Forward-Backward-Forward Dynamics

• Forward-backward-forward dynamics is a continuous-time framework that generalizes traditional splitting schemes for solving monotone mixed variational inequalities.
• It combines forward steps, proximal backward updates, and corrective forward terms to ensure weak or strong convergence under relaxed operator conditions.
• Extensions of FBF include convex regularization, bilevel equilibrium problems, and inertial acceleration techniques, leading to efficient numerical performance.

The forward-backward-forward (FBF) dynamical system is a continuous-time framework for solving monotone and pseudomonotone mixed variational inequalities (MVIs) and related problems in real Hilbert spaces. The system generalizes and refines classical forward-backward and Tseng’s discrete splitting schemes, by providing weak or strong convergence under relaxed operator-theoretic assumptions and efficient per-iteration cost. Key developments include extensions to convex regularization, bilevel equilibrium problems, quasimonotone and non-Lipschitz settings, and inertia- or relaxation-based acceleration schemes. This article systematically synthesizes the rigorous theory underpinning FBF systems, their principal formulations, operator conditions, convergence results, discretizations, and representative applications.

1. Mathematical Formulation of the Forward-Backward-Forward Dynamical System

Consider a real Hilbert space $\mathcal{H}$, a nonempty closed convex subset $\mathcal{C} \subset \mathcal{H}$, a $\beta$-Lipschitz continuous operator $T: \mathcal{H} \to \mathcal{H}$, and a proper, lower semicontinuous (lsc), convex function $h: \mathcal{C} \rightarrow (-\infty, +\infty]$. The mixed variational inequality is to find $\bar{x} \in \mathcal{C}$ such that

$$\langle T(\bar{x}), u - \bar{x} \rangle + h(u) - h(\bar{x}) \geq 0 \quad \forall u \in \mathcal{C}.$$

The FBF dynamical system is then defined as: $$\begin{aligned} & y(t) = \operatorname{prox}_{\lambda h}(x(t) - \lambda T(x(t))), \ & \dot{x}(t) + x(t) = y(t) + \lambda [T(x(t)) - T(y(t))], \ & x(0) = x_0, \end{aligned}$$ where the proximal mapping is

$$\operatorname{prox}_{\lambda h}(u) = \arg\min_{v \in \mathcal{C}} \left\{ \lambda h(v) + \frac{1}{2} \| v - u \|^2 \right\}.$$

This FBF system structurally comprises a forward step, a backward (proximal) step, and a corrective forward term, and coincides with Tseng’s discrete FBF splitting when $h \equiv 0$ or $h \equiv \mathrm{const}$ (Nwakpa et al., 23 Nov 2025).

2. Operator and Regularization Assumptions

The convergence and stability of the FBF system depend critically on the following conditions for $T$ and $h$:

• Lipschitz Continuity: $T$ is $\beta$-Lipschitz, which guarantees the composite $$\operatorname{prox}_{\lambda h} \circ (\mathrm{Id} - \lambda T)$$ is Lipschitz, ensuring existence and uniqueness of continuous trajectories.
• General Monotonicity: For all $v \in \mathcal{C}$ and any solution $\bar{x}$,

$$\langle T(v), v - \bar{x} \rangle + h(v) - h(\bar{x}) \geq 0.$$

• $h$-Pseudomonotonicity: Whenever

$\langle T(u), v - u \rangle + h(v) - h(u) \geq 0,$

then

$\langle T(v), v - u \rangle + h(v) - h(u) \geq 0.$

• $h$-Strong Pseudomonotonicity: If the above holds, there exists $\mu > 0$ such that

$$\langle T(v), v - u \rangle + h(v) - h(u) \geq \mu \|u - v\|^2,$$

which provides error bounds and drives exponential convergence.

• Convexity and Lower Semi-Continuity of $h$: Guarantees uniqueness of the proximal mapping and supports Lyapunov and weak convergence analysis.

These conditions allow the FBF framework to accommodate monotone and strongly pseudomonotone operators, general convex regularizers, and equilibrium problems beyond standard variational inequalities (Nwakpa et al., 23 Nov 2025, Bot et al., 2018).

3. Convergence Properties and Lyapunov Analysis

Convergence of FBF trajectories is established through Lyapunov functionals and energy inequalities. Let $V(t) = \frac{1}{2}\|x(t) - \bar{x}\|^2$ for any solution $\bar{x}$ of the MVI. Under monotonicity and Lipschitz conditions, differentiation yields

$$\frac{d}{dt}\|x(t) - \bar{x}\|^2 \leq -2[1 - \lambda(1 + \beta^2)] \|x(t) - y(t)\|^2,$$

for $\lambda < 1/(1 + \beta^2)$. This ensures that $V(t)$ is nonincreasing, $x(t)$ is bounded, and $\int_0^\infty \|x(t) - y(t)\|^2 dt < \infty$.

Weak convergence is proved via quasi-Fejér monotonicity, the Opial lemma, and the diminishing gap $\|x(t) - y(t)\| \to 0$, so that $x(t) \rightharpoonup x_\infty \in S$ (Nwakpa et al., 23 Nov 2025, Banert et al., 2015). Under $h$-strong pseudomonotonicity,

$\frac{d}{dt} V(t) \leq -\alpha V(t),$

with explicit $\alpha>0$, yielding

$$\|x(t) - \bar{x}\|^2 \leq \|x(0) - \bar{x}\|^2 e^{-\alpha t},$$

demonstrating global exponential stability of the equilibrium.

A plausible implication is that FBF systems, with mildly regular operators and convex regularization, provide robust convergence guarantees under general monotonicity—not requiring strong monotonicity or cocoercivity (Nwakpa et al., 23 Nov 2025, Banert et al., 2015, Bot et al., 2018).

4. Extensions and Discretizations

Discretization of FBF systems, typically via explicit Euler steps,

$$\frac{x_{n+1} - x_n}{h} + x_n = y_n + \lambda [T(x_n) - T(y_n)],$$

with $$y_n = \operatorname{prox}_{\lambda h} (x_n - \lambda T(x_n))$$, recovers Tseng’s FBF splitting for variational inequalities with affine or zero $h$: $$y_n = P_{\mathcal{C}} (x_n - \lambda F(x_n)), \quad x_{n+1} = y_n + \lambda [F(x_n) - F(y_n)],$$ where $F$ may be monotone or pseudo-monotone. Parameter choices (step-size $\lambda$, relaxation $\gamma_n$) enable underrelaxation to ensure stability, and overrelaxation for acceleration subject to explicit bounds (Bot et al., 2018).

Advanced variants incorporate golden-ratio extrapolation, adaptive step-size rules, and Bregman-type projection frameworks to ensure strong convergence under weaker continuity assumptions, including uniform continuity or non-Lipschitz cases (Zhang et al., 26 Aug 2025).

Inertial and relaxed inertial FBF algorithms (RIFBF) introduce second-order dynamics and momentum via damping/relaxation terms, further accelerating convergence and broadening applicability to monotone inclusions and saddle-point problems (Bot et al., 2020).

5. Applications to Mixed Variational Inequalities and Related Problems

FBF systems admit wide deployment:

• Mixed VI with Convex Regularization: Logistic regression with $\ell_1$ penalty, solved by setting $T$ to the gradient of the loss and $h$ as the $\ell_1$ regularization (Nwakpa et al., 23 Nov 2025).
• Affine and Quadratic Constraints: Low-dimensional examples (e.g., $\mathcal{C} = [3,5]$, $T(u) = 4-u$, $h(u)=u^2$) illustrate convergence despite non-classical pseudomonotonicity.
• High-Dimensional Equilibria: Bregman-FBF discretizations solve infinite-dimensional and non-Lipschitz VIs (e.g., $F(x) = (b - \|x\|)x$ in $\ell^2$) with improved iteration complexity and stability (Zhang et al., 26 Aug 2025).
• Bilevel and Saddle-Point Problems: FBF ODEs and discrete procedures extend to noncoercive, monotone-Lipschitz bilinear systems for games, optimization with saddle constraints, and equilibrium programming (Mittal et al., 2024).
• Pseudo-monotone and Fractional Programming VIs: FBF methods surpass Korpelevich extragradient and subgradient-extragradient algorithms in both projection costs and empirical convergence rates (Bot et al., 2018).

6. Computational and Numerical Characteristics

Representative numerical tests for FBF and its discretizations document:

Problem Type	Iterations to Converge	Observed Computational Advantage
$\ell_1$-regularized logistic regression ($n=100$)	30–40	Fast decay of loss and gap (Nwakpa et al., 23 Nov 2025)
Infinite-dimensional VI in $\ell^2$	165–180 (Bregman-FBF), 182–479 (baseline)	FBF achieves faster/steadier convergence (Zhang et al., 26 Aug 2025)
Polyhedral VI (Tseng/Extragradient)	FBF: 0.55s, Extragradient: 1.1s	FBF requires half the time, one projection only (Bot et al., 2018)

This suggests FBF trajectories and algorithms maintain low per-iteration complexity and robust stability, rendering them attractive for large-scale and non-smooth settings.

7. Connections to Operator Splitting, Generalizations, and Future Directions

FBF dynamical systems subsume and extend several important splitting methods, including:

• Classical forward-backward, backward-forward, and extragradient schemes.
• Nonlinear NOFOB and four-operator splittings with projection corrections, which embed FBF as a method with single-projection cost per iteration and more general projection relaxations (Giselsson, 2019).
• Relaxed and inertial variants, handling non-strongly monotone, non-Lipschitz, and composite operator scenarios.
• Bregman and implicit/explicit FBF flows enabling golden-ratio-based step-size adaptation and improved convergence for non-standard geometric structures (Zhang et al., 26 Aug 2025).

A plausible implication is that future research may leverage FBF dynamics as a platform for developing efficient and stable algorithms for a range of noncoercive monotone inclusions, large-scale equilibrium problems, nonconvex composite VIs, and learning-interpretable optimization, while relaxing even further the required operator regularity through advanced discretization and extrapolation techniques.

• "Forward-Backward-Forward Dynamical System for Solving Mixed Variational Inequality Problems" (Nwakpa et al., 23 Nov 2025)
• "Forward-backward-forward dynamics for bilevel equilibrium problem" (Mittal et al., 2024)
• "A Relaxed Inertial Forward-Backward-Forward Algorithm for Solving Monotone Inclusions with Application to GANs" (Bot et al., 2020)
• "Asymptotic Properties of a Forward-Backward-Forward Differential Equation and Its Discrete Version for Solving Quasimonotone Variational Inequalities" (Zhang et al., 26 Aug 2025)
• "A forward-backward-forward differential equation and its asymptotic properties" (Banert et al., 2015)
• "Nonlinear Forward-Backward Splitting with Projection Correction" (Giselsson, 2019)
• "The Forward-Backward-Forward Method from continuous and discrete perspective for pseudo-monotone variational inequalities in Hilbert spaces" (Bot et al., 2018)