Non-Convex Global Optimization as an Optimal Stabilization Problem: Convergence Rates

TL;DR

This paper reformulates non-convex optimization as an optimal control problem, providing explicit convergence rates and pathwise convergence guarantees without requiring convexity or smoothness.

Contribution

It introduces a rigorous framework connecting non-convex optimization to optimal control, establishing explicit convergence rates and trajectory convergence under minimal assumptions.

Findings

Proves exponential convergence rates with computable constants.

Shows convergence of the value function and objective function.

Demonstrates practical effectiveness through numerical experiments.

Abstract

We develop a rigorous framework for global non-convex optimization by reformulating the minimization problem as a discounted infinite-horizon optimal control problem. For non-convex, continuous, and possibly non-smooth objective functions with multiple global minimizers, where classical gradient-based methods lack global convergence guarantees, we establish explicit exponential convergence rates with computable constants. Our analysis proves (i) variational convergence of the value function of the optimal control problem, (ii) convergence in the objective function for the original problem, as well as (iii) pathwise convergence of optimal trajectories to the minimizer set under minimal structural assumptions that require neither convexity, differentiability, nor {\L}ojasiewicz-type conditions on the objective. These quantitative results significantly strengthen the asymptotic theory developed in our previous work (arXiv:2511.10815). Numerical experiments demonstrate the practical effectiveness of the approach on challenging non-convex problems.