Relaxed Inertial Proximal Splitting (RIPSA)

• RIPSA is a proximal splitting method that combines inertial extrapolation with relaxation to improve convergence in solving monotone inclusions.
• The algorithm uses momentum-like inertial updates and resolvent/proximal computations, which have shown to reduce iterations and CPU time in numerical experiments.
• RIPSA generalizes several classic methods (FB, FBF, Chambolle–Pock, Condat–Vu), demonstrating its versatility across convex optimization and equilibrium problems.

Relaxed Inertial Proximal Splitting Algorithm (RIPSA) denotes a class of proximal splitting schemes that combine inertial extrapolation with relaxed updates, and, in several formulations, resolvent or proximal computations for monotone operators or bifunctions. In the 2025 nonlinear Forward–Backward setting, RIPSA is the relaxed inertial specialization of the Nonlinear Forward-Backward (NFB) algorithm, also known as warped resolvent iterations, for finding zeros of sums of monotone operators; in particular cases, this framework reduces to Forward–Backward, Forward–Backward–Forward, Chambolle–Pock, and Condat–Vu (Maulén et al., 25 Jul 2025). The acronym also appears in earlier and later literature for related but non-identical inertial–relaxed proximal schemes, including inexact proximal-point/Douglas–Rachford/ADMM variants, relaxed inertial Forward–Backward–Forward methods, and hierarchical equilibrium algorithms (Alves et al., 2019, Bot et al., 2020, Mazgouri et al., 28 Sep 2025).

1. Canonical monotone-inclusion formulation

In the formulation emphasized in "Relaxed and inertial nonlinear Forward-Backward algorithm" (Maulén et al., 25 Jul 2025), RIPSA is posed on a real Hilbert space $H$ with a maximally monotone operator $A:H\rightrightarrows H$, a $\beta$-cocoercive operator $B:H\to H$ with $\beta>0$, and resolvent

$J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$

The target problem is the monotone inclusion

$0\in A(x)+B(x),$

and, more generally, the NFB framework treats warped resolvent formulations involving sums of maximally monotone, cocoercive, monotone and Lipschitz operators as well as linear compositions terms (Maulén et al., 25 Jul 2025).

Given $x_{-1},x_0\in H$, inertial parameters $(\alpha_n)\subset[0,1[$, relaxation parameters $(\lambda_n)\subset[\underline\lambda,2[$, and a stepsize $A:H\rightrightarrows H$0, the Relaxed Inertial Proximal Splitting Algorithm is

$A:H\rightrightarrows H$1

The standing assumptions are explicit. The operator $A:H\rightrightarrows H$2 is maximally monotone, $A:H\rightrightarrows H$3 is $A:H\rightrightarrows H$4-cocoercive in the sense that

$A:H\rightrightarrows H$5

and the stepsize satisfies $A:H\rightrightarrows H$6, often with $A:H\rightrightarrows H$7. Relaxation obeys $A:H\rightrightarrows H$8 for some $A:H\rightrightarrows H$9, with $\beta$0, while inertia satisfies $\beta$1 for $\beta$2 (Maulén et al., 25 Jul 2025).

The same paper also states a more general warped resolvent setting in which one introduces a linear $\beta$3 and a single-valued $\beta$4 satisfying Lipschitz and strong-monotonicity conditions; in the simplest case one sets $\beta$5 (Maulén et al., 25 Jul 2025).

2. Inertial and relaxation mechanisms

RIPSA is organized around two interacting accelerative devices. The first is inertia, implemented through the extrapolated point

$\beta$6

which uses the previous displacement $\beta$7 as a momentum-like direction. The second is relaxation, implemented through the factor $\beta$8 in the update from $\beta$9 toward the resolvent point $B:H\to H$0. With $B:H\to H$1, the update is unrelaxed; with $B:H\to H$2, the method allows under-relaxation or over-relaxation, provided $B:H\to H$3 (Maulén et al., 25 Jul 2025).

A distinctive feature of the 2025 analysis is that it considers both nondecreasing and decreasing sequences of inertial parameters. The two regimes are:

• Nondecreasing inertia: $B:H\to H$4 and $B:H\to H$5, with constant inertia $B:H\to H$6 as an example.
• Decreasing inertia: $B:H\to H$7 and $B:H\to H$8, which allows larger $B:H\to H$9 in early iterates (Maulén et al., 25 Jul 2025).

The paper identifies the decreasing-inertia regime as a novel approach in the context of inertial algorithms (Maulén et al., 25 Jul 2025). This suggests that RIPSA is not only an abstract extension of NFB but also a parameterized acceleration strategy in which early-iteration aggressiveness and asymptotic stability are separated by design.

3. Convergence theory

The principal convergence statements in the NFB-based RIPSA analysis are weak convergence theorems under both nondecreasing and decreasing inertia. For nondecreasing inertia, Theorem 3.2 assumes that there exist $\beta>0$0, $\beta>0$1, and $\beta>0$2 such that for all $\beta>0$3,

$\beta>0$4

and that the derived quantities $\beta>0$5 satisfy $\beta>0$6, while $\beta>0$7 is nondecreasing. Under these conditions,

$\beta>0$8

for any $\beta>0$9 the sequence $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$0 converges, and

$J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$1

The proof constructs a Lyapunov sequence $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$2 combining $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$3 and inertial cross terms, establishes

$J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$4

and then invokes Fejér monotonicity and Opial’s lemma (Maulén et al., 25 Jul 2025).

Theorem 3.3 treats decreasing inertia. If $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$5, $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$6, and the same stepsize and relaxation conditions hold, then the conclusions of Theorem 3.2 remain valid (Maulén et al., 25 Jul 2025). In other words, the decreasing-inertia strategy preserves the same weak convergence guarantees while changing the transient dynamics.

Related RIPSA-type analyses in adjacent literatures use closely allied proof templates. The inexact 2019 scheme derives a quasi-Fejér monotonicity estimate for a gap measure involving inertia and relaxation, then uses the relative-error condition and a mutual constraint on $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$7 to obtain boundedness and weak convergence to a point in $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$8; moreover, $J_{\gamma A}=(\mathrm{Id}+\gamma A)^{-1}.$9 and $0\in A(x)+B(x),$0 (Alves et al., 2019). The relaxed inertial Forward–Backward–Forward analysis likewise proves boundedness, a discrete descent inequality, and weak convergence via Opial’s lemma under monotonicity and Lipschitz assumptions (Bot et al., 2020). In the hierarchical equilibrium setting, RIPSA yields weak ergodic and weak convergence without a contraction factor, strong convergence under strong monotonicity, and strong convergence via a Browder–Halpern contraction factor (Mazgouri et al., 28 Sep 2025).

4. Reductions and algorithmic relatives

One of the central reasons RIPSA is technically useful is that the NFB formulation subsumes several standard splitting methods as special cases. The reductions stated in the 2025 paper are summarized below.

Method	Specialization	Relation to RIPSA
Forward–Backward (FB)	$0\in A(x)+B(x),$1 maximally monotone, $0\in A(x)+B(x),$2 cocoercive, $0\in A(x)+B(x),$3, $0\in A(x)+B(x),$4	Direct recovery
Forward–Backward–Forward (FBF)	$0\in A(x)+B(x),$5 maximal monotone, $0\in A(x)+B(x),$6	Inertial/relaxed update yields Tseng’s FBF
Chambolle–Pock (CP)	Choose $0\in A(x)+B(x),$7, $0\in A(x)+B(x),$8, and $0\in A(x)+B(x),$9 as block operators	With $x_{-1},x_0\in H$0, general RIPSA recovers CP with $x_{-1},x_0\in H$1 relaxation
Condat–Vu	Preconditioned FB on a primal–dual pair	Particular choice of $x_{-1},x_0\in H$2 and $x_{-1},x_0\in H$3 reduces RIPSA to Condat–Vu with inertial/relaxed steps

These reductions are exact statements of specialization, not merely analogies (Maulén et al., 25 Jul 2025). They place RIPSA inside a broad unifying operator-splitting perspective in which inertial and relaxation terms can be added to established primal, dual, and primal–dual recursions.

The same unifying tendency appears in related literature, but with different base operators. The 2020 relaxed inertial FBF scheme treats the inclusion $x_{-1},x_0\in H$4 when $x_{-1},x_0\in H$5 is maximally monotone and $x_{-1},x_0\in H$6 is monotone and $x_{-1},x_0\in H$7-Lipschitz, and reduces to Tseng’s original FBF when $x_{-1},x_0\in H$8 and $x_{-1},x_0\in H$9 (Bot et al., 2020). The 2019 inexact RIPSA is instead built around an inexact resolvent satisfying a relative-error criterion and serves as the basis for inexact Douglas–Rachford and ADMM variants (Alves et al., 2019). The 2025 hierarchical-equilibrium RIPSA replaces monotone operators by bifunction resolvents $(\alpha_n)\subset[0,1[$0 and $(\alpha_n)\subset[0,1[$1, and also includes a contraction term $(\alpha_n)\subset[0,1[$2 (Mazgouri et al., 28 Sep 2025).

A common source of confusion is therefore terminological rather than mathematical: the acronym RIPSA is used for multiple inertial–relaxed proximal constructions, not for a single universally fixed recursion. This suggests that the stable conceptual core is the combination of inertia, resolvent/proximal structure, and relaxation, while the surrounding operator model varies by problem class.

5. Numerical behavior and parameter selection

The 2025 NFB paper reports two classes of numerical experiments. The first concerns optimization with affine constraints through a Forward-Backward-Half-Forward instance. The tested problem has the form

$(\alpha_n)\subset[0,1[$3

subject to linear inequalities. In these experiments, the Relaxed Inertial FBHF (FBHFRI) and Decreasing-Inertia FBHF (FBHFID) were compared. The reported findings are that standard FBHF often outperforms naive constant-inertia variants in CPU time and iterations, while decreasing inertia sequences such as $(\alpha_n)\subset[0,1[$4 achieve up to 30% fewer iterations and CPU time than nonincreasing or constant-$(\alpha_n)\subset[0,1[$5 schemes, especially on large problems (Maulén et al., 25 Jul 2025).

The second set of experiments concerns image restoration through a Forward-Primal–Dual-Half-Forward instance applied to TV/Huber regularized CT deblurring. Using adaptive initialization with tunable $(\alpha_n)\subset[0,1[$6 via Initialization 4.1, decreasing inertia reduced by 20–50% the number of iterations and wall-clock time compared to classical FPDHF and constant-$(\alpha_n)\subset[0,1[$7 variants, with negligible loss in PSNR (Maulén et al., 25 Jul 2025).

The practical parameter recommendations are also explicit. For primal–dual settings, Initialization 4.1 chooses $(\alpha_n)\subset[0,1[$8 and sets

$(\alpha_n)\subset[0,1[$9

$(\lambda_n)\subset[\underline\lambda,2[$0

Relaxation may be chosen so that $(\lambda_n)\subset[\underline\lambda,2[$1, or simply $(\lambda_n)\subset[\underline\lambda,2[$2 if no over-relaxation is desired. For constant inertia,

$(\lambda_n)\subset[\underline\lambda,2[$3

with

$(\lambda_n)\subset[\underline\lambda,2[$4

For decreasing inertia, one may choose $(\lambda_n)\subset[\underline\lambda,2[$5 decreasing to $(\lambda_n)\subset[\underline\lambda,2[$6 with $(\lambda_n)\subset[\underline\lambda,2[$7, for example $(\lambda_n)\subset[\underline\lambda,2[$8, setting $(\lambda_n)\subset[\underline\lambda,2[$9 near $A:H\rightrightarrows H$00 but tapering off to speed up early iterates. The guideline stated in the paper is to start with moderate $A:H\rightrightarrows H$01 near the upper bound $A:H\rightrightarrows H$02, set $A:H\rightrightarrows H$03 close to $A:H\rightrightarrows H$04, choose $A:H\rightrightarrows H$05, and, if aggressive acceleration is needed, use a decreasing $A:H\rightrightarrows H$06. The recommended diagnostic is to monitor $A:H\rightrightarrows H$07; if stagnation or oscillations occur, reduce $A:H\rightrightarrows H$08 or $A:H\rightrightarrows H$09 (Maulén et al., 25 Jul 2025).

6. Broader variants and scope of application

The acronym RIPSA has been extended beyond the cocoercive NFB setting. In "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," RIPSA blends three effects in each iteration: inertial extrapolation, an inexact proximal subproblem satisfying a relative-error criterion, and relaxation or overrelaxation. With

$A:H\rightrightarrows H$10

the inexact step finds $A:H\rightrightarrows H$11 and $A:H\rightrightarrows H$12 such that

$A:H\rightrightarrows H$13

followed by

$A:H\rightrightarrows H$14

Under maximal monotonicity, $A:H\rightrightarrows H$15, parameter bounds $A:H\rightrightarrows H$16, $A:H\rightrightarrows H$17, $A:H\rightrightarrows H$18, and a mutual bound on $A:H\rightrightarrows H$19, the iterates converge weakly; the same framework yields inexact Douglas–Rachford and inexact ADMM algorithms, with numerical experiments on LASSO and logistic regression problems (Alves et al., 2019).

In "A Relaxed Inertial Forward-Backward-Forward Algorithm for Solving Monotone Inclusions with Application to GANs," the relaxed inertial FBF method—also called RIPSA in some contexts—is

$A:H\rightrightarrows H$20

$A:H\rightrightarrows H$21

for $A:H\rightrightarrows H$22 maximally monotone and $A:H\rightrightarrows H$23 monotone and $A:H\rightrightarrows H$24-Lipschitz. The paper provides a weak-convergence theorem under monotonicity, a variational-inequality result for pseudo-monotone $A:H\rightrightarrows H$25, an adaptive stepsize rule that does not require prior knowledge of $A:H\rightrightarrows H$26, and numerical illustrations on a bilinear saddle-point problem and GAN training (Bot et al., 2020).

In "Weak and strong convergence of a relaxed inertial proximal splitting algorithm for solving hierarchical equilibrium problems," RIPSA is formulated for bifunctions $A:H\rightrightarrows H$27 over a nonempty closed convex set $A:H\rightrightarrows H$28, with iteration

$A:H\rightrightarrows H$29

$A:H\rightrightarrows H$30

where $A:H\rightrightarrows H$31 is a $A:H\rightrightarrows H$32-contraction. The paper establishes weak ergodic and weak convergence without a Browder–Halpern contraction factor, strong convergence under a strong monotonicity assumption, and strong convergence via the contraction factor without strong monotonicity; it also discusses convex minimization, monotone variational inequalities, and fixed-point problems as special cases (Mazgouri et al., 28 Sep 2025).

Taken together, these formulations show that RIPSA functions as a recurrent design template across monotone inclusions, saddle-point systems, inexact splitting, primal–dual imaging models, hierarchical equilibrium problems, and fixed-point formulations. The shared structure is the systematic coupling of inertial extrapolation with a proximal or resolvent step and a relaxed terminal update, while the precise operator model, admissible parameter regime, and convergence mode depend on the surrounding problem class.