), pp. 280–287

    2011

  29. [37]

    Frugal and decentralised resolvent splittings defined by nonexpansive opera- tors

    M. K. Tam. “Frugal and decentralised resolvent splittings defined by nonexpansive opera- tors”. In:Optimization Letters18.7 (2024), pp. 1541–1559

    2024

  30. [38]

    Douglas–Rachford splitting and ADMM for nonconvex opti- mization: Tight convergence results

    A. Themelis and P . Patrinos. “Douglas–Rachford splitting and ADMM for nonconvex opti- mization: Tight convergence results”. In:SIAM Journal on Optimization30.1 (2020), pp. 149– 181

    2020

  31. [39]

    Projections on convex sets in Hilbert space and spectral theory: Part I. projections on convex sets: Part II. spectral theory

    E. H. Zarantonello. “Projections on convex sets in Hilbert space and spectral theory: Part I. projections on convex sets: Part II. spectral theory”. In:Contributions to Nonlinear Functional Analysis. Elsevier, 1971, pp. 237–424. 29

    1971

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.