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Controlled Optimization with a Prescribed Finite-Time Convergence Using a Time Varying Feedback Gradient Flow

Osama F. Abdel Aal, Necdet Sinan Ozbek, Jairo Viola, YangQuan Chen

TL;DR

This paper addresses the need for gradient-based optimization methods that guarantee convergence within a user-defined time independent of initial conditions. It introduces a Prescribed Finite-Time Convergent Gradient Flow (PFT-GF) with time-varying feedback, proving prescribed-time convergence via Lyapunov analysis for functions in the Polyak-Łojasiewicz class and for $\mu$-strongly convex functions. The approach differs from traditional finite-time schemes by using a time-scaling function $\mathcal{T}(t)$ and a dynamic gain $k(t)$ to shape the trajectory, providing explicit convergence timing at $T_p$. Experimental results on Trid and Rosenbrock functions validate exact convergence at the prescribed time and demonstrate robustness across initial conditions, highlighting potential applications in time-critical control, robotics, and distributed optimization.

Abstract

From the perspective of control theory, the gradient descent optimization methods can be regarded as a dynamic system where various control techniques can be designed to enhance the performance of the optimization method. In this paper, we propose a prescribed finite-time convergent gradient flow that uses time-varying gain nonlinear feedback that can drive the states smoothly towards the minimum. This idea is different from the traditional finite-time convergence algorithms that relies on fractional-power or signed gradient as a nonlinear feedback, that is proved to have finite/fixed time convergence satisfying strongly convex or the Polyak-Łojasiewicz (PŁ) inequality, where due to its nature, the proposed approach was shown to achieve this property for both strongly convex function, and for those satisfies Polyak-Łojasiewic inequality. Our method is proved to converge in a prescribed finite time via Lyapunov theory. Numerical experiments were presented to illustrate our results.

Controlled Optimization with a Prescribed Finite-Time Convergence Using a Time Varying Feedback Gradient Flow

TL;DR

This paper addresses the need for gradient-based optimization methods that guarantee convergence within a user-defined time independent of initial conditions. It introduces a Prescribed Finite-Time Convergent Gradient Flow (PFT-GF) with time-varying feedback, proving prescribed-time convergence via Lyapunov analysis for functions in the Polyak-Łojasiewicz class and for -strongly convex functions. The approach differs from traditional finite-time schemes by using a time-scaling function and a dynamic gain to shape the trajectory, providing explicit convergence timing at . Experimental results on Trid and Rosenbrock functions validate exact convergence at the prescribed time and demonstrate robustness across initial conditions, highlighting potential applications in time-critical control, robotics, and distributed optimization.

Abstract

From the perspective of control theory, the gradient descent optimization methods can be regarded as a dynamic system where various control techniques can be designed to enhance the performance of the optimization method. In this paper, we propose a prescribed finite-time convergent gradient flow that uses time-varying gain nonlinear feedback that can drive the states smoothly towards the minimum. This idea is different from the traditional finite-time convergence algorithms that relies on fractional-power or signed gradient as a nonlinear feedback, that is proved to have finite/fixed time convergence satisfying strongly convex or the Polyak-Łojasiewicz (PŁ) inequality, where due to its nature, the proposed approach was shown to achieve this property for both strongly convex function, and for those satisfies Polyak-Łojasiewic inequality. Our method is proved to converge in a prescribed finite time via Lyapunov theory. Numerical experiments were presented to illustrate our results.

Paper Structure

This paper contains 7 sections, 2 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Design of a PFT-GF with time-varying Gradient feedback
  4. The function satisfies the Polyak-Łojasiewicz inequality
  5. The function is $\mu-$strongly convex
  6. Experiments
  7. Conclusion

Key Result

Lemma 1

bhat2000finite Consider system (1), where the equilibrium is finite-time stable if there exists a Lyapunov function to be $V(x,t) \in C^1$, $V(x) > 0, \quad \forall x \neq 0, \quad \text{and} \quad V(0) = 0.$ and defined over the domain $\mathcal{U} \times \mathbb{R}_{\ge0}$, where $\mathcal{U} \sub Under this condition, for $\forall x_0 \in \mathcal{U}$, the system state reaches the origin in a f

Figures (3)

  • Figure 1: Convergent of the proposed GF for the Trid function ($x_1$ solid, $x_2$ dashed) for different prescribed times $T_p=5,10$ and $15$.
  • Figure 2: Convergent of the proposed GF for the Trid function from different initial conditions ($x_1$ solid, $x_2$ dashed), $T_p=10,k=0.1$.
  • Figure 3: Convergent of the proposed GF for the Rosenbrock function to the global minimum $(1,1)$ from different initial conditions ($x_1$ solid, $x_2$ dashed), $T_p=10,k=0.05$.

Theorems & Definitions (9)

  • Definition 1
  • Lemma 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Lemma 2
  • Definition 6
  • Definition 7