Controlled stochastic processes for simulated annealing

Department of Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, Gothenburg, Sweden

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 4 July 2025
Accepted: 8 January 2026

Abstract

Simulated annealing solves global optimization problems by means of a random walk in a cooling energy landscape based on the objective function and a temperature parameter. However, if the temperature is decreased too quickly, this procedure often gets stuck in suboptimal local minima. In this work, we consider the cooling landscape as a curve of probability measures. We prove the existence of a minimal norm velocity field which solves the continuity equation, a differential equation that governs the evolution of the aforementioned curve. The solution is the weak gradient of an integrable function, which is in line with the interpretation of the velocity field as a derivative of optimal transport maps. We show that controlling stochastic annealing processes by superimposing this velocity field would allow them to follow arbitrarily fast cooling schedules. Here we consider annealing processes based on diffusions and piecewise deterministic Markov processes. Based on convergent optimal transport-based approximations to this control, we design a novel interacting particle-based optimization method that accelerates annealing. We validate this accelerating behaviour in numerical experiments.