Optimization via a Control-Centric Framework
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866917354464608256 |
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| author | Mudrik, Liraz Kaminer, Isaac Kragelund, Sean Clark, Abram H. |
| author_facet | Mudrik, Liraz Kaminer, Isaac Kragelund, Sean Clark, Abram H. |
| contents | 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-centric methods are standard: controllers are designed by repeatedly solving optimization problems, as in 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 vector, 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. Within this framework, we introduce three feedback realizations of increasing restrictiveness: the Hessian-gradient, Newton, and gradient dynamics. Each realization shapes the decay of the stationarity vector to achieve the desired rate. These constructions unify unconstrained optimization, extend naturally to constrained problems via Lyapunov-consistent primal-dual dynamics, and broaden the results for minimax and generalized Nash equilibrium seeking problems beyond exponential stability. The framework provides systematic design tools for optimization algorithms in control and game-theoretic problems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05455 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimization via a Control-Centric Framework Mudrik, Liraz Kaminer, Isaac Kragelund, Sean Clark, Abram H. Optimization and Control Systems and Control 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-centric methods are standard: controllers are designed by repeatedly solving optimization problems, as in 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 vector, 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. Within this framework, we introduce three feedback realizations of increasing restrictiveness: the Hessian-gradient, Newton, and gradient dynamics. Each realization shapes the decay of the stationarity vector to achieve the desired rate. These constructions unify unconstrained optimization, extend naturally to constrained problems via Lyapunov-consistent primal-dual dynamics, and broaden the results for minimax and generalized Nash equilibrium seeking problems beyond exponential stability. The framework provides systematic design tools for optimization algorithms in control and game-theoretic problems. |
| title | Optimization via a Control-Centric Framework |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2510.05455 |