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Warped Proximal Iterations for Monotone Inclusions

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Pith's one-line read The paper proposes warped resolvents $(K+\gamma M)^{-1}\circ K$ as a building block for monotone inclusion algorithms, and proves weak and strong convergence for the resulting proximal iterations.

desk verdict Warped resolvents are a real generalization and the convergence theorems are correct, but the advertised breadth rests on a single nonstandard condition that only strong monotonicity is known to trigger. read the letter →

arxiv 1908.07077 v5 pith:XUEPUTRG submitted 2019-08-19 math.OC

Minh N. Bùi , Patrick L. Combettes This is my paper · ORCID
classification math.OC MSC 47J2547N1047H0590C25
keywords monotoneinclusionswarpedresolventoperatorsplittingproximalpointalgorithmprimal-dualalgorithmsstrongconvergenceFejérmonotonicitymaximallyoperators
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's project is to replace the resolvent $(\mathrm{Id}+\gamma M)^{-1}$ inside the proximal point algorithm by a warped resolvent $(K+\gamma M)^{-1}\circ K$, where $K$ is an auxiliary monotone operator chosen to fit the structure of the problem. The central results, Theorems 4.2 and 4.8, state that when each warped resolvent is well defined, the auxiliary evaluation points track the iterates, and a kernel regularity condition holds, the iterates converge weakly to a zero of $M$, or strongly to the projection of the starting point onto the zero set. The payoff is a single abstract principle that specializes to classical proximal point and to forward-backward-forward splitting, and that yields new algorithms, including a forward-backward-forward method with memory and a splitting method for a system of coupled primal-dual inclusions. A sympathetic reader would care because the kernel $K$ can be engineered so that otherwise intractable resolvents become computable while convergence is preserved.

What carries the argument

The central object is the warped resolvent $J^K_{\gamma M}=(K+\gamma M)^{-1}\circ K$ for a monotone kernel $K$; in a Hilbert space it maps $x$ to the unique $p$ satisfying $Kx\in Kp+\gamma Mp$, so it has the fixed-point characterization $\mathrm{Fix}\, J^K_{\gamma M}=\mathrm{zer}\, M$ and produces graph points $(p,\gamma^{-1}(Kx-Kp))\in \mathrm{gra}\, M$. This is what lets the algorithm generate half-spaces that contain the solution set. The convergence mechanism is Fej\'er monotonicity together with condition (ii)[b], the bridge that turns a vanishing inner product into weak convergence of $\widetilde{x}_n-y_n$ and strong convergence of the kernel difference. In the strong-convergence variant, the extra mechanism is the projection $Q(x_0,x_n,x_{n+1/2})$ onto the intersection of two half-spaces, which preserves Fej\'er monotonicity and forces convergence to the best approximation of $x_0$ in $Z$.

What would settle it

Take a maximally monotone operator with a known zero set, for instance the normal cone $N_C$ of a closed convex set $C$ in $\ell^2$, and implement iteration (4.5) with $\widetilde{x}_n=x_n$ and a family of monotone kernels $K_n$ for which the warped resolvents are easy to compute but which are not strongly monotone and Lipschitz. If a family can be found where the inner products in condition (ii)[b] tend to $0$ while the iterates have a weak cluster point outside $C$, or while $K_n\widetilde{x}_n-K_n y_n$ fails to converge strongly to $0$, then Theorem 4.2 as stated fails. A sharper falsifier is a single example satisfying every hypothesis of Theorem 4.2 except (ii)[b], with the inner products tending to $0$ and the iterates not converging to a zero of $M$.

Watch

Extended reading notes

Core claim

The discovery is an abstract convergence principle for zero-finding with maximally monotone operators. On a Hilbert space $X$, take $M$ maximally monotone with nonempty zero set $Z$, a sequence of monotone kernels $K_n$, and auxiliary points $\widetilde{x}_n$ with $\widetilde{x}_n-x_n\to 0$. If each warped resolvent $J^{K_n}_{\gamma_n M}$ is well defined and the regularity condition $\langle \widetilde{x}_n-y_n \mid (K_n\widetilde{x}_n-K_n y_n)^\sharp\rangle\to 0$ implies both $\widetilde{x}_n-y_n\rightharpoonup 0$ and $K_n\widetilde{x}_n-K_n y_n\to 0$, then the iteration (4.5), which evaluates $y_n=J^{K_n}_{\gamma_n M}\widetilde{x}_n$ and takes a relaxed projection onto the half-space containing $Z$, converges weakly to a zero of $M$ (Theorem 4.2). With the additional correction step $x_{n+1}=Q(x_0,x_n,x_{n+1/2})$ projecting onto the intersection of two half-spaces, the same hypotheses give strong convergence to $\mathrm{proj}_Z x_0$ (Theorem 4.8). The proof routes through Fej\'er monotonicity: the iteration is Fej\'er-monotone with respect to $Z$, and the regularity condition upgrades the vanishing inner products to enough subsequential information to apply maximal monotonicity.

Load-bearing premise

The load-bearing premise is condition (ii)[b] of Theorems 4.2 and 4.8: each kernel family must convert a vanishing inner product $\langle \widetilde{x}_n-y_n \mid (K_n\widetilde{x}_n-K_n y_n)^\sharp\rangle\to 0$ into both $\widetilde{x}_n-y_n\rightharpoonup 0$ and $K_n\widetilde{x}_n-K_n y_n\to 0$. The only concrete sufficient condition the paper supplies for this is strong monotonicity plus Lipschitzianity of the kernels, so a kernel family that violates the implication falls outside the theorems.

Editorial extensions

If this is right

  • Setting $K_n=\mathrm{Id}$, $\widetilde{x}_n=x_n$, and $\lambda_n=1$ in (4.5) recovers the classical proximal point algorithm, so the theorems contain that method as a special case.
  • Choosing $K_n=\mathrm{Id}-\gamma_n B$ recovers the forward-backward-forward splitting method for $0\in Ax+Bx$; the proof attributes the required regularity to strong monotonicity and cocoercivity of the kernels.
  • The new perturbed forward-backward-forward algorithm with memory (Corollary 5.2) converges weakly to a zero of $A+B$ under additive perturbations $e_n$ that only tend to $0$ and under inertial weights supported on a bounded window, conditions milder than summability of the errors.
  • For the coupled system of primal-dual inclusions in Problem 5.4, Corollary 5.5 yields a parallel splitting algorithm whose primal and dual iterates converge weakly to a point in the primal-dual set $\mathcal{Z}$; the strong-convergence counterpart follows from Theorem 4.8.
  • The multi-point extension in Remark 4.10 allows several warped resolvent evaluations per step to be combined through weighted half-spaces, so the principle covers algorithms using more than one graph point per iteration.

Reading between the lines

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

  • A testable extension is to replace the strong monotonicity plus Lipschitz sufficient condition of Remark 4.3 with a wider sufficient class for condition (ii)[b], for instance kernels built from strongly monotone operators plus compact perturbations; all Section 5 algorithms would carry over to that class.
  • The static theory already covers reflexive Banach spaces through duality-mapping and distance kernels, but the dynamic convergence theorems are proved only in Hilbert space; a non-Hilbert convergence theorem would be the natural next step and is not established here.
  • The vanishing-error condition $\widetilde{x}_n-x_n\to 0$ without summability suggests that asynchronous or block-coordinate implementations, where the evaluation point lags behind the current iterate by a shrinking error, may still converge under Theorem 4.2.
  • One could test numerically whether non-symmetric linear kernels of the type used in the primal-dual example give faster convergence on ill-conditioned saddle-point problems, since warped resolvents allow couplings that standard resolvents cannot express.
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Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper introduces a generalization of resolvents of monotone operators, called warped resolvents, defined by J^K_M = (K+M)^{-1}∘K for a kernel K and a monotone operator M. After establishing well-definedness criteria and basic properties (Propositions 3.8–3.11), the authors prove two abstract convergence principles: weak convergence (Theorem 4.2) and strong convergence to the projection onto the zero set (Theorem 4.8) for iterations that use warped resolvents with possibly time-varying kernels K_n applied at auxiliary points ~x_n, followed by a Fejér-type update. The main hypotheses are ~x_n−x_n→0 and condition (ii)[b], which states that a vanishing normalized inner product forces both ~x_n−y_n⇀0 and K_n~x_n−K_n y_n→0. The principles are then instantiated to recover known algorithms (e.g., Tseng's forward-backward-forward method, Corollary 5.3) and to design new ones, including a perturbed forward-backward-forward algorithm with memory (Corollary 5.2) and a primal-dual method for a coupled system of monotone inclusions (Problem 5.4 and Corollary 5.5).

Significance. If the results are correct, the warped resolvent framework provides a unified perspective on a broad class of splitting algorithms and a modular design tool for new solution methods. The paper is self-contained, with detailed proofs from standard monotone operator theory, and the applications verify the required hypotheses rather than relying on numerical evidence. The main strength is the breadth of the proposed formalism and the concrete new algorithms derived from it. The main weakness is that the crucial condition (ii)[b] in Theorems 4.2 and 4.8 is only shown to hold for uniformly strongly monotone and Lipschitzian kernels (Remark 4.3), and every Section 5 application uses exactly that regime; no example is given of a non-strongly-monotone kernel satisfying (ii)[b], and a simple rotation kernel shows that the condition is not automatic.

major comments (1)
  1. [Section 4, Theorem 4.2 (and Theorem 4.8), condition (ii)[b]] Condition (ii)[b] is essential to the proof: at (4.10)–(4.13) it is used to convert the vanishing inner product produced by the Fejér-monotone update into weak convergence of ~x_n−y_n and strong convergence of K_n~x_n−K_n y_n, which together with γ_n≥ε yields y*_n→0 and enables the maximal monotonicity argument. The only sufficient condition offered in Remark 4.3 is uniform α-strong monotonicity plus β-Lipschitzianity, and every application in Section 5 relies on exactly that sufficient condition. The paper does not provide an example of a non-strongly-monotone family satisfying (ii)[b], and in fact the condition is not implied by the other hypotheses. Concretely, in X=R^2 take M=Id, let K be the 90-degree rotation, set γ_n=1, λ_n=1, and ~x_n=x_n. Then all hypotheses of Theorem 4.2 except (ii)[b] hold; the antecedent of (ii)[b] holds (the inner product is identically zero) but the consequent fails because ~x_n−y_n is a nonzero constant, and the iterates remain fixed at the starting point, which is not a zero of M. I recommend that the authors add a remark explicitly noting this non-redundancy, and either prove (ii)[b] under more general checkable hypotheses or qualify the abstract principle's scope to the strongly monotone kernel regime.
minor comments (4)
  1. [Section 4, Theorem 4.2, condition (ii)[b]] The implication notation in (ii)[b] uses an arrow followed by a set of two statements; this is nonstandard and should be restated in words, for example: 'if ⟨~x_n−y_n | (K_n~x_n−K_n y_n)^♯⟩→0, then both ~x_n−y_n⇀0 and K_n~x_n−K_n y_n→0 hold.'
  2. [Section 4, proof of Theorem 4.8] The sentence 'this is done as in the proof of Theorem 4.2(ii)' should be expanded, because Theorem 4.2 uses the bound ε^{-1}‖x_{n+1}−x_n‖ with λ_n∈[ε,2−ε], whereas Theorem 4.8 has no relaxation parameter and the corresponding bound is simply ‖x_{n+1/2}−x_n‖; the adaptation is straightforward but should be stated explicitly for clarity.
  3. [Section 3, Proposition 3.9(i)[f]] The coercivity condition leading to (3.6) and the inequality involving inf ⟨x,∂ϕ(x+z)⟩ are stated without proof or reference; one sentence of justification, or a citation to the relevant convex-analytic fact, would improve readability.
  4. [References, [14]] Reference [14] cites the preprint version of this paper; if a published version exists, it should be cited in place of the preprint to avoid a self-citation of an unpublished report.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper proves conditional convergence theorems under explicit hypotheses, and its cited prior results are independent mathematical facts rather than self-derived inputs.

full rationale

The paper is a self-contained mathematical derivation in the monotone operator theory tradition. The central object, the warped resolvent J^K_M = (K + M)^{-1} o K, is defined in Definition 1.1 and its properties (fixed points, graph characterization, Lipschitzianity) are proved from the definition and from standard prior results, e.g., Proposition 3.10 establishes that p = J^K_{gamma M} x implies (p, gamma^{-1}(Kx - Kp)) in gra M. Theorem 4.2 and Theorem 4.8 state convergence of the warped proximal iterations (4.5) and (4.20) under explicit assumptions: Z = zer M nonempty, ran K_n subset ran(K_n + gamma_n M), injectivity, command perturbations tending to zero, and condition (ii)[b]. The proofs do not assume the conclusion. Fejer monotonicity gives summability of the step norms, the Cauchy-Schwarz estimate (4.11) converts the update inequality into the antecedent of condition (ii)[b], and the consequent of (ii)[b] is then used with maximal monotonicity to identify weak cluster points as zeros. This is a genuine proof chain, not a renaming or a fit. The applications in Section 5 are all instantiations of Theorem 4.2 in which the kernel conditions are verified, typically via strong monotonicity plus Lipschitzianity as in Remark 4.3. The heavy self-citation present in the paper cites prior theorems, such as the textbook [7], the Fejer-monotonicity survey [16], and previous Kuhn-Tucker duality work [20]; these are independent mathematical facts with stated assumptions and are not equivalent to the paper's own conclusions. The nonstandard condition (ii)[b] is indeed the load-bearing assumption, and Remark 4.3 notes that the only sufficient condition supplied is uniform strong monotonicity plus Lipschitzianity. A skeptical reviewer could question the breadth of the theory outside that regime, or even exhibit natural kernels for which (ii)[b] fails, but that concern is a correctness or generality risk about a stated assumption, not a circularity. No quantity is fitted to data, no prediction is produced from its own definition, and no central claim is justified solely by a self-citation. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters fitted to data; no new physical entities. The only new object is the warped resolvent itself, a mathematically defined operator, not an entity with independent empirical evidence.

assumptions (4)
  • standard math X is a real Hilbert space (or reflexive real Banach space) with standard duality pairing
    The entire framework is built in this setting; standard results from [7,35] are used without proof.
  • domain assumption M: X→2^X is maximally monotone and Z = zer M ≠ ∅
    The convergence theorems (4.2, 4.8) and applications assume a nonempty solution set; without it no convergence to a solution can be claimed.
  • domain assumption For each n, ran K_n ⊂ ran(K_n+γ_n M) and K_n+γ_n M is injective
    This ensures the warped resolvent J^{K_n}_{γ_n M} is well defined (Def. 1.1, Prop. 3.8).
  • ad hoc to paper Condition (ii)[b]: ⟨~x_n−y_n | (K_n~x_n−K_n y_n)^♯⟩→0 implies ~x_n−y_n ⇀0 and K_n~x_n−K_n y_n→0
    This nonstandard regularity condition is introduced in Theorem 4.2(ii)[b] and Theorem 4.8(ii)[b] and is essential to convert the Fejér type update into weak convergence. Only a sufficient condition (strong monotonicity plus Lipschitzianity) is given in Remark 4.3.

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Pith. "Pith review of Warped Proximal Iterations for Monotone Inclusions." pith.science (2026). https://pith.science/paper/XUEPUTRG

@misc{pith2026190807077,
  author       = {Pith},
  title        = {Pith review of: Warped Proximal Iterations for Monotone Inclusions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XUEPUTRG}},
  note         = {Machine review of arXiv:1908.07077}
}
read the original abstract

Resolvents of set-valued operators play a central role in various branches of mathematics and in particular in the design and the analysis of splitting algorithms for solving monotone inclusions. We propose a generalization of this notion, called warped resolvent, which is constructed with the help of an auxiliary operator. The properties of warped resolvents are investigated and connections are made with existing notions. Abstract weak and strong convergence principles based on warped resolvents are proposed and shown to not only provide a synthetic view of splitting algorithms but to also constitute an effective device to produce new solution methods for challenging inclusion problems.

Figures

Figures reproduced from arXiv: 1908.07077 by the authors.

Figure 1
Figure 1. Warped projections onto the closed unit ball C centered at the origin in the Euclidean plane. Sets of points projecting onto p1, p2, and p3 for the kernels K1 = Id (in green) and K2 : (ξ1, ξ2) 7→ (ξ 3 1 /2 + ξ1/5 − ξ2, ξ1 + ξ2) (in red). Note that K2 is not a gradient. Example 3.3 Suppose that X is strictly convex, let M : X → 2 X∗ be maximally monotone, and let K be the normalized duality mapping of X . Then J K M … view at source ↗

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Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [23]

    Nonlinear Forward-Backward Splitting with Projection Correction

    P. Giselsson, Nonlinear forward-backward splitting w ith projection correction, 2019-08-20. https://arxiv.org/pdf/1908.07449v1

    work page Pith review arXiv 2019

  2. [1]

    Alotaibi, P

    A. Alotaibi, P. L. Combettes, and N. Shahzad, Solving cou pled composite monotone inclusions by succes- sive Fej´ er approximations of their Kuhn-T ucker set,SIAM J. Optim. , vol. 24, pp. 2076–2095, 2014. 19

    work page 2014

  3. [2]

    Alotaibi, P

    A. Alotaibi, P. L. Combettes, and N. Shahzad, Best approx imation from the Kuhn-T ucker set of composite monotone inclusions, Numer . Funct. Anal. Optim., vol. 36, pp. 1513–1532, 2015

    work page 2015

  4. [3]

    Attouch and A

    H. Attouch and A. Cabot, Convergence of a relaxed inertia l proximal algorithm for maximally monotone operators, Math. Program. A , published online 2019-06-29

    work page 2019

  5. [4]

    Baillon, R

    J.-B. Baillon, R. E. Bruck, and S. Reich, On the asymptoti c behavior of nonexpansive mappings and semigroups in Banach spaces, Houston J. Math. , vol. 4, pp. 1–9, 1978

    work page 1978

  6. [5]

    Banert, A

    S. Banert, A. Ringh, J. Adler , J. Karlsson, and O. ¨Oktem, Data-driven nonsmooth optimization, SIAM J. Optim., vol. 30, pp. 102–131, 2020

    work page 2020

  7. [6]

    H. H. Bauschke, J. M. Borwein, and P. L. Combettes, Bregma n monotone optimization algorithms, SIAM J. Control Optim. , vol. 42, pp. 596–636, 2003

    work page 2003

  8. [7]

    H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spa ces, 2nd ed., correct. printing. Springer , New York, 2019

    work page 2019

Show all 39 references
  1. [8]

    H. H. Bauschke, X. Wang, and L. Yao, General resolvents fo r monotone operators: Characterization and extension, in: Biomedical Mathematics: Promising Directions in Imaging, T herapy Planning and Inverse Problems, (Y. Censor , M. Jiang, and G. Wang, eds.), pp. 57–74. Medical P...

    2010

  2. [9]

    R. I. Bot ¸ and E. R. Csetnek, ADMM for monotone operators: Convergence analysis and rates, Adv. Comput. Math., vol. 45, pp. 327–359, 2019

    2019

  3. [10]

    R. I. Bot ¸, E. R. Csetnek, and A. Heinrich, A primal-dual splitting algorithm for finding zeros of sums of maximal monotone operators, SIAM J. Optim. , vol. 23, pp. 2011–2036, 2013

    2011

  4. [11]

    R. I. Bot ¸ and C. Hendrich, A Douglas-Rachford type prim al-dual method for solving inclusions with mix- tures of composite and parallel-sum type monotone operator s, SIAM J. Optim. , vol. 23, pp. 2541–2565, 2013

    2013

  5. [12]

    L. M. Brice˜ no-Arias, Forward-partial inverse-forwa rd splitting for solving monotone inclusions, J. Optim. Theory Appl., vol. 166, pp. 391–413, 2015

    2015

  6. [13]

    M. N. B` ui and P. L. Combettes, Bregman forward-backwar d operator splitting, 2019-09-13. https://arxiv.org/pdf/1908.03878

    2019 arXiv

  7. [14]

    M. N. B` ui and P. L. Combettes, Warped proximal iteratio ns for monotone inclusions, 2019-08-19. https://arxiv.org/pdf/1908.07077v1

    2019 arXiv

  8. [15]

    P. L. Combettes, Construction d’un point fixe commun ` a u ne famille de contractions fermes, C. R. Acad. Sci. Paris S´ er . I Math., vol. 320, pp. 1385–1390, 1995

    1995

  9. [16]

    P. L. Combettes, Fej´ er-monotonicity in convex optimi zation, in: Encyclopedia of Optimization , (C. A. Floudas and P. M. Pardalos, Eds.), vol. 2, Springer-Verlag, New York, 2001, pp. 106–114. (Also avail- able in 2nd ed., pp. 1016–1024, 2009.)

    2001

  10. [17]

    P. L. Combettes, Systems of structured monotone inclus ions: Duality , algorithms, and applications, SIAM J. Optim., vol. 23, pp. 2420–2447, 2013

    2013

  11. [18]

    P. L. Combettes and J. Eckstein, Asynchronous block-it erative primal-dual decomposition methods for monotone inclusions, Math. Program., vol. B168, pp. 645–672, 2018

    2018

  12. [19]

    P. L. Combettes and L. E. Glaudin, Quasinonexpansive it erations on the affine hull of orbits: From Mann’s mean value algorithm to inertial methods, SIAM J. Optim. , vol. 27, pp. 2356–2380, 2017

    2017

  13. [20]

    P. L. Combettes and Q. V. Nguyen, Solving composite mono tone inclusions in reflexive Banach spaces by constructing best Bregman approximations from their Kuh n-T ucker set, J. Convex Anal. , vol. 23, pp. 481–510, 2016

    2016

  14. [21]

    P. L. Combettes and J.-C. Pesquet, Primal-dual splitti ng algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monoton e operators, Set-V alued V ar . Anal., vol. 20, pp. 307–330, 2012. 20

    2012

  15. [22]

    Condat, A primal-dual splitting method for convex op timization involving Lipschitzian, proximable and linear composite terms, J

    L. Condat, A primal-dual splitting method for convex op timization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl. , vol. 158, pp. 460–479, 2013

    2013

  16. [24]

    Haugazeau, Sur les In´ equations V ariationnelles et la Minimisation de Fonctionnelles Convexes

    Y. Haugazeau, Sur les In´ equations V ariationnelles et la Minimisation de Fonctionnelles Convexes. Th` ese, Universit´ e de Paris, Paris, France, 1968

    1968

  17. [25]

    P. R. Johnstone and J. Eckstein, Projective splitting w ith forward steps: Asynchronous and block-iterative operator splitting. https://arxiv.org/pdf/1803.07043.pdf

    arXiv

  18. [26]

    Kassay , The proximal points algorithm for reflexive B anach spaces, Studia Univ

    G. Kassay , The proximal points algorithm for reflexive B anach spaces, Studia Univ. Babes ¸-Bolyai Math., vol. 30, pp. 9–17, 1985

    1985

  19. [27]

    K. C. Kiwiel and B. Łopuch, Surrogate projection method s for finding fixed points of firmly nonexpansive mappings, SIAM J. Optim. , vol. 7, pp. 1084–1102, 1997

    1997

  20. [28]

    J. J. Moreau, Fonctions convexes duales et points proxi maux dans un espace hilbertien, C. R. Acad. Sci. Paris S´ er . A, vol. 255, pp. 2897–2899, 1962

    1962

  21. [29]

    Pennanen, Dualization of generalized equations of m aximal monotone type, SIAM J

    T. Pennanen, Dualization of generalized equations of m aximal monotone type, SIAM J. Optim. , vol. 10, pp. 809–835, 2000

    2000

  22. [30]

    Raguet, A note on the forward-Douglas–Rachford spli tting for monotone inclusion and convex opti- mization, Optim

    H. Raguet, A note on the forward-Douglas–Rachford spli tting for monotone inclusion and convex opti- mization, Optim. Lett., vol. 13, pp. 717–740, 2019

    2019

  23. [31]

    Renaud and G

    A. Renaud and G. Cohen, An extension of the auxiliary pro blem principle to nonsymmetric auxiliary operators, ESAIM Control Optim. Calc. V ar ., vol. 2, pp. 281–306, 1997

    1997

  24. [32]

    S. M. Robinson, Composition duality and maximal monoto nicity ,Math. Program., vol. 85, pp. 1–13, 1999

    1999

  25. [33]

    R. T. Rockafellar , On the maximality of sums of nonlinea r monotone operators, Trans. Amer . Math. Soc., vol. 149, no. 1, pp. 75–88, 1970

    1970

  26. [34]

    R. T. Rockafellar , Monotone operators and the proximal point algorithm, SIAM J. Control Optim. , vol. 14, pp. 877–898, 1976

    1976

  27. [35]

    Simons, From Hahn-Banach to Monotonicity , Lecture Notes in Math

    S. Simons, From Hahn-Banach to Monotonicity , Lecture Notes in Math. 1693, Springer-Verlag, New York, 2008

    2008

  28. [36]

    Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J

    P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Con- trol Optim., vol. 38, pp. 431–446, 2000

    2000

  29. [37]

    B. C. V˜ u, A splitting algorithm for dual monotone inclu sions involving cocoercive operators, Adv. Comput. Math., vol. 38, pp. 667–681, 2013

    2013

  30. [38]

    Z˘ alinescu, Convex Analysis in General V ector Spaces

    C. Z˘ alinescu, Convex Analysis in General V ector Spaces. World Scientific Publishing, River Edge, NJ, 2002

    2002

  31. [39]

    Zeidler , Nonlinear Functional Analysis and Its Applications II/B – N onlinear Monotone Operators, Springer-Verlag, New York, 1990

    E. Zeidler , Nonlinear Functional Analysis and Its Applications II/B – N onlinear Monotone Operators, Springer-Verlag, New York, 1990. 21

    1990

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