An Eikonal equation with vanishing Lagrangian arising in Global Optimization

TL;DR

This paper establishes a novel connection between global optimization of continuous functions and weak KAM theory through an eikonal equation, providing new insights into gradient descent trajectories and convergence to minima.

Contribution

It introduces a new approach linking optimization problems with weak KAM theory and control, including finite-time convergence results for minima.

Findings

Solution v constructed via small discount approximation

Optimal trajectories driven by differential inclusion converge to minima

In some cases, minima are reached in finite time

Abstract

We show a connection between global unconstrained optimization of a continuous function $f$ and weak KAM theory for an eikonal-type equation arising also in ergodic control. A solution $v$ of the critical Hamilton-Jacobi equation is built by a small discount approximation as well as the long time limit of an associated evolutive equation. Then $v$ is represented as the value function of a control problem with target, whose optimal trajectories are driven by a differential inclusion describing the gradient descent of $v$. Such trajectories are proved to converge to the set of minima of $f$, using tools in control theory and occupational measures. We prove also that in some cases the set of minima is reached in finite time.