---
title: Douglas-Rachford Splitting (DRS)
url: https://www.emergentmind.com/topics/douglas-rachford-splitting-drs
type: topic
---

# Douglas-Rachford Splitting (DRS)

Douglas–Rachford splitting (DRS) is an iterative splitting method for solving the monotone inclusion
\[
0\in Ax+Bx,
\]
where \(A\) and \(B\) are maximally monotone operators on a real Hilbert space. Originating in the 1956 affine setting and substantially extended by Lions–Mercier in 1979, DRS has become a standard tool for convex feasibility, convex composite optimization, and operator-theoretic formulations related to ADMM. A characteristic feature of the method is the distinction between the governing fixed-point iterates and the associated “shadow” sequence obtained by a resolvent or projection; in many important settings the shadow carries the primal solution information even when the raw iterates converge to a larger fixed-point set or drift in inconsistent problems [1603.09418][1504.03721].

## 1. Operator-theoretic formulation

For a maximally monotone operator \(A\), the resolvent and reflected resolvent are
\[
J_A:=(\operatorname{Id}+A)^{-1},\qquad R_A:=2J_A-\operatorname{Id}.
\]
The Douglas–Rachford operator for the ordered pair \((A,B)\) is
\[
T_{(A,B)}:=\frac12(\operatorname{Id}+R_BR_A)=\operatorname{Id}-J_A+J_BR_A.
\]
Starting from \(x_0\in X\), the DRS iteration is
\[
x_{n+1}=Tx_n,\qquad x_n=T^n x_0.
\]
The associated shadow sequence is
\[
J_A T^n x_0,
\]
which reduces to a projection sequence in feasibility problems [1504.03721].

This operator form encodes the solution set through fixed points. In the consistent case,
\[
Z:=(A+B)^{-1}(0)=J_A(\operatorname{Fix}T),
\]
so fixed points of \(T\) are not themselves the primal solutions in general; rather, their resolvent images are. Classical consistent-case theory gives weak convergence of the raw iterates to some fixed point of \(T\), and weak convergence of the shadow sequence to an element of \(Z\) [1604.04603][1504.03721].

When \(A=\partial f\) and \(B=\partial g\) for proper lower semicontinuous convex functions, DRS becomes a proximal splitting method:
\[
T=\operatorname{Id}-\operatorname{prox}_f+\operatorname{prox}_g\,R_f
=\tfrac12(\operatorname{Id}+R_gR_f),
\]
with \(R_f=2\operatorname{prox}_f-\operatorname{Id}\). In this form, DRS sits at the center of a large family of primal, dual, and primal-dual algorithms, including ADMM through standard reformulations [2106.11547][1407.6723].

## 2. Convex feasibility and the shadow sequence

A particularly important specialization is the convex feasibility problem
\[
\text{find }x\in U\cap V,
\]
obtained by choosing
\[
A=N_U,\qquad B=N_V,
\]
with \(U,V\subseteq X\) nonempty closed convex sets. Then
\[
J_{N_U}=P_U,\qquad J_{N_V}=P_V,
\]
and the Douglas–Rachford operator becomes
\[
T=\operatorname{Id}-P_U+P_VR_U
=\operatorname{Id}-P_U-P_V+2P_VP_U,
\qquad R_U=2P_U-\operatorname{Id}.
\]
The shadow sequence is then
\[
P_U T^n x.
\]
This is the sequence that typically represents the primal feasibility candidates [1504.03721].

The geometry of DRS in subspace problems is especially explicit. For closed subspaces \(U,V\), one has
\[
T=P_VP_U+P_{V^\perp}P_{U^\perp},
\]
and the fixed-point set is exactly
\[
\operatorname{Fix}T=(U\cap V)\oplus(U^\perp\cap V^\perp).
\]
Consequently, the raw iterates do not generally converge to \(P_{U\cap V}x\); rather,
\[
T^n x \to P_{\operatorname{Fix}T}x
= P_{U\cap V}x + P_{U^\perp\cap V^\perp}x,
\]
whereas the shadows satisfy
\[
P_U T^n x \to P_{U\cap V}x,\qquad
P_V T^n x \to P_{U\cap V}x.
\]
In this precise sense, DRS becomes a best-approximation method through its shadow sequence rather than through its governing iterates [1309.4709].

This distinction resolves a common misconception. DRS is often informally described as “converging to the projection onto the solution set,” but that statement is generally correct only for the shadow sequence. The governing fixed-point iteration and the primal approximation sequence are different objects, and the difference is structurally significant already in the simplest subspace setting [1309.4709].

## 3. Exact geometry, strong convergence, and rates

For two closed subspaces, the convergence theory is exceptionally sharp. Let \(c_F\) denote the cosine of the Friedrichs angle between \(U\) and \(V\). Then
\[
U+V \text{ closed } \Longleftrightarrow c_F<1,
\]
and the DRS operator satisfies the exact norm identity
\[
\|T^n-P_{\operatorname{Fix}T}\|=c_F^n.
\]
Likewise, the shadow sequence has the exact rate
\[
\|P_UT^n-P_{U\cap V}\|
=
\|P_VT^n-P_{U\cap V}\|
=
c_F^n.
\]
Thus strong convergence always holds in Hilbert space, and linear convergence occurs exactly under the closed-sum condition, with rate equal to the cosine of the Friedrichs angle [1309.4709].

The subspace theory also admits explicit model examples. For two lines in \(\mathbb{R}^2\),
\[
U=\mathbb{R}e_0,\qquad V=\mathbb{R}e_\theta,
\]
the DRS operator is
\[
T=\cos(\theta)R_\theta,
\]
so
\[
T^n=\cos^n(\theta)R_{n\theta}.
\]
This makes the role of the Friedrichs angle completely transparent: the asymptotic contraction factor is precisely \(\cos(\theta)\) [1309.4709].

The affine setting is broader than subspaces. For maximally monotone affine relations, the DRS operator is affine nonexpansive, and this rigidity yields strong convergence of the raw iterates themselves:
\[
T_{\mathrm{DR}}^n x \to P_{\operatorname{Fix}T_{\mathrm{DR}}}x.
\]
In finite dimensions, this convergence is linear. Under additional assumptions such as paramonotonicity and the orthogonality condition \(K\perp(Z-Z)\), the shadow sequence converges strongly to the metric projection onto the primal solution set,
\[
J_A T_{\mathrm{DR}}^n x \to P_Z x.
\]
This sharpens the general Lions–Mercier picture, where one usually has only weak convergence [1603.09418].

Recent rate analysis has added a complementary operator-theoretic perspective. For the relaxed DRS operator
\[
T=\Bigl(1-\frac{\lambda}{2}\Bigr)I+\frac{\lambda}{2}R_{\gamma A}R_{\gamma B},
\]
an exact worst-case fixed-point residual rate was established for \(\lambda=1\):
\[
\|T(w^N)-w^N\|^2 \le \frac{(N-1)^{N-1}}{N^N}\|w^1-w^\star\|^2,
\]
and this bound is sharp already in convex feasibility. The same work characterizes linear convergence in terms of the error bound
\[
\operatorname{dist}_{W^\star}(w)\le \mu\|(I-T)w\|,
\]
which is both sufficient and necessary for linear convergence in distance to the fixed-point set [2509.06676].

## 4. Inconsistent problems, infimal displacement, and normal solutions

The inconsistent case, where \(Z=(A+B)^{-1}(0)=\varnothing\), is one of the defining subtleties of DRS. A central object is the infimal displacement vector
\[
v:=P_{\overline{\operatorname{ran}(\operatorname{Id}-T)}}0.
\]
In convex feasibility with sets \(U\) and \(V\), this becomes
\[
v=P_{\overline{U-V}}0.
\]
The associated normal problem shifts one operator, or one set, by the minimal displacement:
\[
0\in -v+Ax+B(x-v).
\]
In the normal-cone feasibility setting the normal solution set is
\[
Z_v=U\cap (v+V).
\]
Thus the original inconsistent problem is replaced by a canonically shifted consistent one [1504.03721][2106.11547].

For two affine subspaces, this structure becomes exact. One has
\[
T_{-v}=v+T=T_{U,v+V},
\]
so the inconsistent original problem is converted into a consistent DRS problem for \(U\) and \(v+V\). The main convergence theorem then states that for every \(x\),
\[
P_UT^n x
=
P_U(T^n x+nv)
=
P_U((T_{-v})^n x)
\to
P_{U\cap(v+V)}x.
\]
The convergence is strong, and if \(\operatorname{par}U+\operatorname{par}V\) is closed, it is linear with rate \(c_F(\operatorname{par}U,\operatorname{par}V)\). By contrast, the raw iterates \(T^n x\) need not converge and typically drift to infinity in inconsistent convex feasibility [1504.03721].

Beyond affine subspaces, the inconsistent convex-feasibility shadow sequence also admits a general weak-convergence theorem. If
\[
U\cap(v+V)\neq\varnothing,
\]
then
\[
P_U T^n x \rightharpoonup \bar z\in U\cap(v+V).
\]
This completes the weak-convergence picture for convex feasibility shadows in the inconsistent case [1604.04603].

For general convex optimization
\[
\min_x f(x)+g(x),
\]
the corresponding normal problem is
\[
\min_x f(x)+g(x-v),
\]
with \(v=P_{\operatorname{ran}(\operatorname{Id}-T)}(0)\). Under assumptions including \(v\in\operatorname{ran}(\operatorname{Id}-T)\) and solvability of the normal problem, the primal shadow satisfies
\[
\operatorname{prox}_f T^n x \rightharpoonup z,
\]
where \(z\) minimizes \(f(x)+g(x-v)\), while the companion shadow satisfies
\[
\operatorname{prox}_gR_fT^n x \rightharpoonup z-v,
\]
and the objective values converge to the optimal value of the shifted problem. The same analysis decomposes the minimal displacement as
\[
v=v_D+v_R,\qquad \langle v_D,v_R\rangle=0,
\]
linking domain mismatch and range mismatch to asymptotic behavior of primal and dual shadows [2106.11547].

## 5. Envelope viewpoints and nonconvex DRS

A major development in DRS theory is the introduction of the Douglas–Rachford envelope (DRE). For convex composite minimization
\[
\min_x f(x)+g(x),
\]
with smooth \(f\), the DRE is
\[
F^{\mathrm{DR}}_\gamma(x)
=
f^\gamma(x)-\gamma\|\nabla f^\gamma(x)\|^2
+
g^\gamma\bigl(x-2\gamma\nabla f^\gamma(x)\bigr).
\]
This function is continuously differentiable even when \(g\) is nonsmooth, and for \(\gamma\in(0,1/L_f)\) it satisfies
\[
\inf F=\inf F^{\mathrm{DR}}_\gamma,\qquad
\arg\min F = P_\gamma(\arg\min F^{\mathrm{DR}}_\gamma).
\]
When \(f\) is convex quadratic, DRS is exactly a scaled gradient method on the DRE, which permits transfer of smooth optimization techniques to DRS, including explicit parameter tuning and accelerated variants. In that setting the recommended tuning is
\[
\gamma_\star=\frac{\sqrt2-1}{L_f},
\]
and an accelerated DRS variant achieves an \(O(1/k^2)\) objective bound [1407.6723].

The DRE also plays a central role in nonconvex analysis. For
\[
\min_s \varphi_1(s)+\varphi_2(s),
\]
with \(\varphi_1\in C^{1,1}\) and \(\varphi_2\) proper lsc, the DRE yields a sufficient decrease inequality for DRS and PRS, leading to global residual convergence and subsequential convergence to stationary points under explicit parameter ranges. The same analysis shows that the parameter bounds are tight whenever the relaxation parameter lies in \((0,2]\) [1709.05747].

In weakly convex optimization, the envelope viewpoint becomes even closer to the Moreau-envelope interpretation of the proximal point method. If \(f\) is \(L_f\)-smooth and \(g\) is proper lsc and \(\rho\)-weakly convex, then the Douglas–Rachford envelope epi-approximates the original objective as \(\gamma\downarrow 0\). Under boundedness of the generated sequence, DRS yields convergence of DRE values and convergence of cluster points to critical points of \(f+g\). Under a local error bound and the proper separation of isocost surfaces, the DRE gap converges \(Q\)-linearly and the iterates converge \(R\)-linearly to a critical point [2303.16394].

This suggests a unifying interpretation: in convex settings the DRE exposes best-approximation and rate structure, while in weakly convex settings it supplies a descent function strong enough to replace the convex fixed-point geometry that classical DRS analysis relies on.

## 6. Generalizations, variants, and graph-structured extensions

Modern work has pushed DRS far beyond its classical form without abandoning its basic resolvent architecture. One direction is parameter generalization. In the convex optimization setting, all frugal, no-lifting, fixed-point-encoding resolvent splittings have the three-line form
\[
x_1=\prox_{\alpha f}(z),\qquad
x_2=\prox_{\beta g}\!\left(\left(1+\frac{\beta}{\alpha}\right)x_1-\frac{\beta}{\alpha}z\right),\qquad
z^+=z+\theta(x_2-x_1),
\]
and unconditional convergence holds exactly when
\[
\theta\in(0,\min\{2,2\alpha/\beta\}).
\]
Classical relaxed DRS is the special case \(\alpha=\beta\), \(\theta\in(0,2)\). This identifies a larger family of unconditionally convergent DRS-type methods specific to convex optimization [2511.19637].

A second direction is robustness beyond monotonicity. For the inclusion
\[
0\in Ax+Bx,
\]
with \(A\) maximally \(\alpha\)-monotone and \(B\) maximally \(\beta\)-monotone, including the regime “strongly monotone + weakly monotone,” an adaptive DRS scheme uses
\[
R_1=(1-\lambda)I+\lambda J_{\gamma A},\qquad
R_2=(1-\mu)I+\mu J_{\delta B},\qquad
T=(1-\kappa)I+\kappa R_2R_1,
\]
with the coupling
\[
(\lambda-1)(\mu-1)=1,\qquad \delta=(\lambda-1)\gamma.
\]
If \(\alpha+\beta\ge 0\) and the admissibility conditions of the paper hold, the iterates converge weakly to a fixed point and the shadow recovers the unique primal solution when \(\alpha+\beta>0\); if one operator is Lipschitz, global linear convergence follows [1809.00761].

Inexactness is another major extension. A fully inexact DRS method with relative error tolerance allows both proximal subproblems to be solved approximately within \(\varepsilon\)-enlargements and residual tests, while still guaranteeing weak convergence of the generated sequences to a point in the extended solution set, provided a solution exists. A semi-inexact variant solves the first subproblem exactly and the second inexactly under a related criterion [1809.02312].

Finally, DRS has acquired graph-based and decentralized forms. For sums of \(N\) maximal monotone operators, bilevel-graph constructions generate unconditionally stable frugal resolvent splittings with the minimal \((N-1)\)-fold lifting required by general lower bounds. These schemes are instances of degenerate preconditioned proximal point methods, and they recover Ryu-type and Malitsky–Tam-type multi-operator splittings as special cases [2211.04782]. In decentralized smooth optimization over compact submanifolds, a decentralized DRS algorithm and its inexact variant integrate nonconvex DRS, gradient tracking, and projection onto a manifold; under proximal smoothness of the manifold constraint, both methods achieve an \(\mathcal O(1/K)\) rate for consensus and stationarity measures [2311.16399].

Across these variants, the persistent structural ideas are resolvent-based decomposition, shadow recovery of primal information, and fixed-point iterations whose convergence is often best understood through auxiliary geometry: Friedrichs angles, infimal displacement vectors, envelopes, or graph-induced preconditioners.

Source: https://www.emergentmind.com/topics/douglas-rachford-splitting-drs