Inexact Inertial Projective Splitting

• Inexact inertial projective splitting is a method for solving monotone inclusions by decomposing operators and employing affine separators with inertial extrapolation.
• The approach augments standard projective splitting with relative-error inexact subproblem solves and relaxed projection steps, ensuring both weak and strong convergence under appropriate parameter settings.
• Applications include optimization problems like LASSO, with recent developments providing pointwise complexity bounds and structured variants through forward-backward processing.

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 (Alves et al., 2020). The 2020 paper “A relative-error inertial-relaxed inexact projective splitting algorithm” introduced a relative-error inertial-relaxed inexact PS method with weak convergence (Alves et al., 2020), 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 (Alves et al., 7 Jul 2025). 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 (Johnstone et al., 2018).

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, $T_i$0, and the solution set is nonempty (Alves et al., 2020). Using $T_i$1, the inclusion is equivalently

$T_i$2

which is the standard structured form handled by projective splitting (Alves et al., 2020).

The method is formulated in a product space

$T_i$3

equipped with the weighted inner product and norm

$T_i$4

for some $T_i$5 (Alves et al., 2020). The associated extended solution set, also described as a generalized Kuhn–Tucker set, is

$T_i$6

Then $T_i$7 solves the original inclusion if and only if there exist $T_i$8 such that $T_i$9, and $G_i$0 is nonempty, closed, and convex (Alves et al., 2020).

This product-space embedding is fundamental. Projective splitting methods iteratively construct affine separators whose nonpositive half-spaces contain $G_i$1, then project or relax toward those half-spaces (Alves et al., 2020). 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 (Johnstone et al., 2018). A plausible implication is that the natural objects of inexact inertial PS are not only the primal iterates $G_i$2, but the full product-space tuples $G_i$3, 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 $G_i$4 built from per-operator graph points. In the 2020 inertial-relaxed inexact PS method, with

$G_i$5

the algorithm first forms an extrapolated point $G_i$6, then computes graph points $G_i$7, and defines

$G_i$8

equivalently,

$G_i$9

For all $G_n=I$0, $G_n=I$1, so $G_n=I$2 defines a separating half-space containing $G_n=I$3 (Alves et al., 2020).

Its gradient in $G_n=I$4 is

$G_n=I$5

and therefore

$G_n=I$6

(Alves et al., 2020). These quantities encode the natural residuals of the split formulation: block consistency through $G_n=I$7, and primal-dual balance through $G_n=I$8.

The relaxed projection/correction step is

$G_n=I$9

with $0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$0 (Alves et al., 2020). 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” (Johnstone et al., 2018). There, one constructs $0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$1 so that $0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$2 and $0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$3 for all $0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$4, then performs the update

$0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$5

That paper treats exact and inexact proximal computations on backward blocks, but it does not include inertial terms (Johnstone et al., 2018). 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 (Alves et al., 2020). Its inertial step chooses $0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$6 and defines

$0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$7

with

$0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$8

In product form,

$0\in \sum_{i=1}^{n} G_i^*T_iG_i(z),$9

(Alves et al., 2020). If $n\ge 2$0, the method reduces to a noninertial PS method of classical type; if additionally $n\ge 2$1, it becomes exact PS (Alves et al., 2020).

The inexact subproblems are resolved blockwise. For each $n\ge 2$2, one computes $n\ge 2$3 such that

$n\ge 2$4

with relative error condition

$n\ge 2$5

(Alves et al., 2020). In the exact case $n\ge 2$6, $n\ge 2$7, and the step becomes a standard resolvent evaluation. The paper emphasizes that no $n\ge 2$8-enlargement or approximate graph formalism is introduced there; the approximate point must still satisfy the exact graph inclusion $n\ge 2$9, and the inexactness lies only in the resolvent equation (Alves et al., 2020).

A key consequence is the positivity estimate

$H_0,H_1,\dots,H_n$0

assuming bounded stepsizes $H_0,H_1,\dots,H_n$1 (Alves et al., 2020). 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 $H_0,H_1,\dots,H_n$2 with

$H_0,H_1,\dots,H_n$3

and imposes the relative-error condition

$H_0,H_1,\dots,H_n$4

(Alves et al., 7 Jul 2025). 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 (Alves et al., 2020) and enlargement-based relative-error PS (Alves et al., 7 Jul 2025).

The 2025 paper also introduces two inertial sequences rather than one: $H_0,H_1,\dots,H_n$5 and similarly for $H_0,H_1,\dots,H_n$6 (Alves et al., 7 Jul 2025). The first is a standard momentum-like extrapolation, while the second is an anchoring extrapolation relative to the initial point. The paper requires $H_0,H_1,\dots,H_n$7 and $H_0,H_1,\dots,H_n$8 (Alves et al., 7 Jul 2025). 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 (Alves et al., 2020). For arbitrary $H_0,H_1,\dots,H_n$9, defining

$H_0=H_n$0

the paper proves the quasi-Fejér estimate

$H_0=H_n$1

where

$H_0=H_n$2

(Alves et al., 2020). A second estimate is

$H_0=H_n$3

with

$H_0=H_n$4

(Alves et al., 2020). These inequalities replace the exact Fejér monotonicity available in noninertial PS by a Lyapunov-type recursion.

Weak convergence follows if

$H_0=H_n$5

and the stepsizes are bounded (Alves et al., 2020). The paper then gives an explicit sufficient parameter condition: if there exists $H_0=H_n$6 such that

$H_0=H_n$7

and one chooses

$H_0=H_n$8

then

$H_0=H_n$9

hence convergence follows (Alves et al., 2020).

The inertia-relaxation tradeoff is explicit. If $T_i:H_i\rightrightarrows H_i$0, then $T_i:H_i\rightrightarrows H_i$1. If $T_i:H_i\rightrightarrows H_i$2, then $T_i:H_i\rightrightarrows H_i$3, so overrelaxation is allowed. As inertial effect grows, admissible relaxation shrinks (Alves et al., 2020). 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 (Alves et al., 2020). 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 $T_i:H_i\rightrightarrows H_i$4 objective rate in convex optimization, an ergodic $T_i:H_i\rightrightarrows H_i$5 distance rate under strong monotonicity, and linear convergence under strong monotonicity plus cocoercivity (Johnstone et al., 2018). 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

$T_i:H_i\rightrightarrows H_i$6

an anchoring half-space

$T_i:H_i\rightrightarrows H_i$7

and updates by

$T_i:H_i\rightrightarrows H_i$8

(Alves et al., 7 Jul 2025). Under the standing assumptions, bounded stepsizes, and inertial conditions, the sequence converges strongly to

$T_i:H_i\rightrightarrows H_i$9

(Alves et al., 7 Jul 2025). 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 (Alves et al., 7 Jul 2025). 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 $G_i:H_0\to H_i$0 residual bounds and ergodic $G_i:H_0\to H_i$1 residual and enlargement bounds for exact and inexact noninertial PS, using decomposable separators and $G_i:H_0\to H_i$2-enlargements (Machado, 2017). On the other side, the 2020 inertial-relaxed inexact PS method proves only asymptotic weak convergence (Alves et al., 2020).

The 2025 strong-convergence paper closes part of this gap by proving a pointwise iteration-complexity result. It defines a $G_i:H_0\to H_i$3-approximate solution by requiring

$G_i:H_0\to H_i$4

$G_i:H_0\to H_i$5

$G_i:H_0\to H_i$6

(Alves et al., 7 Jul 2025). It then proves that for every $G_i:H_0\to H_i$7, there exists $G_i:H_0\to H_i$8 such that

$G_i:H_0\to H_i$9

$T_i$00

$T_i$01

(Alves et al., 7 Jul 2025). The abstract explicitly states that these iteration-complexity results likewise hold without requiring inertial terms (Alves et al., 7 Jul 2025). 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

$T_i$02

where $T_i$03 is single-valued and monotone, $T_i$04 is maximal monotone, and $T_i$05 with $T_i$06 nonempty closed convex (Alves et al., 7 Jul 2025).

When each $T_i$07 is $T_i$08-cocoercive, a forward-backward variant is available with

$T_i$09

$T_i$10

$T_i$11

(Alves et al., 7 Jul 2025). The paper proves

$T_i$12

and that the inexact criterion of the main framework holds (Alves et al., 7 Jul 2025).

When each $T_i$13 is merely $T_i$14-Lipschitz, a forward-backward-forward or Tseng variant is obtained with

$T_i$15

$T_i$16

$T_i$17

(Alves et al., 7 Jul 2025). 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 (Johnstone et al., 2018). Within that lineage, the 2020 paper positions itself as the first PS paper to incorporate inertial effects (Alves et al., 2020). Its novelty is not merely momentum, but the joint incorporation of inertial extrapolation, relaxation or overrelaxation, and relative-error inexact subproblem solves (Alves et al., 2020).

The paper’s numerical experiments focus on the LASSO problem

$T_i$18

rewritten in PS form by partitioning the rows of $T_i$19 (Alves et al., 2020). The reported parameters are

$T_i$20

with $T_i$21 obtained from the admissibility formula using $T_i$22 (Alves et al., 2020). 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 $T_i$23 and $T_i$24, respectively (Alves et al., 2020). The gains are not uniform: Crime slightly worsens in iterations, and Mug32 and Wisconsin show only marginal improvements (Alves et al., 2020). The paper also notes that the experiments do not ablate the separate effects of inertia, relaxation, and inexactness (Alves et al., 2020). 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 (Alves et al., 2019). “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 $T_i$25 and ergodic $T_i$26 residual behavior in an ADMM setting (Alves et al., 2024). 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 (Johnstone et al., 2018). 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 (Alves et al., 7 Jul 2025). 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 (Alves et al., 7 Jul 2025). Relative to standard noninertial PS rate papers, it adds inertia and strong convergence while retaining pointwise residual decay (Alves et al., 7 Jul 2025).

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 (Alves et al., 2020). 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 (Alves et al., 7 Jul 2025). Standard noninertial PS remains the benchmark for ergodic $T_i$27, strong-monotonicity $T_i$28, and linear-convergence regimes (Johnstone et al., 2018). 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 (Alves et al., 2020).