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

TL;DR

This paper recasts non-convex optimization as a discounted infinite-horizon optimal control problem, introducing the value function $u_{\lambda}$ that solves the Hamilton–Jacobi–Bellman equation $\lambda u_{\lambda} + \tfrac12|Du_{\lambda}|^2 = f$. It proves explicit exponential convergence rates: along optimal trajectories, $u_{\lambda}$ approaches the minimum value at rate $e^{-(K-\lambda)t}$, and the trajectories converge exponentially to the minimizer set $\mathfrak{M}$; these results extend to quasi-optimal trajectories with robust bounds. A Riccati-type structure appears for a distance-based distance-to-set objective, yielding a closed-form quadratic form for the associated value function. The work further establishes Turnpike-type behavior, discusses controllability conditions ensuring the main assumption, and situates the framework relative to Polyak–Łojasiewicz and metric-regularity conditions, with a range of examples illustrating applicability to nonconvex landscapes in optimization and learning.

Abstract

We propose a discounted infinite-horizon optimal control formulation that generates trajectories converging to the set of global minimizers of a continuous, non-convex function. The analysis provides explicit convergence rates for both the variational behavior of the value function and the pathwise convergence of the optimal trajectories. This paper is a companion to our previous work, where a more general framework was introduced; here, we focus on a specific setting in which sharper and more detailed results can be established.

Theorems & Definitions (50)

• Remark 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 2.5
• Definition 2.6
• Proposition 3.1
• proof
• Theorem 3.2
• proof