A new continuous-time optimization model is designed to do two jobs at once: reduce the gap between a function value and its best possible value at a theoretical rate proportional to 1/t², and, when the nonempty minimizer set is an affine subspace, determine which solution to choose. The model uses a weighted memory of past gradients as an evolving anchor, together with nonlinear feedback based on squared, coordinate-by-coordinate displacement. It does this without explicit Hessian information.
The work is an arXiv preprint, version 1, dated 26 Aug 2026.
What the analysis establishes
Under the stated assumptions on the objective, parameters and existence of a minimizer, the proposed system has a unique global strong solution for every specified initial condition. In practical terms, the mathematical trajectory is defined for all future times and uniquely determined by how it starts.
The central rate result says that the objective residual, meaning the remaining gap between the current objective value and the optimum, is O(t^-2). The paper presents the restoring and damping feedback as adding dissipation while preserving that theoretical rate. This is a theorem-level bound for the model, not a rate estimated from experimental data.
The analysis also states that, when γ is greater than 0 under the smooth convex assumptions, the main trajectory converges strongly to a minimizer. That result establishes convergence to a solution, while the identity of the particular minimizer requires the additional selection conditions described in the paper.
A sharper selection rule appears when the full set of minimizers is a nonempty affine subspace. In that case, the trajectory converges to the Euclidean projection of the initial anchor onto the solution set, meaning the point in the set closest to that anchor. For rank-deficient least-squares problems, the corresponding statement is that the dynamics selects the least-squares minimizer closest to the initial anchor.
The proposed control is meant to curb lingering motion
The paper gives a coordinatewise dissipation result for the system's phase variables, the position-and-motion quantities used to track each coordinate's state. It says that large phase amplitudes cannot persist on a set of infinite logarithmic measure while the coordinate remains separated from the anchor. The result is weighted, so it does not say that every phase coordinate decreases smoothly at every moment.
The analysis further finds that an active coordinate that stays separated from its anchor receives an additional polynomial damping factor in its homogeneous phase response, on top of the linear damping factor. The extra attenuation applies only while that coordinate-separation condition holds.
Small numerical tests illustrate the mechanism
One numerical example examined anchor-dependent selection. It used three initial anchors that had the same signed normal distance from the solution set but different tangential components. The proposed dynamics ended at states within 7 × 10^-4 of the projections predicted by the theory. Because this was a deterministic finite-interval calculation, the result illustrates the selection rule but does not establish it for every problem or parameter choice.
A scalar quartic example compared different settings of η. For every tested setting, the objective residuals were smooth and monotonically decreasing over the displayed interval from 1 to 100. That observation is limited to the reported trajectories and interval.
The reported integrations used a high-order adaptive DOP853 Runge–Kutta method, a numerical solver that adjusts its step size, with relative tolerance 10^-10 and absolute tolerance 10^-12. The calculations were used to illustrate the model's behavior.
In a two-dimensional quadratic comparison, the proposed dynamics recorded 0 zero crossings in both listed coordinates. The Hessian-driven reference recorded 2 and 6 crossings, while Nesterov's ODE recorded 2 and 7. The tally is a limited indicator of transient overshoot, not a general performance ranking.
What remains unresolved
The main conclusions are conditional on the theorem's smooth convex assumptions, parameter restrictions and minimizer-existence requirements. Point convergence requires γ to be greater than 0. Projection selection is proved only for nonempty affine minimizer sets. The stated Lyapunov argument also does not provide an accelerated objective-value estimate at the critical endpoint a = 1/2, which the paper leaves open.
The numerical examples are illustrative rather than a broad benchmark. They do not show superiority over Nesterov's ODE or Hessian-driven damping across general optimization problems, and the analysis notes that the Hessian-driven reference reached a smaller terminal residual in the quadratic comparison. The calculations also do not show that the reported transient regularity persists for arbitrary objectives, parameters or intervals.
The authors identify several next questions: whether anchor-based selection extends beyond affine solution sets, whether additional geometric assumptions can produce sharper rates, and how to build discrete-time algorithms that preserve the continuous-time selection and restoring-damping properties. No funding statement is reported, and the authors declare no conflict of interest.
Paper data and sources
Original title: Accelerated Gradient Flow with Endogenous Gradient-Memory Anchor: Selection and Restoring Damping
Authors: Chinedu Izuchukwu, Ernest Obini, Jen-Chih Yao, Shengda Zeng
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text