--- title: Composite Monotone Inclusion Problems url: https://www.emergentmind.com/topics/composite-monotone-inclusion-problems-cmips type: topic --- # Composite Monotone Inclusion Problems 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 [2202.09665]. 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 [1107.0081]. ## 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 \(L_i\colon H\to G_i\) is bounded linear with adjoint \(L_i^*\) [2202.09665]. 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 \(\mu\)-Lipschitzian operator \(C\colon\HH\to\HH\), monotone \(\nu_i\)-Lipschitzian operators \(D_i\colon\GG_i\to\GG_i\), shifts \(z\in\HH\) and \(r_i\in\GG_i\), and the parallel sum \[ B_i\parallel D_i=(B_i^{-1}+D_i^{-1})^{-1}. \] The primal inclusion becomes \[ z\in A x+\sum_{i=1}^m L_i^*\bigl[(B_i\parallel D_i)(L_i x-r_i)\bigr]+C x, \] with a corresponding dual inclusion in the product space \(\GG_1\times\cdots\times\GG_m\) [1107.0081]. This formulation covers mixtures of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. Another recent broad formulation is \[ 0\in A\,x+B\,x+L^*\,C(Lx), \] where \(A\) and \(C\) are maximally monotone, \(B\) is \(\tfrac1\beta\)-cocoercive, and \(L\) is bounded linear [2606.28010]. 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 \[ J_{\gamma M}=(\Id+\gamma M)^{-1}, \] which is single-valued and everywhere defined whenever \(M\) is maximally monotone [2202.09665]. In convex-analytic specializations with \(M=\partial f\), 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 \[ 0\in A x+L^*B(Lx), \] the corresponding dual inclusion is \[ 0\in -L\,A^{-1}(-L^*v)+B^{-1}v, \] and the coupled system can be reformulated on the product space \(K=H\oplus G\) as the problem of finding a zero of the sum of a maximally monotone operator and a linear skew-adjoint operator [1011.5517]. 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 \[ 0\in A x+L^*B(Lx), \] the Kuhn-Tucker set is \[ Z=\{(x,v)\in\mathcal H\times\mathcal G\mid -L^*v\in A x,\;Lx\in B^{-1}v\}, \] so any point of \(Z\) provides a primal solution and a dual solution simultaneously [1401.8005]. 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 [1312.6696]. 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 [1312.6696]. 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 \(P_Z(x_0,v_0)\) without requiring knowledge of \(\|L\|\) or inversion of \(L\) or \(L^*L\) [1401.8005]. 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 [1505.07946]. ## 3. Splitting architectures The dominant algorithmic paradigm for CMIPs is primal-dual splitting. In the 2011 fully split framework, the single-valued Lipschitzian operators \(C\) and \(D_i\) are processed individually via explicit steps, while the set-valued operators \(A\) and \(B_i\) are processed individually via their resolvents; most of the branchwise operations over \(i=1,\dots,m\) can be executed simultaneously [1107.0081]. This architecture became a reference point because it handles parallel sums implicitly by splitting \(B_i\) and \(D_i\), rather than forming \(B_i\parallel D_i\) 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 \(A\) and \(B^{-1}\) are computed separately, while the linear coupling through \(L\) and \(L^*\) is treated explicitly [1011.5517]. 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 [1212.0326]. 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 \(H^{n-1}\times G\), rather than \(H^n\times G\), while still using one resolvent \(J_{A_i}\) for each \(A_i\), one \(J_{B/\gamma}\), and one forward pass of \(L\) and \(L^*\) [2202.09665]. The Split–Douglas–Rachford algorithm introduces primal and dual non-standard metrics and requires only the metric-coupling condition \(\|\Sigma^{1/2}L\Upsilon^{1/2}\|\le 1\), reducing to classical Douglas-Rachford when \(\operatorname{ran}L=\mathcal G\) and \(\Sigma=(L\Upsilon L^*)^{-1}\) [2101.11683]. Partial-inverse methods constitute another branch: Spingarn’s method of partial inverses is used to solve \[ 0\in A x+L^*B Lx \] after embedding the problem into a subspace-constrained product space, producing a primal-dual splitting in which \(A\), \(B\), and \(L\) are employed separately [1310.1152]. 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 \[ \gamma\,L^2<(2-\theta)(2-\eta), \] with sublinear convergence rate results for both general CMIPs and structured convex optimization problems [2510.00437]. For inclusions of the form \[ 0\in A(x)+B(x)+C(x)+N_V(x), \] where \(B\) is monotone-Lipschitzian, \(C\) is cocoercive, and \(V\) 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 [2305.17500]. ## 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 \[ z\in\operatorname{ran}\bigl(A+\sum_iL_i^*(B_i\parallel D_i)L_i+C\bigr) \] holds, and the step-size satisfies \(0<\gamma<1/(\beta+1)\), then the primal sequence converges weakly to a solution of the primal inclusion and the dual variables converge weakly to a dual solution; if \(A\) or \(C\) is uniformly monotone at \(\bar x\), then \(x^n\to\bar x\) strongly, and analogous statements hold on the dual side when \(B_i\) or \(D_i\) is uniformly monotone [1107.0081]. 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 \(P_Z(x_0,v_0)\) [1401.8005]. 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 [1312.6696]. 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 [1508.05156]. By contrast, the 2025 convex-combination/extrapolation primal-dual algorithm proves fixed-point residual decay \[ \|(v_N,u_N)-(v_{N-1},u_{N-1})\|_M^2=\mathcal O(1/N),\quad\text{indeed }o(1/N), \] and, in the convex optimization case, ergodic bounds \[ |\Phi(\hat x_N,\hat w_N)-\Phi(x^*,w^*)|=\mathcal O(1/N),\qquad \|K\hat x_N-\hat w_N\|=\mathcal O(1/N) \] [2510.00437]. 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 \(\varepsilon(t)\) vanishes slowly enough, typically with \(\int_0^\infty \varepsilon(t)\,dt=+\infty\) [1911.04758]. 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 \[ A=\partial f,\qquad B=\partial g,\qquad 0\in\partial f(x)+L^*\partial g(Lx) \] recovers Fenchel–Rockafellar primal-dual structure and the minimization problem \[ \min_x f(x)+g(Lx) \] [1107.0081]. More elaborate models arise when \[ A=\partial f,\quad C=\nabla h,\quad B_i=\partial g_i,\quad D_i=\partial \ell_i, \] with each \(\ell_i\) strongly convex, yielding the fully nonsmooth problem \[ \min_x\Bigl\{f(x)+h(x)+\sum_{i=1}^m (g_i\infconv \ell_i)(L_i x-r_i)\Bigr\}, \] for which the iterates admit closed-form prox steps on \(f\), \(g_i\), and \(\ell_i\) [1107.0081]. 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 \[ \min_{x_1,\dots,x_m}\; \sum_{i=1}^m\bigl[f_i(x_i)-\langle x_i,z_i\rangle\bigr]+p(x_1,\dots,x_m)+\sum_{k=1}^s (g_k\infconv \ell_k)\Bigl(\sum_{i=1}^mL_{k,i}x_i-r_k\Bigr) \] [1308.2778]. Constrained variants emphasize feasibility of the iterates. The projected primal-dual splitting for constrained composite primal-dual monotone inclusions introduces a primal constraint set \(X\), a dual constraint \(V\), and a projection step so that \(x^{k+1}=T p\in X\) and \(u^{k+1}=P_V\eta\in V\) at every iteration [1805.11687]. In the convex optimization case, this produces a projected Chambolle–Pock type method. Numerical experiments on \[ \min \|x\|_1\quad\text{s.t.}\quad R x=c,\;S x=d \] with \(n=1000\) report improvements of approximately \(5\%\) when \(m=1\), \(25\%-55\%\) when \(m=10\), and \(40\%-75\%\) when \(m=30\), with larger gains as the number of projected constraints increases and as tolerance tightens [1805.11687]. Imaging provides the most detailed benchmark class. The minimal-lifting CMIP algorithm is applied to \[ \min_{s\in\mathbb R^n}\;\|A s-b\|_1+\alpha_1\|W s\|_1+\alpha_2 TV(s)+\delta_{[0,1]^n}(s), \] with Gaussian blur \(9\times 9\), noise \(\sigma=10^{-3}\), \(\alpha_1=5\times10^{-3}\), and \(\alpha_2=9\times10^{-3}\); the reported outcome is convergence in roughly \(60\%\) of the time of the Douglas–Rachford-type primal-dual method of Bot–Hendrich (2013), equal or better ISNR after \(400\) iterations, and \(25\%\) less memory in the lifted primal variables [2202.09665]. The 2026 primal-dual splitting algorithm is tested on image deblurring and denoising, achieving PSNR up to \(23.65\) dB on the “Building” image, compared with PFDR’s \(23.53\) dB, and reporting \(0.01\)–\(0.02\) dB PSNR gains over PFDR in MC-TV denoising at comparable CPU costs [2606.28010]. Other CMIP-based imaging studies include Heron location and image deblurring via Douglas-Rachford-type methods [1212.0326], total-variation image restoration and sparse minimization via Split–Douglas–Rachford and Split-ADMM [2101.11683], and constrained fused-lasso and computed tomography inverse problems via FRB–PI and SDR–PI, with PSNR up to \(32\) dB in the CT experiments [2305.17500]. ## 6. Minimal lifting, nonlinear compositions, and interpretive issues One contemporary direction is dimensional optimality of the lifted space. For the sum of \(n\) maximally monotone operators with one composite term, the minimal-lifting theorem states that any frugal parametrized-resolvent splitting must lift into at least \(H^{n-1}\), so no fixed-point algorithm that evaluates each resolvent exactly once per iteration can have \(d\)-fold lifting with \(d