---
title: Inexact Inertial Projective Splitting
url: https://www.emergentmind.com/topics/inexact-inertial-projective-splitting-ps
type: topic
---

# Inexact Inertial Projective Splitting

Searching arXiv for recent and foundational papers on inexact inertial projective splitting and closely related projective splitting analyses.
Inexact inertial projective splitting (PS) is a class of operator-splitting methods for monotone inclusions in which the sum of finitely many maximal monotone operators is handled by separately generating operatorwise graph information, constructing an affine separator whose nonpositive half-space contains the extended solution set, and then updating by a relaxed projection step. In the inexact inertial setting, this architecture is augmented by inertial extrapolation and by approximate subproblem solves satisfying a relative-error criterion. A representative formulation solves
\[
0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),
\]
with \(T_i\) maximal monotone, \(G_i\) bounded linear, \(G_n=I\), and a nonempty solution set [2002.07878]. The 2020 paper “A relative-error inertial-relaxed inexact projective splitting algorithm” introduced a relative-error inertial-relaxed inexact PS method with weak convergence [2002.07878], while the 2025 paper “An inexact inertial projective splitting algorithm with strong convergence” developed a strongly convergent inexact inertial PS scheme with pointwise iteration-complexity bounds [2507.05382]. These works place inexact inertial PS within the broader projective-splitting literature, where standard noninertial PS already had asymptotic convergence and, in several regimes, ergodic or linear rates [1806.03920].

## 1. Problem class and product-space formulation

The canonical structured monotone inclusion treated by projective splitting is
\[
0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),
\]
under the assumptions that \(n\ge 2\), \(H_0,H_1,\dots,H_n\) are real Hilbert spaces with \(H_0=H_n\), each \(T_i:H_i\rightrightarrows H_i\) is maximal monotone, each \(G_i:H_0\to H_i\) is bounded linear, \(G_n=I\), and the solution set is nonempty [2002.07878]. Using \(G_n=I\), the inclusion is equivalently
\[
0\in \sum_{i=1}^{n-1}G_i^*T_iG_i(z)+T_n(z),
\]
which is the standard structured form handled by projective splitting [2002.07878].

The method is formulated in a product space
\[
\boldsymbol H:=H_0\times \cdots \times H_{n-1},
\]
equipped with the weighted inner product and norm
\[
\langle (z,w),(z',w')\rangle_\gamma = \gamma\langle z,z'\rangle+\sum_{i=1}^{n-1}\langle w_i,w_i'\rangle,
\qquad
\|(z,w)\|_\gamma^2 = \gamma\|z\|^2+\sum_{i=1}^{n-1}\|w_i\|^2,
\]
for some \(\gamma>0\) [2002.07878]. The associated extended solution set, also described as a generalized Kuhn–Tucker set, is
\[
\mathcal S
:=
\left\{
(z,w_1,\dots,w_{n-1})\in \boldsymbol H
\;\middle|\;
w_i\in T_i(G_iz)\ \ (i=1,\dots,n-1),\
-\sum_{i=1}^{n-1}G_i^*w_i\in T_n(z)
\right\}.
\]
Then \(z\) solves the original inclusion if and only if there exist \(w_1,\dots,w_{n-1}\) such that \((z,w_1,\dots,w_{n-1})\in\mathcal S\), and \(\mathcal S\) is nonempty, closed, and convex [2002.07878].

This product-space embedding is fundamental. Projective splitting methods iteratively construct affine separators whose nonpositive half-spaces contain \(\mathcal S\), then project or relax toward those half-spaces [2002.07878]. The same separator-projector viewpoint underlies standard PS convergence-rate analyses, where the extended solution set is likewise treated as a closed convex target in product space [1806.03920]. A plausible implication is that the natural objects of inexact inertial PS are not only the primal iterates \(z^k\), but the full product-space tuples \(p^k=(z^k,w_1^k,\dots,w_{n-1}^k)\), because both separator geometry and inertial extrapolation operate at that level.

## 2. Separator-projector architecture of projective splitting

The defining geometric object of PS is an affine separator \(\varphi_k\) built from per-operator graph points. In the 2020 inertial-relaxed inexact PS method, with
\[
p^k=(z^k,w_1^k,\dots,w_{n-1}^k),
\qquad
w_n^k:=-\sum_{i=1}^{n-1}G_i^*w_i^k,
\]
the algorithm first forms an extrapolated point \(\widehat p^{\,k}\), then computes graph points \((x_i^k,y_i^k)\in \Graph(T_i)\), and defines
\[
\varphi_k(p)=\sum_{i=1}^{n-1}\langle G_iz-x_i^k,\,y_i^k-w_i\rangle
+\left\langle G_nz-x_n^k,\,
y_n^k+\sum_{i=1}^{n-1}G_i^*w_i\right\rangle,
\]
equivalently,
\[
\varphi_k(p)=\sum_{i=1}^n \langle G_iz-x_i^k,\ y_i^k-w_i\rangle.
\]
For all \(p\in\mathcal S\), \(\varphi_k(p)\le 0\), so \(\varphi_k\) defines a separating half-space containing \(\mathcal S\) [2002.07878].

Its gradient in \((\boldsymbol H,\langle\cdot,\cdot\rangle_\gamma)\) is
\[
\nabla\varphi_k
=
\left(
\frac1\gamma\Big(\sum_{i=1}^{n-1}G_i^*y_i^k+y_n^k\Big),
\ x_1^k-G_1x_n^k,\ \dots,\ x_{n-1}^k-G_{n-1}x_n^k
\right),
\]
and therefore
\[
\|\nabla\varphi_k\|_\gamma^2
=
\gamma^{-1}\left\|\sum_{i=1}^{n-1}G_i^*y_i^k+y_n^k\right\|^2
+
\sum_{i=1}^{n-1}\|x_i^k-G_ix_n^k\|^2
\]
[2002.07878]. These quantities encode the natural residuals of the split formulation: block consistency through \(x_i^k-G_ix_n^k\), and primal-dual balance through \(\sum_{i=1}^{n-1}G_i^*y_i^k+y_n^k\).

The relaxed projection/correction step is
\[
p^{k+1}
=
\widehat p^{\,k}
-
\frac{\beta_k\max\{0,\varphi_k(\widehat p^{\,k})\}}
{\|\nabla\varphi_k\|_\gamma^2}\nabla\varphi_k,
\]
with \(\beta_k\in[\underline\beta,\overline\beta]\subset(0,2)\) [2002.07878]. Thus the 2020 method is explicitly a relaxed projection onto the separator half-space, specialized to graph points of maximal monotone operators.

The separator-projector view also underpins the standard synchronous noninertial PS framework analyzed in “Convergence Rates for Projective Splitting” [1806.03920]. There, one constructs \(\varphi_k\) so that \(\varphi_k(p^k)>0\) and \(\varphi_k(p)\le 0\) for all \(p\in\mathcal S\), then performs the update
\[
p^{k+1}=p^k-\alpha_k\nabla\varphi_k(p^k).
\]
That paper treats exact and inexact proximal computations on backward blocks, but it does not include inertial terms [1806.03920]. By contrast, inexact inertial PS modifies precisely the point at which the separator is evaluated and the direction from which the projection is taken. This suggests that inertia changes not the separator itself, but the geometry of how the current iterate approaches it.

## 3. Inertial and inexact extensions

The 2020 algorithm introduced three features simultaneously: inertial extrapolation, relaxation or overrelaxation, and inexact subproblem solves with relative error [2002.07878]. Its inertial step chooses \(\alpha_k\in[0,\alpha]\) and defines
\[
\widehat z^{\,k}=z^k+\alpha_k(z^k-z^{k-1}),
\qquad
\widehat w_i^{\,k}=w_i^k+\alpha_k(w_i^k-w_i^{k-1}),
\]
with
\[
\widehat w_n^{\,k}=-\sum_{i=1}^{n-1}G_i^*\widehat w_i^{\,k}.
\]
In product form,
\[
\widehat p^{\,k}=p^k+\alpha_k(p^k-p^{k-1})
\]
[2002.07878]. If \(\alpha_k\equiv 0\), the method reduces to a noninertial PS method of classical type; if additionally \(\sigma=0\), it becomes exact PS [2002.07878].

The inexact subproblems are resolved blockwise. For each \(i=1,\dots,n\), one computes \((x_i^k,y_i^k)\) such that
\[
y_i^k\in T_i(x_i^k),
\qquad
\rho_i^k y_i^k+x_i^k = G_i\widehat z^{\,k}+\rho_i^k\widehat w_i^{\,k}+e_i^k,
\]
with relative error condition
\[
\|e_i^k\|^2 \le \sigma^2\left( \|G_i\widehat z^k-x_i^k\|^2 + \|\rho_i^k(\widehat w_i^k-y_i^k)\|^2 \right),
\qquad
0\le \sigma<1
\]
[2002.07878]. In the exact case \(\sigma=0\), \(e_i^k=0\), and the step becomes a standard resolvent evaluation. The paper emphasizes that no \(\varepsilon\)-enlargement or approximate graph formalism is introduced there; the approximate point must still satisfy the exact graph inclusion \(y_i^k\in T_i(x_i^k)\), and the inexactness lies only in the resolvent equation [2002.07878].

A key consequence is the positivity estimate
\[
\varphi_k(\widehat p^k)\ge
\frac{(1-\sigma^2)\min\{\overline\rho^{-1},\underline\rho\}}{2}
\sum_{i=1}^n
\left(
\|G_i\widehat z^k-x_i^k\|^2+\|\widehat w_i^k-y_i^k\|^2
\right)\ge 0,
\]
assuming bounded stepsizes \(0<\underline\rho\le \rho_i^k\le \overline\rho<\infty\) [2002.07878]. This lower bound is central because it guarantees that the separator value is nonnegative at the extrapolated point and quantitatively controls local inexact residuals.

The 2025 strong-convergence paper adopts a more expansive inexactness model based on enlargements. It computes triples \((x_i^k,y_i^k,\varepsilon_i^k)\) with
\[
y_i^k \in T_i^{[\varepsilon_i^k]}(x_i^k),
\]
and imposes the relative-error condition
\[
\|\lambda_i^k y_i^k + x_i^k - (G_i\widetilde z^{\,k} + \lambda_i^k \widetilde w_i^{\,k})\|^2
+ 2\lambda_i^k \varepsilon_i^k
\le
\sigma^2\Big(
\|G_i\widetilde z^{\,k}-x_i^k\|^2
+
\|\lambda_i^k(\widetilde w_i^{\,k}-y_i^k)\|^2
\Big)
\]
[2507.05382]. This is a different inexactness mechanism from the 2020 algorithm: the graph point itself may now lie in an enlarged graph. A plausible implication is that inexact inertial PS has at least two distinct methodological branches: exact-graph relative-error PS [2002.07878] and enlargement-based relative-error PS [2507.05382].

The 2025 paper also introduces two inertial sequences rather than one:
\[
\widehat z^{\,k}=z^k+\alpha_k(z^k-z^{k-1}),
\qquad
\widetilde z^{\,k}=\widehat z^{\,k}+\beta_k(\widehat z^{\,k}-z^0),
\]
and similarly for \(w_i\) [2507.05382]. The first is a standard momentum-like extrapolation, while the second is an anchoring extrapolation relative to the initial point. The paper requires \(\sup_k\alpha_k<\infty\) and \(\sum_k\beta_k^2<\infty\) [2507.05382]. This suggests that strong-convergence inertial PS can tolerate bounded first-order inertia provided the second inertial mechanism is square summable.

## 4. Convergence theory and parameter couplings

The convergence analysis of relative-error inertial-relaxed inexact PS in 2020 is built in two layers: a general inertial-relaxed separator-projection framework and a verification that the PS separator satisfies the framework assumptions [2002.07878]. For arbitrary \(p\in\mathcal S\), defining
\[
h_k=\|p^k-p\|^2,
\]
the paper proves the quasi-Fejér estimate
\[
h_{k+1}-h_k-\alpha_k(h_k-h_{k-1})
\le
\alpha_k(1+\alpha_k)\|p^k-p^{k-1}\|^2-s_{k+1},
\]
where
\[
s_{k+1}:=\beta_k(2-\beta_k)\|\widehat p^{\,k}-\widetilde p^{\,k+1}\|^2
\]
[2002.07878]. A second estimate is
\[
h_{k+1}-h_k-\alpha_k(h_k-h_{k-1})
\le
\gamma_k\|p^k-p^{k-1}\|^2
-
(2-\overline\beta)\overline\beta^{-1}(1-\alpha_k)\|p^{k+1}-p^k\|^2,
\]
with
\[
\gamma_k := 2\left(1-\overline\beta^{-1}\right)\alpha_k^2 + 2\overline\beta^{-1}\alpha_k
\]
[2002.07878]. These inequalities replace the exact Fejér monotonicity available in noninertial PS by a Lyapunov-type recursion.

Weak convergence follows if
\[
\sum_{k=0}^\infty \alpha_k\|p^k-p^{k-1}\|_\gamma^2<\infty
\]
and the stepsizes are bounded [2002.07878]. The paper then gives an explicit sufficient parameter condition: if there exists \(\overline\alpha>0\) such that
\[
0\le \alpha_k\le \alpha_{k+1}\le \alpha<\overline\alpha<1,
\]
and one chooses
\[
\overline\beta=\frac{2(\overline\alpha-1)^2}{2(\overline\alpha-1)^2+3\overline\alpha-1},
\]
then
\[
\sum_{k=0}^{\infty}\|p^k-p^{k-1}\|_\gamma^2<\infty,
\]
hence convergence follows [2002.07878].

The inertia-relaxation tradeoff is explicit. If \(\overline\alpha=1/3\), then \(\overline\beta=1\). If \(\overline\alpha<1/3\), then \(\overline\beta>1\), so overrelaxation is allowed. As inertial effect grows, admissible relaxation shrinks [2002.07878]. This is one of the main quantitative characterizations of inertial PS stability.

The 2020 paper proves only asymptotic convergence; it does not provide iteration-complexity bounds or ergodic convergence rates for the inertial inexact PS method [2002.07878]. This is an important distinction from standard noninertial PS theory. In “Convergence Rates for Projective Splitting,” standard synchronous noninertial PS with inexact backward steps yields an ergodic \(O(1/k)\) objective rate in convex optimization, an ergodic \(O(1/\sqrt{k})\) distance rate under strong monotonicity, and linear convergence under strong monotonicity plus cocoercivity [1806.03920]. A plausible interpretation is that adding inertia and relative-error subproblem solves complicates the Lyapunov structure enough that asymptotic convergence was established before nonasymptotic rates.

The 2025 paper changes the convergence target from weak convergence to strong convergence. It defines, besides the separator half-space
\[
H_k=\{p\in \boldsymbol{\mathcal H}\mid \varphi_k(p)\le 0\},
\]
an anchoring half-space
\[
W_k
=
\left\{
p\in \boldsymbol{\mathcal H}
\;\middle|\;
\langle p^0-p^k,\; p-p^k\rangle_\gamma \le 0
\right\},
\]
and updates by
\[
p^{k+1}=P_{H_k\cap W_k}(p^0)
\]
[2507.05382]. Under the standing assumptions, bounded stepsizes, and inertial conditions, the sequence converges strongly to
\[
p^\ast := P_{\mathcal S_e}(p^0)
\]
[2507.05382]. The paper stresses that the strong convergence mechanism comes from the Solodov–Svaiter-style projection onto the intersection of two half-spaces and remains effective even without inertial effects [2507.05382]. This corrects a common misconception that inertia itself is the source of improved convergence mode; in this setting, inertia is auxiliary, whereas the projection architecture produces strong convergence.

## 5. Complexity and structured variants

Inexact inertial PS sits between two complexity traditions. On one side, noninertial projective splitting has pointwise and ergodic complexity results. For the two-operator case, “On the complexity of the projective splitting and Spingarn's methods for the sum of two maximal monotone operators” gives pointwise \(O(1/\sqrt{k})\) residual bounds and ergodic \(O(1/k)\) residual and enlargement bounds for exact and inexact noninertial PS, using decomposable separators and \(\epsilon\)-enlargements [1707.08655]. On the other side, the 2020 inertial-relaxed inexact PS method proves only asymptotic weak convergence [2002.07878].

The 2025 strong-convergence paper closes part of this gap by proving a pointwise iteration-complexity result. It defines a \(\rho\)-approximate solution by requiring
\[
y_i \in T_i^{[\varepsilon_i]}(x_i), \quad i=1,\dots,n,
\]
\[
\sum_{i=1}^n \varepsilon_i \le \rho,
\qquad
\left\|\sum_{i=1}^{n-1}G_i^\ast y_i + y_n\right\|\le \rho,
\]
\[
\|x_i-G_ix_n\|\le \rho,\quad i=1,\dots,n-1
\]
[2507.05382]. It then proves that for every \(k\ge 0\), there exists \(\ell\in\{0,\dots,k\}\) such that
\[
\left\|\sum_{i=1}^{n-1}G_i^\ast y_i^\ell + y_n^\ell\right\| = \mathcal O\!\left(\frac{1}{\sqrt{k+1}}\right),
\]
\[
\|x_i^\ell-G_ix_n^\ell\| = \mathcal O\!\left(\frac{1}{\sqrt{k+1}}\right),
\qquad i=1,\dots,n-1,
\]
\[
\sum_{i=1}^n \varepsilon_i^\ell = \mathcal O\!\left(\frac{1}{k+1}\right)
\]
[2507.05382]. The abstract explicitly states that these iteration-complexity results likewise hold without requiring inertial terms [2507.05382]. This suggests that, at least for that algorithm, inertia is not needed to preserve the classical pointwise PS residual orders.

The same paper derives two structured variants for
\[
0 \in \sum_{i=1}^n G_i^\ast(F_i+B_i)G_i(z),
\]
where \(F_i\) is single-valued and monotone, \(B_i\) is maximal monotone, and \(\operatorname{Dom}B_i\subseteq C_i\subseteq \operatorname{Dom}F_i\) with \(C_i\) nonempty closed convex [2507.05382].

When each \(F_i\) is \(L_i^{-1}\)-cocoercive, a forward-backward variant is available with
\[
\lambda_i = \frac{2\sigma^2}{L_i},
\]
\[
x_i^k = (\lambda_i B_i+I)^{-1}
\left(
G_i\widetilde z^{\,k}+\lambda_i\widetilde w_i^{\,k}-\lambda_i F_i(\bar z_i^{\,k})
\right),
\]
\[
y_i^k=\frac{1}{\lambda_i}
\left(
G_i\widetilde z^{\,k}+\lambda_i\widetilde w_i^{\,k}-x_i^k
\right),
\qquad
\varepsilon_i^k = \frac{L_i\|x_i^k-\bar z_i^{\,k}\|^2}{4}
\]
[2507.05382]. The paper proves
\[
y_i^k \in (F_i^{[\varepsilon_i^k]}+B_i)(x_i^k)\subseteq (F_i+B_i)^{[\varepsilon_i^k]}(x_i^k),
\]
and that the inexact criterion of the main framework holds [2507.05382].

When each \(F_i\) is merely \(L_i\)-Lipschitz, a forward-backward-forward or Tseng variant is obtained with
\[
\lambda_i=\frac{\sigma}{L_i},
\]
\[
x_i^k = (\lambda_i B_i+I)^{-1}
\left(
G_i\widetilde z^{\,k}+\lambda_i\widetilde w_i^{\,k}-\lambda_i F_i(\bar z_i^{\,k})
\right),
\]
\[
y_i^k =
\frac{G_i\widetilde z^{\,k}+\lambda_i\widetilde w_i^{\,k}-x_i^k}{\lambda_i}
+
F_i(x_i^k)-F_i(\bar z_i^{\,k}),
\qquad
\varepsilon_i^k=0
\]
[2507.05382]. Thus structured inexact inertial PS can subsume FB and FBF processing without leaving the separator-projection framework.

## 6. Position in the literature and computational behavior

Projective splitting was introduced by Eckstein and Svaiter and later diversified into variants with forward steps, asynchronous or block updates, warped metrics, and saddle formulations [1806.03920]. Within that lineage, the 2020 paper positions itself as the first PS paper to incorporate inertial effects [2002.07878]. Its novelty is not merely momentum, but the joint incorporation of inertial extrapolation, relaxation or overrelaxation, and relative-error inexact subproblem solves [2002.07878].

The paper’s numerical experiments focus on the LASSO problem
\[
\min_{x\in\mathbb R^d} \left\{ \frac12\|Qx-b\|_2^2+\lambda\|x\|_1 \right\},
\]
rewritten in PS form by partitioning the rows of \(Q\) [2002.07878]. The reported parameters are
\[
\alpha_k\equiv \alpha=0.1,\qquad \sigma=0.99,\qquad \gamma=1,\qquad \rho_i^k\equiv 1,
\qquad
\beta_k\equiv 1.5519,
\]
with \(\beta_k\) obtained from the admissibility formula using \(\overline\alpha=0.17>\alpha\) [2002.07878]. The experiments compare a previous noninertial exact projective splitting baseline against the proposed inertial-relaxed relative-error PS. Across 9 instances, the proposed method usually improves both iterations and time; the geometric means of the outer-iteration and runtime ratios are about \(0.6883\) and \(0.6793\), respectively [2002.07878]. The gains are not uniform: Crime slightly worsens in iterations, and Mug32 and Wisconsin show only marginal improvements [2002.07878]. The paper also notes that the experiments do not ablate the separate effects of inertia, relaxation, and inexactness [2002.07878]. This limits causal interpretation of the empirical gains.

Several neighboring literatures illuminate aspects of inexact inertial PS without being PS papers themselves. “Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms” develops an inexact inertial-relaxed hybrid proximal projection method with projective correction, relative-error approximate resolvents, and an explicit inertia-relaxation tradeoff [1904.10502]. “A relative-error inexact ADMM splitting algorithm for convex optimization with inertial effects” provides an inertial product-space update, a separator-type inequality with error term, and pointwise \(O(1/\sqrt{k})\) and ergodic \(O(1/k)\) residual behavior in an ADMM setting [2409.10311]. These are not projective splitting algorithms, but they are technically adjacent. This suggests that some of the analytical machinery of inexact inertial PS belongs to a broader family of inertial relative-error splitting and HPE-style methods, even when the explicit separator geometry is different.

A further clarification concerns forward processing. “Projective Splitting with Forward Steps only Requires Continuity” shows that some forward-step PS blocks need only continuity plus full domain in finite-dimensional spaces, with backtracking linesearch replacing Lipschitz assumptions [1809.07180]. That work is noninertial, but it indicates that projective splitting can accommodate richer blockwise processing rules than exact resolvents. A plausible implication is that future inertial inexact PS variants may combine extrapolation with continuity-based forward blocks, though the corresponding inertial analysis is not supplied there.

Finally, the 2025 strong-convergence paper materially changes the state of the subject by providing a genuine inexact inertial PS algorithm with strong convergence and pointwise complexity bounds [2507.05382]. Relative to the 2020 weak-convergence method, it replaces a single relaxed projection onto one separator half-space by a projection onto the intersection of a separator half-space and an anchoring half-space [2507.05382]. Relative to standard noninertial PS rate papers, it adds inertia and strong convergence while retaining pointwise residual decay [2507.05382].

In summary, inexact inertial projective splitting denotes a family of separator-projector methods for monotone inclusions in which blockwise graph information is generated approximately under relative-error control, the current product-space iterate is extrapolated inertially, and the ensuing affine separator guides a relaxed or anchored projection step. The 2020 formulation established weak convergence under explicit inertia-relaxation couplings [2002.07878]. The 2025 formulation established strong convergence to the projection of the initial point onto the extended solution set, together with pointwise complexity bounds and FB/FBF structured variants [2507.05382]. Standard noninertial PS remains the benchmark for ergodic \(O(1/k)\), strong-monotonicity \(O(1/\sqrt{k})\), and linear-convergence regimes [1806.03920]. The resulting picture is that inexact inertial PS is no longer merely a heuristic acceleration of classical PS, but a technically distinct branch of projective splitting with its own inexactness models, parameter couplings, convergence modes, and emerging complexity theory [2002.07878].

Source: https://www.emergentmind.com/topics/inexact-inertial-projective-splitting-ps