Energy stability of the Semi and Forward–Backward (FB) algorithms

Establish energy stability for the two explicit proximal schemes derived from the underdamped inertial dynamics ddot{x}(t)+ (α/t) dot{x}(t) − (α/t) ⟨∇F(x(t)), dot{x}(t)⟩ 1 + γ(t) ∇^2F(x(t)) dot{x}(t) + β(t) ∇F(x(t)) = 0, namely (i) the semi‑discretized algorithm defined by y^k = x^k + (k/(k+α))(x^k − x^{k−1}) + (α/(k+α))⟨∇F(x^k), x^k − x^{k−1}⟩ 1 + (k h/(k+α)) γ_k ∇F(x^k), μ_k = (k h/(k+α))(γ_k + β_k h), and x^{k+1} = prox_{μ_k F}(y^k), and (ii) the forward–backward algorithm defined by y^k = x^k + (k/(k+α))(x^k − x^{k−1}) + (α/(k+α))(F(x^k) − F(x^{k−1})) 1 + (k h/(k+α)) γ_k ∇F(x^k), μ_k = (k h/(k+α))(γ_k + β_k h), and x^{k+1} = prox_{μ_k F}(y^k). Specifically, prove that a suitable discrete Lyapunov/energy functional is nonincreasing along the iterations of each algorithm under appropriate regularity and convexity assumptions on F and appropriate choices of α, γ_k, β_k, and h.

Background

The paper introduces a new underdamped inertial dynamics incorporating gradient and Hessian information and proves an energy dissipation law and function-value decay for the continuous-time system. Structure-preserving discretizations (FD and IMEX‑RB) are then developed, for which discrete energy stability and convergence rate bounds are established.

Two additional explicit schemes are proposed to avoid implicit solves: the semi‑discretized algorithm and the forward–backward algorithm. While these methods are derived from the same inertial ODE and use proximal updates, the authors explicitly state that they have not established energy stability for them. Proving energy stability would align these schemes with the theoretical guarantees shown for FD and IMEX‑RB and would strengthen the framework’s rigor for explicit implementations.