Malitsky-Tam FRB Splitting Method

Updated 7 May 2026

The paper introduces an FRB splitting method that decomposes operator sums using a reflected forward step, enabling convergence under mild assumptions.
FRB requires only resolvent evaluations and a single evaluation of the forward operator per iteration, often permitting larger step sizes than traditional methods.
Extensions to three-operator, inertial, Bregman, and stochastic frameworks broaden FRB’s applications in convex, nonconvex, and large-scale optimization problems.

The Malitsky–Tam Forward-Reflected-Backward (FRB) scheme is a splitting method for solving monotone inclusion problems, variational inequalities, and structured convex or nonconvex minimization, especially in the presence of monotone-Lipschitzian but not necessarily cocoercive operators. FRB enables the decomposition of operator sums into steps requiring only resolvent evaluations and a single call to the forward operator per iteration. It has been extended to stochastic, primal-dual, Bregman, inertial, nonconvex, and strongly convergent frameworks. The method is characterized by reflected forward steps which use information from prior iterates, facilitating convergence under mild assumptions and often admitting larger stepsizes than classic operator splitting schemes.

1. Problem Setting, Algorithmic Formulation, and Uniqueness

The canonical problem addressed by FRB is the monotone inclusion

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

where $A: H \to 2^H$ is maximally monotone, $B: H \to H$ is monotone and $L$ -Lipschitz on a real Hilbert space $H$ . The solution set is $\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ , assumed non-empty.

The forward-reflected-backward iteration is given by

$x_{k+1} = J_{\lambda A}\Big(x_k - 2\lambda B(x_k) + \lambda B(x_{k-1})\Big),$

where $J_{\lambda A} = (I + \lambda A)^{-1}$ denotes the resolvent and $\lambda$ is the step-size parameter. The key feature is the "reflected" forward argument involving both $B(x_k)$ and $A: H \to 2^H$ 0. An equivalent form is

$A: H \to 2^H$ 1

For structured problems, e.g., when $A: H \to 2^H$ 2 and $A: H \to 2^H$ 3, this reduces to the proximal step

$A: H \to 2^H$ 4

No cocoercivity of $A: H \to 2^H$ 5 is required, distinguishing FRB from classical forward-backward splitting (Roldán, 2023, Soe et al., 2 Sep 2025, Dung et al., 2021).

2. Convergence Theory and Step-Size Constraints

FRB achieves weak convergence under minimal assumptions without requiring $A: H \to 2^H$ 6 to be cocoercive. The principal stepsize condition is

$A: H \to 2^H$ 7

with variations depending on the problem structure: for instance, the deterministic monotone-inclusion case allows $A: H \to 2^H$ 8 (Dung et al., 2021), while more general three-operator extensions and schemes with additional correction terms have further restrictions (Rieger et al., 2020).

The convergence proof proceeds by construction of a Lyapunov or Fejér functional, typically of the form

$A: H \to 2^H$ 9

which, under monotonicity and Lipschitz continuity, yields a decrement in $B: H \to H$ 0 up to an explicit error term. Standard Opial's lemma arguments promote weak convergence of the sequence $B: H \to H$ 1 to a solution $B: H \to H$ 2 (Soe et al., 2 Sep 2025, Roldán, 2023, Dung et al., 2021).

For certain linear or affine problems—most notably, the sum of normal cone operators to subspaces—the iterates actually converge strongly to the projection of the initial point onto the intersection subspace (Bauschke et al., 2021, Bauschke et al., 2022).

3. Extensions: Three-Operator, Bregman, Inertial, and Accelerated Schemes

Three-Operator Generalization

For problems $B: H \to H$ 3, with $B: H \to H$ 4 and $B: H \to H$ 5 maximally monotone, $B: H \to H$ 6 monotone-Lipschitzian, the so-called "Backward–Forward–Reflected–Backward" (BFRB) extends the core FRB mechanics. Each iterate involves resolvents for $B: H \to H$ 7 and $B: H \to H$ 8, as well as reflection and update steps: $B: H \to H$ 9 This framework collapses to classic Douglas-Rachford when $L$ 0, and to two-operator FRB when $L$ 1 (Rieger et al., 2020).

Inertial, Nesterov, and Bregman Generalizations

Bregman and inertial extensions allow for non-Euclidean geometry, variable step-size, and explicit momentum:

BiFRB (Bregman-Inertial FRB) solves

$L$ 2

where $L$ 3 is prox-bounded and $L$ 4 has $L$ 5-Lipschitz gradient, by

$L$ 6

and an inertial Bregman-proximal step

$L$ 7

Here, $L$ 8 is the Bregman divergence (Wang et al., 2022, Wang et al., 2022).

Accelerated FRB variants, such as "Fast RFB," incorporate Nesterov-type extrapolation and correction terms, achieving improved rates for certain convex optimization and saddle-point problems, with proven $L$ 9 last-iterate rates for the tangent residual and discrete velocity (Bot et al., 2024).

Stochastic and Variance-Reduced Variants

FRB has been extended to address stochastic monotone inclusions, using unbiased or biased variance-reduced estimators in the forward-reflected direction. Under suitable sampling and variance bounds, the expected residuals achieve $H$ 0 best-iterate rates and almost-sure convergence (Tran-Dinh et al., 16 Mar 2026, Dung et al., 2021).

Compared to classic forward-backward splitting (FB), Tseng’s forward-backward-forward (FBF), and projected reflected gradient methods:

FB requires $H$ 1 to be cocoercive and uses a single call to $H$ 2 per iteration.
FBF can handle merely monotone-Lipschitz $H$ 3 but requires two $H$ 4 evaluations per step.
FRB, with a single evaluation of $H$ 5 (by reusing $H$ 6 and $H$ 7), matches the per-iteration cost of FB but attains feasibility for wider classes absent cocoercivity (Roldán, 2023, Cevher et al., 2019).
FRB and its three-operator analogs handle complex operator sums, including non-cocoercive and non-monotone components in specific settings, often with larger permissible stepsizes than Tseng-style methods (Rieger et al., 2020, Ryu et al., 2019).

5. Convergence Rates and Practical Recommendations

Under monotonicity and Lipschitzian conditions, FRB achieves weak convergence; in special cases (e.g., uniform monotonicity, subspace intersection) it attains strong convergence and linear rates (Bauschke et al., 2021, Bauschke et al., 2022). In nonconvex settings or with Kurdyka–Łojasiewicz (KL) structure, convergence rates depend on the KL exponent, enabling linear or sublinear convergence of function values and iterates (Wang et al., 2021, Wang et al., 2022, Wang et al., 2022):

KL $H$ 8: finite-step convergence.
$H$ 9: linear rate $\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ 0.
$\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ 1: sublinear rate $\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ 2.

Empirical studies reveal that FRB usually requires fewer operator calls than FBF, and its robust step-size selection ( $\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ 3 as close as possible to $\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ 4) leads to favorable practical performance (Roldán, 2023, Bot et al., 2024). Adaptive stepsize and linesearch strategies further enhance stability and speed, especially when the Lipschitz constant is unknown (Soe et al., 2 Sep 2025, Izuchukwu et al., 2022).

6. Applications and Empirical Behavior

FRB and its variants have been applied to:

Composite convex or nonconvex minimization, including $\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ 5-constrained and $\operatorname{zer}(A+B)=\{x : 0\in A(x)+B(x)\}$ 6-regularized problems, and minimax saddle-point models (Wang et al., 2021, Wang et al., 2022, Wang et al., 2022).
Convex optimization with linear cone constraints and large-scale data-driven monotone inclusions (Bot et al., 2024, Tran-Dinh et al., 16 Mar 2026).
Imaging and optimal control (e.g., total variation restoration, Pontryagin-type inclusions), where anchored and inertial FRB outperform classical methods in both iteration count and solution accuracy (Izuchukwu et al., 2022).
Intersections of affine or subspace constraints, where FRB coincides with relaxed projection-reflection and achieves strong, linear convergence to the projection point (Bauschke et al., 2021, Bauschke et al., 2022).

Empirical benchmarks consistently show that FRB and its adaptive or accelerated variants either outperform, or are at least competitive with, other state-of-the-art splitting schemes in iteration complexity and per-iteration cost.

7. Extensions, Open Directions, and Theoretical Significance

Recent work on generalized forward-reflected-backward (GFRB) allows for non-constant coefficients on the reflection terms, increasing algorithmic flexibility and enabling stepsize rules which do not require knowledge of the Lipschitz constant (Soe et al., 2 Sep 2025). Linesearch and variable-metric extensions, Bregman geometry, and inertial/acceleration schemes further broaden the applicability.

Open questions relate to the optimality of convergence rates, global linear convergence outside special cases, and unification with more general monotone inclusion frameworks. The intrinsic capability of FRB-type methods to handle non-cocoercive, nonconvex, and composite structures positions them as a central tool in both monotone operator theory and modern large-scale optimization.