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Optimization via a Control-Centric Framework

Liraz Mudrik, Isaac Kaminer, Sean Kragelund, Abram H. Clark

TL;DR

The paper introduces a control-centric framework for optimization based on optimization Lyapunov functions (OLFs) and a stationarity vector $\mathbf{S}(\mathbf{z})$, enabling optimizer design with explicit convergence timing (exponential, FT, FxT, PT). It develops three continuous-time realizations—Hessian-Gradient Dynamics, Newton Dynamics, and Gradient Dynamics—that enforce a chosen decay law while leveraging varying levels of curvature information. The framework extends naturally to constrained optimization via Lyapunov-consistent primal-dual dynamics, to constrained minimax problems, and to generalized Nash equilibrium seeking, providing convergence guarantees beyond classical asymptotic results. Numerical examples on unconstrained problems, network utility maximization, and Cournot GNE demonstrate the uniform applicability and practical performance of the approach, with discussion on discrete-time discretizations that preserve Lyapunov properties.

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-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.

Optimization via a Control-Centric Framework

TL;DR

The paper introduces a control-centric framework for optimization based on optimization Lyapunov functions (OLFs) and a stationarity vector , enabling optimizer design with explicit convergence timing (exponential, FT, FxT, PT). It develops three continuous-time realizations—Hessian-Gradient Dynamics, Newton Dynamics, and Gradient Dynamics—that enforce a chosen decay law while leveraging varying levels of curvature information. The framework extends naturally to constrained optimization via Lyapunov-consistent primal-dual dynamics, to constrained minimax problems, and to generalized Nash equilibrium seeking, providing convergence guarantees beyond classical asymptotic results. Numerical examples on unconstrained problems, network utility maximization, and Cournot GNE demonstrate the uniform applicability and practical performance of the approach, with discussion on discrete-time discretizations that preserve Lyapunov properties.

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-centric methods are standard: controllers are designed by repeatedly solving optimization problems, as in linear quadratic regulation, 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.

Paper Structure

This paper contains 26 sections, 12 theorems, 75 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical Background
  3. Stationarity Vector
  4. Desired Convergence Properties
  5. Control-Centric Framework
  6. Optimization Lyapunov Function (OLF)
  7. Nonsingularity Condition
  8. Feedback Designs Enforcing the Convergence Law
  9. Numerical Example: Unconstrained Problem
  10. Constrained Optimization
  11. Problem Setup and Preliminaries
  12. Fischer-Burmeister Function and Its Usage for Stationarity
  13. Results
  14. Illustrative Example: Network Utility Maximization
  15. Constrained Minimax Problems
  16. ...and 11 more sections

Key Result

Lemma 2.4

If there exists $\alpha > 0$ such that then $V(\mathbf{z})$ converges to zero exponentially, that is, $V(\mathbf{z}(t)) \leq V(\mathbf{z}(0)) e^{-\alpha t}$.

Figures (3)

  • Figure 1: Gradient norm trajectories $\|\nabla J(\mathbf{x})\|$ for $n=50$ under the exponential (Exp), finite-time (FT), fixed-time (FxT), and prescribed-time (PT) laws, denoted by blue, red, magenta, and black lines, respectively. Three realizations are compared: GD, ND, and HGD, denoted by solid, dashed, and dashed-dotted lines, respectively. The test function is the log-sum-exp plus quadratic model \ref{['eq:logsumexp_cost_nd']}, initialized at $\mathbf{x}_0 = [1,\ldots, 1]^\top$.
  • Figure 2: Decay of the optimality Lyapunov function $V(z(t)) = \tfrac{1}{2}\|S(z(t))\|^2$ for the Network Utility Maximization (NUM) problem under the four convergence laws. Exp converges asymptotically, FT and FxT converge in bounded time, and PT enforces convergence exactly at the prescribed horizon.
  • Figure 3: Decay of the optimality Lyapunov function $V(z(t)) = \tfrac{1}{2}\|S(z(t))\|^2$ for the Cournot game under the four convergence laws. Exp converges asymptotically, FT and FxT converge in bounded time, and PT enforces convergence exactly at the prescribed horizon.

Theorems & Definitions (30)

  • Example 2.3: Stationarity in Unconstrained Optimization
  • Lemma 2.4: Exponential Stability khalil_nonlinear_2002
  • Lemma 2.5: FT Stability bhat_finite-time_2000
  • Lemma 2.6: FxT Stability polyakov_nonlinear_2012
  • Lemma 2.7: PT Stability song_time-varying_2017
  • Definition 3.1: Optimization Lyapunov Function
  • Lemma 3.3: Hessian-Gradient Dynamics (HGD)
  • proof
  • Example 3.4: HGD for Unconstrained Optimization
  • Lemma 3.5: Newton Dynamics (ND)
  • ...and 20 more