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Nonlinear Forward-Backward Splitting with Projection Correction

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Pith's one-line read A two-step nonlinear splitting method (NOFOB) subsumes forward-backward-forward, forward-backward-half-forward, AFBA, and synchronous projective splitting under one convergence and linear-rate theorem.

desk verdict Solid unified splitting framework whose weak convergence core holds up, but Proposition 8's proof gap leaves Theorem 5's step-size range unproven as written. read the letter →

arxiv 1908.07449 v3 pith:JZMA56ME submitted 2019-08-20 math.OC

Pontus Giselsson This is my paper · ORCID
classification math.OC MSC 90C2565K0590C30
keywords nonlinearforward-backwardsplittingmonotoneinclusionsresolventseparatinghalfspacesforward-backward-forwardforward-backward-half-forwardasymmetricforward-backward-adjointsynchronousprojective
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 proposes nonlinear forward-backward splitting (NOFOB), a two-step method for monotone inclusions $0\in Ax+Cx$: first apply the nonlinear forward-backward map with kernel $M_k$, then take a relaxed projection onto the halfspace that map generates. The paper's central claim is that this scheme converges weakly to a solution under a shared-metric strong-monotonicity and cocoercivity assumption, and that under metric subregularity the distance to the solution set converges Q-linearly while iterates converge R-linearly. The same construction, specialized to four operators, is shown to contain forward-backward-forward splitting, forward-backward-half-forward splitting, asymmetric forward-backward-adjoint splitting, and synchronous projective splitting as special cases. A sympathetic reader would care because one convergence proof and one rate argument then cover algorithms that previously required separate analyses, and the framework yields new long-step variants with relaxed step-size restrictions.

What carries the argument

The load-bearing object is the nonlinear forward-backward map $T_{\mathrm{FB}}=(M+A)^{-1}\circ(M-C)$, whose kernel $M$ is any maximally monotone, single-valued, Lipschitz operator that is 1-strongly monotone in the $\|\cdot\|_P$ metric; when $C=0$ the map reduces to the nonlinear resolvent $(M+A)^{-1}M$, generalizing both the standard resolvent and the Bregman resolvent. The argument is carried by the affine separating function $\psi_x(z)=\langle Mx-MT_{\mathrm{FB}}x, z-T_{\mathrm{FB}}x\rangle-\frac{\beta}{4}\|x-T_{\mathrm{FB}}x\|_P^2$: strong monotonicity of $M$ makes $\psi_x(x)>0$ away from solutions, monotonicity of $A$ plus cocoercivity of $C$ with $\beta\in[0,4)$ makes $\psi_x(z)\le 0$ for every solution $z$, and the relaxed projection onto the halfspace $\{z:\psi_x(z)\le 0\}$ converts that separation into Fej{\'e}r monotonicity. In the four-operator specialization $M_k=Q_k-D-K$ moves $D$ and the skew operator $K$ out of the inversion, which is why one forward evaluation of the resolvent plus one extra evaluation of $M_k$ is enough to run the method.

What would settle it

Run Algorithm 3.1 on the inclusion $0\in \partial |x| + \frac12 x$ on the real line, with kernel $M(x)=2x+0.1\sin x$ and projection metric $S=\mathrm{Id}$; if from $x_0=1$ the gap $\|x_k-\hat x_k\|$ fails to converge to $0$, then Theorem 2 is false, because the theorem asserts this gap always vanishes under the stated assumptions.

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Extended reading notes

Core claim

The discovery is that iterating a nonlinear resolvent alone need not converge, but appending a projection correction makes the whole map converge and linearize. For kernels $M_k$ that are maximally monotone, single-valued, 1-strongly monotone with respect to $\|\cdot\|_P$, and Lipschitz, the point $\hat x_k=(M_k+A)^{-1}(M_k-C)x_k$ creates a halfspace that strictly separates $x_k$ from the solution set whenever $x_k$ is not already a solution. Projecting onto that halfspace gives Fej{\'e}r monotonicity, so $\|x_k-\hat x_k\|\to 0$ and every weak cluster point solves the inclusion. When $A+C$ is metrically subregular at all solutions and the space is finite-dimensional (or the subregularity neighborhood is the whole space), the same inequality contracts the distance to the solution set, yielding local Q-linear convergence in distance and R-linear convergence of the iterates.

Load-bearing premise

The scheme's convergence rests on the existence of one way of measuring distances in which every kernel is strongly monotone while the cocoercive term stays below constant 4; if no such common measure exists, the separating-halfspace step can fail.

Editorial extensions

If this is right

  • Under Assumptions 1 and 2, NOFOB (Algorithm 3.1) generates iterates that converge weakly to a solution of $0\in Ax+Cx$, and the gap $\|x_k-\hat x_k\|_P$ converges to zero.
  • If $A+C$ is metrically subregular at every solution and the space is finite-dimensional (or the subregularity neighborhoods are global), the solution-distance converges locally Q-linearly and the iterates converge locally R-linearly.
  • The four-operator method (Algorithm 6.1), and its scalar-step versions, recover FBF, FBHF, AFBA, and synchronous projective splitting as special cases, so all inherit the weak and linear convergence guarantees.
  • Standard FBF and FBHF formulations are conservative short-step instances of the long-step Algorithm 6.2, and Algorithm 6.2 permits larger step-sizes than the standard analyses.
  • In the projective-splitting specialization with $D=E=0$, the step-size parameters are only required to be bounded above and below, and metric subregularity yields a linear convergence rate.

Reading between the lines

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

  • A testable extension the paper leaves implicit: run the long-step variant of FBHF on any cocoercive monotone inclusion and compare the largest admissible step-size against the standard short-step rule; the theory predicts the long-step bound is larger by a factor that depends only on $\beta_E$ and $L_D$.
  • Because the separating halfspace needs only the two evaluations $M_k x_k$ and $M_k \hat x_k$, the same convergence machinery should carry over to randomized or block-coordinate choices of the kernel, provided each block preserves strong monotonicity in the same metric; the paper does not analyze this.
  • The $\beta\in[0,4)$ threshold suggests that the strict separation argument is tight: at $\beta=4$ the lower bound $\psi_x(x)\ge(1-\beta/4)\|x-\hat x\|_P^2$ collapses to zero, so any convergence proof at the endpoint would need a mechanism other than the paper's separating-halfspace construction.
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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

2 major / 3 minor

Summary. The paper introduces NOFOB, a projection-correction algorithm for monotone inclusions of the form 0 in Ax + Cx. Each iteration evaluates a nonlinear forward-backward map T^k_FB = (M_k + A)^{-1}(M_k - C), constructs a separating halfspace H_k, and performs a relaxed projection onto H_k. The main convergence result (Theorem 2) proves weak convergence to zer(A + C) under Assumptions 1 and 2 via Fejer monotonicity, and Theorem 3 adds local Q-linear convergence of the distance to the solution set under metric subregularity. The paper then derives a four-operator splitting method (Algorithm 6.1) as a special case and shows that FBF, FBHF, AFBA, and synchronous projective splitting are recovered, with a short-step variant (Algorithm 6.3) recovering standard FBF/FBHF formulations. The paper also claims new linear convergence results for FBHF and projective splitting under metric subregularity.

Significance. The framework is conceptually interesting and, if the technical results are correct, provides a genuine unification: the nonlinear resolvent with an arbitrary maximal monotone kernel is more general than standard or Bregman resolvents, and the special-case catalog is substantial. The core convergence proof is a clean reduction to Fejer-monotone projections and standard monotone operator theory, and the linear-convergence analysis under metric subregularity is a useful contribution. The paper is also careful to state limitations, such as the coupling of P in Assumptions 1 and 2. However, the validity of the conservative FBF/FBHF specializations and the new linear-rate claims for them rests on Proposition 8, whose proof contains a significant gap.

major comments (2)
  1. [Appendix A.3, Proposition 8] The proof of Proposition 8 does not establish the stated bound on the stated step-size range. The step from (A.3) to (A.2) is invalid when gamma^{-1} - L_D > 1. Rearranging (A.2) and using the displayed bound on the delta-terms gives the sufficient condition A <= -gamma^{-1}((gamma^{-1}-L_D) beta_E/2 + bar_epsilon/2), whereas (A.3) is A <= -gamma^{-1}(beta_E/2 + bar_epsilon/2). When gamma^{-1} - L_D > 1, the former is strictly stronger and is not implied. For example, with L_D = 0, K = 0, beta_E = 3, epsilon = 0.01, gamma = 0.1 (which satisfies the stated hypotheses), (A.3) holds but the rearranged (A.2) fails. In addition, the proof concludes with beta_E/4 ||x-y||^2_P in the final inner-product inequality, while the proposition's statement uses beta/4 ||x-y||^2_P with beta as defined in Lemma 1; this mismatch must be resolved. Because Theorem 5 and the conservative FBF/FBHF claims rely on Proposition 8, this proof gap is load-bearing and must be corrected.
  2. [Section 5.1, Theorem 3] The proof of Theorem 3 uses a union U_star = union_z U_z and nu_star = inf_z nu_z to obtain a single metric-subregularity constant kappa valid on U_star. Metric subregularity at every point of the solution set does not, without further compactness or uniformity assumptions, yield a common kappa and nu on the union; the infimum of the nu_z over an unbounded solution set can be zero, and the constants kappa_z can vary. This can likely be repaired by localizing at the actual limit point bar{x} after strong convergence in finite dimension, but as written the argument is not complete.
minor comments (3)
  1. [Sections 6.1.1 and Appendix A.3] The symbol beta is used inconsistently: in Lemma 1 it denotes beta_E/(gamma^{-1} - L_D), while in Proposition 8 and the proof it appears to denote beta_E in some places and the Lemma 1 beta in others. Please unify the notation and state explicitly which quantity appears in (6.6).
  2. [Appendix A.3, final paragraph] The 'straight-forward but somewhat tedious algebra' verifying the identity between (A.4) and the claimed step-size bound should be written out in full, since this is exactly the kind of omitted algebra that can hide the kind of mismatch noted in the major comments.
  3. [Section 3.2, Eq. (3.4)] The definition of hat_mu_k says the inequality must hold for all x,y in H with x != y, but the surrounding text could more clearly separate this global assumption from the local definition of mu_k in Eq. (3.3). A short clarification would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: NOFOB's convergence and linear-rate proofs are self-contained reductions to standard monotone-operator facts, and the special-case inclusions are exact identifications rather than fitted predictions.

full rationale

The paper's central claims are Theorem 2 (weak convergence of NOFOB), Theorem 3 (linear convergence under metric subregularity), and the four-operator specialization with its special cases. The proofs proceed from Assumptions 1 and 2 to construct a separating halfspace using monotonicity of A, cocoercivity of C, and strong monotonicity of the kernel M_k; convergence then follows from Fejér monotonicity of relaxed projections and a standard weak-cluster-point argument. No parameter is fitted to data, and no convergence conclusion is imported from the papers that introduced the special cases. The statements that FBF, FBHF, AFBA, and synchronous projective splitting are special cases are explicit algebraic identifications: for example, Algorithm 6.1 is obtained from Algorithm 3.2 by substituting A = B + D + K, C = E, and M_k = Q_k - D - K, and the equivalence of Algorithm 7.2 with Algorithm 7.1 is verified in Appendix B by direct computation of the resolvent step and the projection coefficient. These are definitions and algebraic reductions, not circular derivations. The only self-citation, reference [23] in Section 6.3, appears in a side remark about the extended step-size range already established by [37] and is not used to prove any theorem in this paper. A reviewer-identified gap in the proof of Proposition 8 in Appendix A.3 is a possible correctness issue about the stated step-size range, not a circularity: the claimed reduction from (A.2) to (A.3) may be invalid, but this does not mean the theorem is assumed as an input. Thus the derivation chain is not circular, and the appropriate score is 0.

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

No numbers are fitted to data; the step sizes and relaxation parameters are user choices with proven feasible ranges, not fitted constants. The paper introduces no new physical or ontological entities: the nonlinear resolvent and four-operator splitting method are mathematical constructions defined explicitly. The axioms listed are the standard monotone operator facts and the explicit domain assumptions on the operators M_k, C, and A + C.

assumptions (6)
  • standard math Real Hilbert space setting and the standard theory of maximal monotone operators from Bauschke and Combettes [4].
    Used throughout, for example in Proposition 4 where M + A is shown to be maximally monotone and surjective via Corollaries 25.5, 25.28, and Example 22.7 of [4].
  • standard math The sum of maximal monotone operators with a full-domain single-valued maximal monotone operator is maximal monotone.
    Invoked in Corollary 3 and Proposition 6 to justify maximal monotonicity of B + D + K and of A + C.
  • standard math Relaxed projections onto halfspaces are averaged operators and satisfy the Fejer-type inequality (5.3).
    Used in the proof of Theorem 2 via Propositions 4.16, 4.35, and 4.41 of [4].
  • domain assumption C is 1/beta-cocoercive with respect to a metric ||.||_P with beta in [0, 4).
    Assumption 1(ii); this is required for the strict separation property in Theorem 1 and for the lower bound on the step parameter in Proposition 7.
  • domain assumption Each kernel M_k is maximally monotone, single-valued, 1-strongly monotone with respect to ||.||_P, and L_M-Lipschitz continuous.
    Assumption 2; this ensures T_FB has full domain, is single-valued and Lipschitz, and that the separating hyperplane is strictly separating.
  • domain assumption A + C is metrically subregular at every solution point for 0, with finite-dimensional H or global neighborhoods.
    Definition 1, used in Theorem 3 for local Q-linear and R-linear convergence of the distance to the solution set.

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Pith. "Pith review of Nonlinear Forward-Backward Splitting with Projection Correction." pith.science (2026). https://pith.science/paper/JZMA56ME

@misc{pith2026190807449,
  author       = {Pith},
  title        = {Pith review of: Nonlinear Forward-Backward Splitting with Projection Correction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JZMA56ME}},
  note         = {Machine review of arXiv:1908.07449}
}
read the original abstract

We propose and analyze a versatile and general algorithm called nonlinear forward-backward splitting (NOFOB). The algorithm consists of two steps; first an evaluation of a nonlinear forward-backward map followed by a relaxed projection onto the separating hyperplane it constructs. The key of the method is the nonlinearity in the forward-backward step, where the backward part is based on a nonlinear resolvent construction that allows for the kernel in the resolvent to be a nonlinear single-valued maximal monotone operator. This generalizes the standard resolvent as well as the Bregman resolvent, whose resolvent kernels are gradients of convex functions. This construction opens up for a new understanding of many existing operator splitting methods and paves the way for devising new algorithms. In particular, we present a four-operator splitting method as a special case of NOFOB that relies nonlinearity and nonsymmetry in the forward-backward kernel. We show that forward-backward-forward splitting (FBF), forward-backward-half-forward splitting (FBHF), asymmetric forward-backward-adjoint splitting (AFBA) with its many special cases, as well as synchronous projective splitting are special cases of the four-operator splitting method and hence of NOFOB. We also show that standard formulations of FB(H)F use smaller relaxations in the projections than allowed in NOFOB. Besides proving convergence for NOFOB, we show linear convergence under a metric subregularity assumption, which in a unified manner shows (in some cases new) linear convergence results for its special cases.

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Cited by 1 Pith paper

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

  1. Warped Proximal Iterations for Monotone Inclusions

    math.OC 2019-08 accept novelty 7.0 of 10

    Warped resolvents generalize classical resolvents, and new convergence theorems unify and extend operator-splitting algorithms for monotone inclusions.

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    We will finish the proof by proving that ( A.2) holds for all allowed γ

    Therefore, if ( A.2) holds, also the result in ( 6.6) holds for all x⁄=y by dividing both sides by ‖Mx−My‖2> 0 and (2−δ)γ−1> 0. We will finish the proof by proving that ( A.2) holds for all allowed γ. Since the terms involving δ in ( A.2) satisfy δγ −1((γ−1 +LD +‖K‖)− βE (γ− 1−...

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