Optimization via a Control-Centric Framework

Abstract

Optimization plays a central role in intelligent systems and cyber-physical technologies, where speed and reliability of convergence directly impact performance. In control theory, optimization is the foundation of widely-used design methodologies such as linear quadratic regulation, $H_{\infty }$ control, and model predictive control. In contrast, this paper develops a control-centric framework for optimization itself, where algorithms are constructed directly from Lyapunov stability principles rather than being proposed first and analyzed afterward. A key element is the stationarity mapping, which encodes first-order optimality conditions and enables Lyapunov-based convergence analysis. By pairing a Lyapunov function with a selectable decay law, we obtain continuous-time dynamics with guaranteed exponential, finite-time, fixed-time, or prescribed-time convergence. The convergence regime thus becomes a design input: choosing among these regimes changes only the decay law, not the algorithm or its analysis. Within this framework, we introduce three feedback realizations of increasing restrictiveness: the Hessian-gradient, Newton, and gradient dynamics. These constructions unify unconstrained optimization, extend to constrained problems via Lyapunov-consistent primal-dual dynamics, and broaden results for minimax and generalized Nash equilibrium seeking problems beyond exponential stability, the last two through centralized solvers. In total, the framework spans six problem classes and four convergence regimes, yielding a unified design recipe across twenty-four combinations, some of which, to the best of our knowledge, have no direct continuous-time counterpart in the prior literature.

arXiv commentThis work has been submitted to the IEEE for possible publication. 25 pages, 6 figures