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

TL;DR

This paper demonstrates that optimal control trajectories for non-convex functions can be designed to converge globally and practically to the set of global minimizers, using a control-theoretic approach without solving complex equations.

Contribution

It introduces a novel control-theoretic framework for global optimization of non-convex functions, establishing convergence properties without solving ergodic Hamilton-Jacobi-Bellman equations.

Findings

Trajectories converge to global minimizers within any desired tolerance.

Convergence is achieved without solving ergodic Hamilton-Jacobi-Bellman equations.

Results apply to multiple problem formulations including discounted and non-discounted cases.

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 $\eta > 0$, there exist parameters $\lambda$ (discount rate) and $t$ (time horizon) such that trajectories remain within an $\eta$-neighborhood of the global minimizers after some finite time $\tau$. 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.