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RESEARCH ARTICLE

Beyond Nesterov: Dynamical systems perspective with time-dependent inertia and conformable Bregman flows

Osama F. Abdel Aal1† ,  Necdet Sinan Ozbek1,2† ,  Jairo Viola1† ,  YangQuan Chen1†*
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1 Department of Mechanical and Aerospace Engineering, School of Engineering, University of California Merced, California, United States of America
2 Department of Electrical and Electronic Engineering, Faculty of Engineering, Adana Alparslan Turkes Science and Technology University, Adana, Turkiye
†These authors contributed equally to this work.
IJOCTA, 025390163 https://doi.org/10.36922/IJOCTA025390163
Received: 25 September 2025 | Revised: 6 November 2025 | Accepted: 10 November 2025 | Published online: 16 December 2025
© 2025 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

We present a Lyapunov-based framework for analyzing continuous-time accelerated optimization dynamics with time-dependent inertia and damping. By explicitly designing Lyapunov functions that account for varying inertia, we rigorously characterize convergence rates of the objective function, achieving exponential or polynomial acceleration beyond the classical O(1/t2), even in the absence of strong convexity. Building on this foundation, we introduce a variational extension using conformable (fractional) derivatives in the Lagrangian formulation, replacing the classical velocity term with a time-weighted fractional velocity. This approach systematically modulates the system’s effective inertia and damping, providing a principled mechanism to balance acceleration and stability, reduce oscillations, and interpolate smoothly between strongly damped gradient flows and momentum-driven dynamics. The resulting framework unifies Lyapunov analysis and fractional variational modeling, offering flexible, theoretically grounded design principles for fast and stable accelerated optimization.

Keywords
Accelerated Gradient Methods
Continuous optimization
Bregman lagrangian
Calculus of variation
Gradient algorithms
Conformable derivatives
Funding
None.
Conflict of interest
YangQuan Chen is an Editorial Board Member of this journal, but was not in any way involved in the editorial and peer-review process conducted for this paper, directly or indirectly. Separately, other authors declared that they have no known competing financial interests or personal relationships that could have influenced the work reported in this paper.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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