Composite Monotone Inclusion Problems

• Composite monotone inclusion problems (CMIPs) are operator-theoretic frameworks in real Hilbert spaces, combining monotone, Lipschitzian, and cocoercive operators through linear compositions and resolvent operations.
• They employ primal-dual reformulations and splitting methods to separately process set-valued and single-valued operators, facilitating efficient algorithmic implementations.
• CMIPs underpin advanced applications in convex optimization and imaging, offering convergence guarantees and design flexibility through techniques like minimal lifting and non-standard metric preconditioning.

Searching arXiv for recent and foundational papers on composite monotone inclusion problems. Composite monotone inclusion problems (CMIPs) are a family of operator-theoretic problems in real Hilbert spaces whose objective is to find a point $x$ satisfying a monotone inclusion with additive, linearly composed, and, in many formulations, parallel-sum or cocoercive/Lipschitzian components. A canonical form is

$$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$

with $A$ and $B_i$ maximally monotone and $L_i$ bounded linear, while major extensions incorporate monotone Lipschitzian operators, cocoercive terms, and operators of parallel-sum type (Aragón-Artacho et al., 2022). In the literature, CMIPs are treated primarily through primal-dual reformulations, product-space embeddings, and splitting methods that activate monotone operators through resolvents and smooth or linear couplings through explicit steps (Combettes et al., 2011).

1. Canonical formulations and operator structure

The simplest CMIP formulation in the cited literature is

$$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$

where $H$ and each $G_i$ are real Hilbert spaces, $A\colon H\rightrightarrows H$ is maximally monotone, $B_i\colon G_i\rightrightarrows G_i$ is maximally monotone, and $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$0 is bounded linear with adjoint $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$1 (Aragón-Artacho et al., 2022). This form isolates the defining composite feature of the class: monotone operators are not merely summed on a common space, but also composed with linear maps and pulled back by adjoints.

A broader formulation, due to Combettes and Pesquet, adds a monotone $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$2-Lipschitzian operator $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$3, monotone $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$4-Lipschitzian operators $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$5, shifts $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$6 and $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$7, and the parallel sum

$$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$8

The primal inclusion becomes

$$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$9

with a corresponding dual inclusion in the product space $A$0 (Combettes et al., 2011). This formulation covers mixtures of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators.

Another recent broad formulation is

$A$1

where $A$2 and $A$3 are maximally monotone, $A$4 is $A$5-cocoercive, and $A$6 is bounded linear (Huang et al., 26 Jun 2026). This emphasizes the now-standard decomposition between backward steps for maximally monotone operators and forward steps for single-valued cocoercive parts.

Across these formulations, the central computational primitive is the resolvent

$A$7

which is single-valued and everywhere defined whenever $A$8 is maximally monotone (Aragón-Artacho et al., 2022). In convex-analytic specializations with $A$9, resolvents become proximal operators, which is why CMIPs serve simultaneously as an inclusion-theoretic and an optimization-theoretic framework.

2. Primal-dual duality and the Kuhn-Tucker set

A persistent structural theme in CMIPs is the simultaneous treatment of primal and dual inclusions. For the basic composite model

$B_i$0

the corresponding dual inclusion is

$B_i$1

and the coupled system can be reformulated on the product space $B_i$2 as the problem of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator (Briceno-Arias et al., 2010). This “monotone + skew” reduction is foundational because it converts primal-dual coupling into a form amenable to forward-backward-forward and related splittings.

The Kuhn-Tucker set gives a more geometric representation of primal-dual solvability. For

$B_i$3

the Kuhn-Tucker set is

$B_i$4

so any point of $B_i$5 provides a primal solution and a dual solution simultaneously (Alotaibi et al., 2014). In the multi-operator setting, the same logic persists: primal and dual variables are coupled through graph conditions that define an extended solution set or Kuhn-Tucker set (Alotaibi et al., 2013).

This representation has algorithmic consequences. One line of work constructs half-spaces containing the Kuhn-Tucker set and then projects onto those half-spaces to obtain Fejér monotonicity with respect to the solution set (Alotaibi et al., 2013). Another line seeks the best approximation from a reference point to the Kuhn-Tucker set, combining resolvent steps with Haugazeau’s method; this yields strong convergence to the unique point $B_i$6 without requiring knowledge of $B_i$7 or inversion of $B_i$8 or $B_i$9 (Alotaibi et al., 2014). The 2015 splitting method of Bot and Csetnek explicitly states that its main idea is to represent the solution set of both the primal and dual problems using their associated Kuhn-Tucker set and then develop a projected method to successively approximate a feasible point of that set (Tran-Dinh et al., 2015).

3. Splitting architectures

The dominant algorithmic paradigm for CMIPs is primal-dual splitting. In the 2011 fully split framework, the single-valued Lipschitzian operators $L_i$0 and $L_i$1 are processed individually via explicit steps, while the set-valued operators $L_i$2 and $L_i$3 are processed individually via their resolvents; most of the branchwise operations over $L_i$4 can be executed simultaneously (Combettes et al., 2011). This architecture became a reference point because it handles parallel sums implicitly by splitting $L_i$5 and $L_i$6, rather than forming $L_i$7 explicitly.

A closely related architecture is the monotone-plus-skew model, which applies Tseng’s forward-backward-forward method in a product space. In the primal-dual interpretation, the resolvents of $L_i$8 and $L_i$9 are computed separately, while the linear coupling through $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$0 and $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$1 is treated explicitly (Briceno-Arias et al., 2010). Douglas-Rachford-type variants extend this decomposition further: two algorithms proposed in 2012 allow the bounded linear operators and the set-valued operators occurring in the formulation to be processed separately at each iteration, with weak convergence of primal-dual sequences under range conditions and step-size inequalities (Bot et al., 2012).

Several later developments alter the product-space geometry rather than the individual updates. The minimal-lifting algorithm for CMIPs establishes a primal-dual splitting with ambient space $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$2, rather than $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$3, while still using one resolvent $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$4 for each $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$5, one $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$6, and one forward pass of $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$7 and $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$8 (Aragón-Artacho et al., 2022). The Split–Douglas–Rachford algorithm introduces primal and dual non-standard metrics and requires only the metric-coupling condition $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$9, reducing to classical Douglas-Rachford when $H$0 and $H$1 (Briceño-Arias et al., 2021). Partial-inverse methods constitute another branch: Spingarn’s method of partial inverses is used to solve

$H$2

after embedding the problem into a subspace-constrained product space, producing a primal-dual splitting in which $H$3, $H$4, and $H$5 are employed separately (Alghamdi et al., 2013).

Recent variants modify the explicit part of the iteration. The 2025 primal-dual splitting algorithm for CMIPs adds both an extrapolation step and a convex combination step, and proves convergence under the relaxed condition

$H$6

with sublinear convergence rate results for both general CMIPs and structured convex optimization problems (Chang et al., 1 Oct 2025). For inclusions of the form

$H$7

where $H$8 is monotone-Lipschitzian, $H$9 is cocoercive, and $G_i$0 is a closed vector subspace, the FRB–PI and SDR–PI methods combine partial inverse techniques with forward-reflected-backward and shadow-Douglas–Rachford, respectively; each iteration uses one activation of the Lipschitzian operator, one activation of the cocoercive operator, two projections onto the closed vector subspace, and one calculation of the resolvent of the maximally monotone operator (Roldán, 2023).

4. Convergence regimes and rate theory

Weak convergence is the default guarantee in much of the CMIP literature. In the 2011 primal-dual splitting algorithm, if the error sequences are absolutely summable, the qualification

$G_i$1

holds, and the step-size satisfies $G_i$2, then the primal sequence converges weakly to a solution of the primal inclusion and the dual variables converge weakly to a dual solution; if $G_i$3 or $G_i$4 is uniformly monotone at $G_i$5, then $G_i$6 strongly, and analogous statements hold on the dual side when $G_i$7 or $G_i$8 is uniformly monotone (Combettes et al., 2011).

Other constructions are explicitly designed to obtain strong convergence without uniform monotonicity assumptions. The best-approximation framework from the Kuhn-Tucker set combines a proximal step that generates a separating half-space containing the solution set with Haugazeau’s projection of the reference point onto the intersection of two half-spaces, and the resulting sequence converges strongly to $G_i$9 (Alotaibi et al., 2014). A related Fejér-projection framework guarantees Fejér monotonicity with respect to the Kuhn-Tucker set and weak convergence under a demiclosedness-type condition, while retaining freedom in the step-size parameters and avoiding explicit norm estimates on the linear operator (Alotaibi et al., 2013).

Rate theory for CMIPs is heterogeneous. Under metric subregularity of the composite operator, the generalized proximal point algorithm, the over-relaxed forward-backward splitting algorithm, the generalized Douglas-Rachford splitting algorithm, and Davis’s three-operator splitting all enjoy local linear convergence; this condition is stated as weaker than strong monotonicity, and sufficient conditions include strong monotonicity, polyhedral monotonicity plus affine structure, and several structured subdifferential-plus-smooth cases (Shen et al., 2015). By contrast, the 2025 convex-combination/extrapolation primal-dual algorithm proves fixed-point residual decay

$A\colon H\rightrightarrows H$0

and, in the convex optimization case, ergodic bounds

$A\colon H\rightrightarrows H$1

(Chang et al., 1 Oct 2025).

Continuous-time analyses supply another mechanism for strong convergence. Tikhonov-regularized Krasnosel’skii–Mann, forward-backward, and forward-backward-forward dynamics are shown to converge strongly to the minimum norm solution of the underlying inclusion when the regularization $A\colon H\rightrightarrows H$2 vanishes slowly enough, typically with $A\colon H\rightrightarrows H$3 (Boţ et al., 2019). This suggests a general distinction in the CMIP literature: weak convergence is typical for unregularized fixed-point iterations, whereas strong convergence is usually obtained either from additional monotonicity assumptions or from architectures built around best approximation, projection, or vanishing Tikhonov regularization.

5. Convex optimization specializations

A large share of CMIP research is driven by convex optimization models. The basic identification

$A\colon H\rightrightarrows H$4

recovers Fenchel–Rockafellar primal-dual structure and the minimization problem

$A\colon H\rightrightarrows H$5

(Combettes et al., 2011). More elaborate models arise when

$A\colon H\rightrightarrows H$6

with each $A\colon H\rightrightarrows H$7 strongly convex, yielding the fully nonsmooth problem

$A\colon H\rightrightarrows H$8

for which the iterates admit closed-form prox steps on $A\colon H\rightrightarrows H$9, $B_i\colon G_i\rightrightarrows G_i$0, and $B_i\colon G_i\rightrightarrows G_i$1 (Combettes et al., 2011). A multivariate extension with several primal blocks and several dual blocks appears in the system of composite monotone inclusions of Dũng and Vũ, where the inclusion system encodes the first-order conditions of

$B_i\colon G_i\rightrightarrows G_i$2

(Dung et al., 2013).

Constrained variants emphasize feasibility of the iterates. The projected primal-dual splitting for constrained composite primal-dual monotone inclusions introduces a primal constraint set $B_i\colon G_i\rightrightarrows G_i$3, a dual constraint $B_i\colon G_i\rightrightarrows G_i$4, and a projection step so that $B_i\colon G_i\rightrightarrows G_i$5 and $B_i\colon G_i\rightrightarrows G_i$6 at every iteration (Briceño-Arias et al., 2018). In the convex optimization case, this produces a projected Chambolle–Pock type method. Numerical experiments on

$B_i\colon G_i\rightrightarrows G_i$7

with $B_i\colon G_i\rightrightarrows G_i$8 report improvements of approximately $B_i\colon G_i\rightrightarrows G_i$9 when $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$00, $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$01 when $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$02, and $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$03 when $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$04, with larger gains as the number of projected constraints increases and as tolerance tightens (Briceño-Arias et al., 2018).

Imaging provides the most detailed benchmark class. The minimal-lifting CMIP algorithm is applied to

$$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$05

with Gaussian blur $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$06, noise $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$07, $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$08, and $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$09; the reported outcome is convergence in roughly $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$10 of the time of the Douglas–Rachford-type primal-dual method of Bot–Hendrich (2013), equal or better ISNR after $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$11 iterations, and $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$12 less memory in the lifted primal variables (Aragón-Artacho et al., 2022). The 2026 primal-dual splitting algorithm is tested on image deblurring and denoising, achieving PSNR up to $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$13 dB on the “Building” image, compared with PFDR’s $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$14 dB, and reporting $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$15–$$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$16 dB PSNR gains over PFDR in MC-TV denoising at comparable CPU costs (Huang et al., 26 Jun 2026). Other CMIP-based imaging studies include Heron location and image deblurring via Douglas-Rachford-type methods (Bot et al., 2012), total-variation image restoration and sparse minimization via Split–Douglas–Rachford and Split-ADMM (Briceño-Arias et al., 2021), and constrained fused-lasso and computed tomography inverse problems via FRB–PI and SDR–PI, with PSNR up to $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$17 dB in the CT experiments (Roldán, 2023).

6. Minimal lifting, nonlinear compositions, and interpretive issues

One contemporary direction is dimensional optimality of the lifted space. For the sum of $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$18 maximally monotone operators with one composite term, the minimal-lifting theorem states that any frugal parametrized-resolvent splitting must lift into at least $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$19, so no fixed-point algorithm that evaluates each resolvent exactly once per iteration can have $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$20-fold lifting with $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$21 (Aragón-Artacho et al., 2022). In that context, the $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$22-fold lifting scheme is presented as the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting.

A second direction is the extension from linear to nonlinear compositions. The perturbation framework of Briceño-Arias and Combettes treats Banach-space monotone inclusions involving nonlinear maps $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$23, introduces a maximally monotone Kuhn–Tucker operator decomposable as the sum of simpler monotone operators, and, in the Hilbertian setting, derives a block-iterative primal-dual algorithm that fully splits all the components of the problem; it is described there as the first proximal splitting algorithm for handling nonlinear composite problems (Briceño-Arias et al., 2021). This broadens the meaning of “composite” beyond linear pullbacks $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$24.

Several recurring misconceptions are corrected by the literature. CMIPs are not synonymous with a single template $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$25; they include mixtures of sums, linear compositions, parallel sums, Lipschitzian operators, cocoercive operators, and normal cones (Combettes et al., 2011). They are not restricted to convex optimization, although convex minimization provides many of the most explicit special cases (Briceno-Arias et al., 2010). Nor is product-space lifting merely an implementation detail: the choice between full lifting, minimal lifting, partial inverse reformulations, or metric preconditioning determines memory usage, admissible step sizes, and the form of the fixed-point map (Aragón-Artacho et al., 2022).

The step-size question remains a major point of comparison. Standard primal-dual hybrid gradient requires $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$26, golden-ratio variants allow approximately $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$27, while the 2025 CMIP algorithm allows $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$28 up to nearly $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$29 under $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$30; the He–Yuan example is used to show that the boundary case $$\text{find }x\in H\text{ such that }0\in A(x)+\sum_{i=1}^m L_i^*\,B_i(L_i x),$$31 is sharp, since the iterates fail to converge there (Chang et al., 1 Oct 2025). A plausible implication is that modern CMIP research increasingly treats operator structure, lifted dimension, and admissible parameter region as coequal design variables rather than as secondary details of convergence proofs.