Fast Optimization via Inertial Dynamics with Closed-Loop Damping

The paper’s Theorem 10.3 states exactly the claims at issue for (ADIGE–VGH): the energy-like quantity t ↦ 1/2||ẋ+β∇f||^2 + f(x) is nonincreasing, and ∫φ(ẋ+β∇f) + β∫||∇f||^2 < ∞; under sharpness φ(u) ≥ r||u||, one obtains weak convergence to a minimizer and strong decay of both ∇f(x(t)) and ẋ(t) to 0 (see (10.6)–(10.7) and Theorem 10.3) . The candidate reproduces the energy estimate and integrability conclusions, and even the Opial-based weak convergence route. However, their key step to derive y(t)→0 (with y=ẋ+β∇f) is flawed: they “pick” the minimal-norm selection p(t)=(∂φ)0(y(t)) to claim p(t) is bounded and hence ẏ is bounded, but the differential inclusion fixes a particular p(t) (namely, p(t)=-ẍ-β∇²f ẋ-∇f), and one cannot replace it by a different subgradient element in general. Property (iii) of damping potentials (boundedness of the minimal section on bounded sets) does not imply the actual p(t) entering the dynamics is bounded in infinite dimensions . Without a valid bound on ẏ (or ẍ), the step “y ∈ L¹ and uniformly continuous ⇒ y→0” is unjustified, so the conclusion ẋ→0 is not rigorously proved by the candidate. The paper avoids this pitfall by treating ẋ+β∇f as an L¹ perturbation of the gradient flow and by ensuring boundedness of the relevant time-derivatives via the structural estimates and approximations (Theorem 10.2 and the proof strategy surrounding (10.4)–(10.8)) .