A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators

Bang Cong Vu; Volkan Cevher

A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For $0\in Ax+Bx$ with $A$ maximally monotone and $B$ monotone and $\mu$-Lipschitzian, the reflected forward-backward iteration converges weakly to a zero for every step size below $(\sqrt{2}-1)/\mu$.

desk verdict A sound, honest generalization of Malitsky's reflected gradient method to monotone inclusions; the three-operator variant is genuinely new but leans on global cocoercivity, which is a limitation to state, not a flaw. read the letter →

arxiv 1908.05912 v1 pith:HEOZOUBH submitted 2019-08-16 math.OC

Volkan Cevher , Bang Cong Vu This is my paper · ORCID

classification math.OC MSC 47H0549M2949M2790C25

keywords monotoneinclusionsoperatorsplittingreflectedforward-backwardcocoerciveoperatorsweakconvergenceLipschitzianprimal-dualalgorithmscomposite

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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 claims that a 'reflected' version of forward-backward splitting—where the Lipschitzian operator is evaluated at the reflected point $y_n=2x_n-x_{n-1}$ instead of at $x_n$—converges weakly to a zero of $A+B$, for any maximally monotone operator $A$ and any monotone $\mu$-Lipschitzian operator $B$, as long as the step size is below $(\sqrt{2}-1)/\mu$. This matters because it gives a provably convergent splitting that needs only one evaluation of the Lipschitzian operator per iteration, unlike older forward-backward-forward schemes that required two. The same idea is combined with ordinary forward-backward splitting to handle sums of three operators $A+B+C$, and is applied to composite primal-dual inclusions. A careful reader would care because these methods are the workhorses of large-scale convex optimization, where operator evaluations dominate cost.

What carries the argument

The load-bearing object is the reflected extrapolation $y_n=2x_n-x_{n-1}$, which lets the method update with one evaluation of $B$ per step instead of two. The proof tracks a Lyapunov function that is small at a solution: bounds on $\|x_n-x\|^2$, $\|x_{n-1}-x_n\|^2$, $\|p_n+\gamma Bx\|^2$, and related reflected-difference terms. A telescoping inequality forces $\sum_n\|x_n-y_n\|^2<\infty$ and $\sum_n\|x_{n+1}-y_n\|^2<\infty$, hence $x_{n+1}-x_n\to0$. In the three-operator case, cocoercivity of $C$ contributes the dissipative term $-\beta\gamma\|Cx_n-Cx\|^2$, and the Lyapunov function $\alpha_{n+1}$ is shown nonincreasing and coercive, so the iterates stay bounded. The argument finishes with the standard lemma that bounded iterates with converging norms and only zeros of $A+B$ (or $A+B+C$) as weak cluster points must converge weakly.

What would settle it

Set $H=\mathbb{R}^2$, take $A$ to be the normal cone of the positive orthant, and take $B$ to be the 90-degree rotation matrix; run the two-operator iteration with $\gamma$ just below $(\sqrt{2}-1)/\mu$ from several starting points—if any such run fails to converge weakly, the theorem's central bound is unsound. For the three-operator claim, replace $C$ by the same rotation matrix (monotone Lipschitzian but not cocoercive) and run the semi-reflected scheme under (3.3); divergence would show the cocoercivity assumption is doing essential work.

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Extended reading notes

Core claim

For the inclusion $0\in Ax+Bx$ with $A$ maximally monotone and $B$ monotone and $\mu$-Lipschitzian, the reflected forward-backward splitting $x_{n+1}=(I+\gamma A)^{-1}(x_n-\gamma B(2x_n-x_{n-1}))$ produces a sequence that converges weakly to some solution for every $\gamma\in(0,(\sqrt{2}-1)/\mu)$. When $B$ is $\beta$-cocoercive the admissible range enlarges to $\gamma\in(0,\beta/2)$, matching the natural forward-backward range. For the three-operator inclusion $0\in Ax+Bx+Cx$ with $C$ $\beta$-cocoercive, the semi-reflected iteration $x_{n+1}=J_{\gamma A}(x_n-\gamma B(2x_n-x_{n-1})-\gamma Cx_n)$ also converges weakly under explicit step-size conditions. The paper further embeds the two-operator method in a product space to solve composite inclusions involving parallel sums, where the iteration remains a single evaluation of each Lipschitzian component per step.

Load-bearing premise

The proof for the three-operator method and the enlarged-stepsize case requires the operator to be cocoercive on the entire space, a property strictly stronger than both monotonicity and Lipschitz continuity; if that assumption fails, the step-size ranges and convergence guarantees in those results have no basis.

Editorial extensions

If this is right

Any $\beta$-cocoercive $B$ can be handled with step sizes up to $\beta/2$, the same range as ordinary forward-backward splitting, while still using only one evaluation of $B$ per iteration.
For convex minimization $\min f+h$ with $h$ convex and $\mu$-Lipschitz-smooth, the reflected proximal-gradient method converges for $\gamma<1/(2\mu)$.
The three-operator method solves $0\in Ax+Bx+Cx$ with $A$ maximally monotone, $B$ monotone $\mu$-Lipschitzian, and $C$ $\beta$-cocoercive, under the explicit bounds in (3.3).
In the product-space formulation of Section 4, the same iteration yields weak convergence of primal and dual sequences for composite inclusions involving parallel sums, with one evaluation of each Lipschitzian operator per step.

Reading between the lines

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

Inference: the two-operator step-size bound $\gamma<(\sqrt{2}-1)/\mu$ may be sharp for general monotone Lipschitzian $B$; constructing a divergence example at the critical value would test that.
Inference: the cocoercivity of $C$ in Theorem 3.3 is used only through a quadratic dissipation term; one could try to replace it by a weaker 'dissipation at the solution' condition, but the paper does not.
Inference: because each iteration uses one evaluation of each operator, variance-reduced or stochastic variants of $B$ might inherit the same Lyapunov structure; the paper does not address randomness.
Inference: the reflected construction could likely be combined with variable-metric resolvents to give a preconditioned method, but no such extension is claimed here.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper does what it says: it takes the projected reflected gradient method from variational inequalities and proves weak convergence for the general monotone inclusion 0 ∈ Ax + Bx, with A maximally monotone and B monotone and Lipschitzian. The two-operator result is a legitimate extension of Malitsky's work, and when B is linear it reduces exactly to the forward-reflected-backward method of Malitsky–Tam. The improved stepsize under cocoercivity is a small but useful addition. Section 3's SRFB method for three operators is the most novel piece; it combines the reflected idea with forward-backward in a way I have not seen before, and the convergence conditions are explicit and checkable. The proofs are self-contained, the Lyapunov arguments are standard, and I found no circularity. The citations to [14,15,16] are appropriate and not doing load-bearing work beyond context and special cases.

The soft spots are real but mostly minor. The cocoercivity assumption in Theorem 2.1(i) and Theorem 3.3 is explicit and used correctly to produce the dissipative term, but it does restrict the practical range: many monotone Lipschitz operators of interest are not globally cocoercive, and for the three-operator method you need C to be cocoercive on the whole space. That is a genuine limitation of scope, not a hidden flaw, and it should be stated more prominently. The proof also invokes maximal monotonicity of A+B and A+B+C in the limit steps without stating the standard sum theorem; that is a small gap that a referee should ask the authors to fix. The definition of T_n after (2.9) has a misindexed subscript; again minor. No numerical experiments or code are provided, which is fine for a theory paper but leaves the practical behavior of SRFB untested.

I agree with the stress-test note more than with the reader's conditional verdict: the central claims hold up, and the concerns are presentation-level rather than load-bearing. This is a useful paper for people working on operator splitting and nonsmooth optimization. It deserves a serious referee and, after minor revision, publication. I would take it to reading group and would probably cite it for the three-operator method.

Referee Report

0 major / 6 minor

Summary. The paper studies reflected forward-backward splitting (RFBS) for monotone inclusions on a Hilbert space. For the inclusion 0 in Ax + Bx, with A maximally monotone and B monotone and mu-Lipschitzian, it proves weak convergence of the one-evaluation-per-iteration iteration (1.6) for gamma in (0, (sqrt(2)-1)/mu), and improves the stepsize range to gamma in (0, beta/2] when B is beta-cocoercive. It then proposes a semi-reflected forward-backward (SRFB) method for the three-operator inclusion 0 in Ax + Bx + Cx, where C is cocoercive, and proves weak convergence under explicit stepsize bounds. A primal-dual application to composite monotone inclusions is given in Section 4. The main results are Theorem 2.1, Theorem 3.3, and Corollary 4.2.

Significance. If correct, the paper contributes a simple, cheap-to-implement splitting method that extends Malitsky's projected reflected gradient method from normal-cone operators to general maximal monotone operators. The improved cocoercive stepsize and the three-operator SRFB combination are useful additions to the operator-splitting toolkit. The proofs are self-contained and the Lyapunov algebra is coherent; I checked the key inequalities (2.14), (2.19), and (3.26) and found them consistent. The main limitation, namely the explicit reliance on global cocoercivity in Theorem 2.1(i) and Theorem 3.3, is a real restriction on applicability but it is not hidden and is used correctly to provide the dissipative terms. The paper is an incremental but solid theoretical contribution; it does not require numerical experiments for its claims to be credible.

minor comments (6)

[Section 2, after Eq. (2.9)] The definition of T_n is misindexed. As written, T_n = ||x_{n+1}-x||^2 - 2 <p_{n+1}+gamma Bx | x_{n+1}-x_n>, but the inequality (2.10) requires T_n = ||x_n-x||^2 - 2 <p_n+gamma Bx | x_n-x_{n-1}>. Please adjust the index so that the subsequent inequality reads consistently.
[Theorem 2.1(i), proof after Eq. (2.14)] The theorem allows gamma = beta(1-epsilon)/2, but the proof says 'Since gamma < (1-epsilon)beta/2, we obtain ...'. The endpoint is harmless because the coefficient 2beta - 4gamma/(1-epsilon) vanishes there, but the proof should use 'less than or equal to' or explain that the strict inequality is not essential.
[Theorem 2.1(ii) and Theorem 3.3, limit passages] The proofs invoke maximal monotonicity of A+B (in Theorem 2.1(ii)) and of A+B+C (in Theorem 3.3) without stating why these sums are maximally monotone. This is a standard consequence of the sum theorem for a maximally monotone operator and a Lipschitz monotone (hence maximally monotone, full-domain) operator, but it should be stated explicitly or cited, since it is load-bearing for the cluster-point argument.
[Theorem 2.1(i), cluster-point step] In the sentence 'Since B is maximally monotone, its graph is closed ..., we obtain Bx = Bx and thus By_{k_n} -> Bx', the equality 'Bx = Bx' appears to be a typo for 'B xbar = Bx' where xbar is the weak cluster point. As written it is tautological.
[Theorem 3.3, inequality (3.24)] The inequality (3.24) uses the constant (1+sqrt(2)) but the Young-expansion details that lead to the three quadratic terms with coefficients gamma mu(1+sqrt(2)), gamma mu, and gamma mu sqrt(2) are not shown. Please spell out the bound on ||y_n - y_{n-1}|| in terms of the increments y_n - x_n and y_{n-1} - x_{n-1}, so that the reader can verify the coefficients.
[Throughout] There are several typographical and presentation issues: 'Lipschizian' in the abstract, 'Optial's result' in the proof of Theorem 3.3, 'sing' for 'using' in the proof of Corollary 4.2, and the reference [16] is listed as 'Y. Malitsky, Y. and M. K.Tam'. In Corollary 4.2, the formula for mu in (4.4) is stated without derivation; a brief justification or explicit citation to [8] for this Lipschitz bound would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence proofs in Theorems 2.1 and 3.3 are self-contained algebraic arguments built on explicit monotonicity, Lipschitz, and cocoercivity assumptions; cited works are context or special cases, not load-bearing premises.

full rationale

The paper's central claims are proved from scratch: Theorem 2.1 establishes weak convergence of the reflected forward-backward splitting method by constructing Lyapunov-type inequalities (2.14) and (2.21) directly from monotonicity, Lipschitz continuity, and cocoercivity, and then invoking Opial's lemma. No fitted parameter is renamed as a prediction, and the stepsize ranges are derived from the inequalities rather than imposed to force convergence. Theorem 3.3 similarly proves convergence of the semi-reflected forward-backward splitting via the descent inequality (3.26), with the cocoercivity of C used explicitly to generate the dissipative term -beta gamma ||Cx_n - Cx||^2 in (3.15); this is an explicit hypothesis, not a hidden input. The references to [14], [15], and [16] are used to situate the method and to identify special cases (e.g., Remark 2.2(i)-(iii)), but the convergence arguments do not depend on those papers. Corollary 4.2 uses the standard primal-dual reduction technique from [8] to rewrite the composite problem in product-space form and then applies Theorem 2.1; this is an external, verifiable transformation, not a self-citation chain. No uniqueness theorem is imported from the authors, no ansatz is smuggled in via citation, and no known result is merely renamed. The minor notational and indexing slips noted in the proofs do not create circularity. The derivation chain therefore remains self-contained, and no step reduces by construction to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper is a proof-based theory contribution. It introduces no fitted parameters and no new postulated objects. The central claims rest on standard monotone operator theory, particularly maximal monotonicity, resolvent calculus, Lyapunov arguments, and Opial's lemma. The main domain assumption is existence of a zero point, which is standard for operator splitting.

assumptions (5)

domain assumption There exists x in H such that 0 in Ax plus Bx (and 0 in Ax plus Bx plus Cx in Section 3).
Stated at the start of Problem 3.1 and implicitly assumed throughout; the Lyapunov analysis fixes an arbitrary zero point x and would not apply otherwise.
standard math A is maximally monotone and J_{gamma A} is single-valued and everywhere defined.
A standard consequence of Minty's theorem; every iteration requires the resolvent to be well-defined.
standard math The sum of a maximally monotone operator and a monotone Lipschitzian operator is maximally monotone.
Invoked in the final limit steps of Theorem 2.1(ii) and Theorem 3.3 to pass from approximate inclusions to 0 in (A+B)x or 0 in (A+B+C)x; not cited explicitly.
standard math Opial's lemma, which turns boundedness and unique sequential limits into weak convergence.
Used at the end of both proofs without proof; standard in Hilbert space fixed point theory.
standard math Baillon-Haddad theorem in Example 2.3, implying the gradient of a convex differentiable function with mu-Lipschitz gradient is (1/mu)-cocoercive.
Basis for applying Theorem 2.1(i) to convex minimization.

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Cite this review

Pith. "Pith review of A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators." pith.science (2026). https://pith.science/paper/HEOZOUBH

@misc{pith2026190805912,
  author       = {Pith},
  title        = {Pith review of: A reflected forward-backward splitting method for monotone inclusions involving Lipschitzian operators},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HEOZOUBH}},
  note         = {Machine review of arXiv:1908.05912}
}

read the original abstract

The proximal extrapolated gradient method \cite{Malitsky18a} is an extension of the projected reflected gradient method \cite{Malitsky15}. Both methods were proposed for solving the classic variational inequalities. In this paper, we investigate the projected reflected gradient method, in the general setting, for solving monotone inclusions involving Lipschitzian operators. As a result, we obtain a simple method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. We also show that one can improve the range of the stepsize of this method for the case when the Lipschitzian operator is restricted to be cocoercive. A nice combination of this method and the forward-backward splitting was proposed. As a result, we obtain a new splitting method for finding a zero point of the sum of three operators ( maximally monotone + monotone Lipschitzian + cocoercive). Application to composite monotone inclusions are demonstrated.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

H. H. Bauschke, P. L. Combettes , Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, 2nd ed., 2017

work page 2017

[2] [2]

R. I. Bot ¸ and C. Hendrich, A Douglas–Rachford type primal-dual method for solving inc lu- sions with mixtures of composite and parallel-sum type mono tone operators, SIAM J. Optim., 23 (2013), pp. 2541–2565

work page 2013

[3] [3]

R. I. Bot ¸ and C. Hendrich , Convergence Analysis for a Primal-Dual Monotone + Skew Splitting Algorithm with Applications to Total Variation M inimization, J. Math. Imaging Vis., 49 (2014), pp. 551–568

work page 2014

[4] [4]

L. M. Brice ˜no-Arias, Forward–Douglas–Rachford splitting and forward–partial inverse method for solving monotone inclusions , Optimization, 64 (2015), pp. 1239–1261

work page 2015

[5] [5]

L. M. Brice ˜no-Arias and P. L. Combettes , A monotone +skew splitting model for com- posite monotone inclusions in duality , SIAM J. Optim., 21 (2011), pp. 1230–1250

work page 2011

[6] [6]

L. M. Brice ˜no-Arias and D. Davis , Forward-Backward-Half Forward Algorithm for Solving Monotone Inclusions , SIAM J. Optim., 28 (2018), pp. 2839–2871

work page 2018

[7] [7]

P. L. Combettes , Systems of structured monotone inclusions: duality, algor ithms, and ap- plications, SIAM J. Optim. , 23 (2013), pp. 2420–2447

work page 2013

[8] [8]

P. L. Combettes and J.-C. Pesquet , Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators, Set-Valued Var. Anal., 20 (2012), pp. 307–330

work page 2012