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Non-Convex Global Optimization as an Optimal Stabilization Problem: Dynamical Properties

Yuyang Huang, Dante Kalise, Hicham Kouhkouh

TL;DR

The paper casts global non-convex optimization as a deterministic optimal stabilization problem, introducing three connected control formulations (evolutive discounted, stationary discounted, evolutive non-discounted) and studying their trajectories via occupation measures. It proves that (quasi-)optimal trajectories globally and practically converge to the set of global minimizers $\mathfrak{M}$: for any tolerance $\eta$, suitable $\lambda$ and horizon $t$ exist such that trajectories stay within $\eta$ of $\mathfrak{M}$ after time $\tau$, with the non-minimizer time vanishing as $\lambda\to0$ or $t\to\infty$. The results hold across all three problem variants, and are complemented by explicit bounds on the time spent outside neighborhoods of $\mathfrak{M}$ via occupation measures, as well as Lyapunov-type stability guarantees. The approach eschews ergodic Hamilton–Jacobi–Bellman limits, offering a direct dynamical mechanism for global optimization grounded in deterministic control and measure-theoretic analysis. This provides a principled interpretation of optimization dynamics and potential guidance for designing robust global-search algorithms grounded in optimal-control theory.

Abstract

We study global optimization of non-convex functions through optimal control theory. Our main result establishes that (quasi-)optimal trajectories of a discounted control problem converge globally and practically asymptotically to the set of global minimizers. Specifically, for any tolerance $η> 0$, there exist parameters $λ$ (discount rate) and $t$ (time horizon) such that trajectories remain within an $η$-neighborhood of the global minimizers after some finite time $τ$. This convergence is achieved directly, without solving ergodic Hamilton-Jacobi-Bellman equations. We prove parallel results for three problem formulations: evolutive discounted, stationary discounted, and evolutive non-discounted cases. The analysis relies on occupation measures to quantify the fraction of time trajectories spend away from the minimizer set, establishing both reachability and stability properties.

Non-Convex Global Optimization as an Optimal Stabilization Problem: Dynamical Properties

TL;DR

The paper casts global non-convex optimization as a deterministic optimal stabilization problem, introducing three connected control formulations (evolutive discounted, stationary discounted, evolutive non-discounted) and studying their trajectories via occupation measures. It proves that (quasi-)optimal trajectories globally and practically converge to the set of global minimizers : for any tolerance , suitable and horizon exist such that trajectories stay within of after time , with the non-minimizer time vanishing as or . The results hold across all three problem variants, and are complemented by explicit bounds on the time spent outside neighborhoods of via occupation measures, as well as Lyapunov-type stability guarantees. The approach eschews ergodic Hamilton–Jacobi–Bellman limits, offering a direct dynamical mechanism for global optimization grounded in deterministic control and measure-theoretic analysis. This provides a principled interpretation of optimization dynamics and potential guidance for designing robust global-search algorithms grounded in optimal-control theory.

Abstract

We study global optimization of non-convex functions through optimal control theory. Our main result establishes that (quasi-)optimal trajectories of a discounted control problem converge globally and practically asymptotically to the set of global minimizers. Specifically, for any tolerance , there exist parameters (discount rate) and (time horizon) such that trajectories remain within an -neighborhood of the global minimizers after some finite time . This convergence is achieved directly, without solving ergodic Hamilton-Jacobi-Bellman equations. We prove parallel results for three problem formulations: evolutive discounted, stationary discounted, and evolutive non-discounted cases. The analysis relies on occupation measures to quantify the fraction of time trajectories spend away from the minimizer set, establishing both reachability and stability properties.

Paper Structure

This paper contains 18 sections, 17 theorems, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contribution
  3. Related literature
  4. Preliminaries
  5. Useful estimates
  6. Occupation measures
  7. Trajectories
  8. Reachability of the set of global minimizers
  9. The finite-horizon discounted problem
  10. The infinite-horizon discounted problem
  11. The finite-horizon un-discounted problem
  12. Stability of the set of global minimizers
  13. The finite-horizon discounted problem
  14. The infinite-horizon discounted problem
  15. The finite-horizon un-discounted problem
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let Assumptions (A) be satisfied. Let $y_{x}^{\alpha}(\cdot)$ be an optimal trajectory for problem (OCP intro). Then for all $\eta>0$, there exist $\lambda>0$ small, $t>0$ large, and $\tau = \tau(\eta, \lambda,t)>0$ such that In other words, $y_{x}^{\alpha}(s) \in \left\{z\in \mathbb{R}^{n}: \operatorname{dist}(z,\mathfrak{M})\leq \eta\right\}$ for all $s\in [\tau, t]$. The same conclusion holds

Figures (1)

  • Figure 1: Illustration of the convergence relationships: The dashed arrows "$\dashrightarrow$" are those in the present manuscript. The other arrows "$\rightarrow$" are those studied in bardi2023eikonal.

Theorems & Definitions (35)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Theorem 3.1
  • proof
  • ...and 25 more