A Consensus-based optimization algorithm using Gaussian processes for global optimization problems in Sobolev spaces

TL;DR

This paper introduces a novel global optimization algorithm in Sobolev spaces leveraging Gaussian processes, enabling the approximation of solutions for complex boundary value and control problems.

Contribution

It presents a new consensus-based optimization method in infinite-dimensional Sobolev spaces using Gaussian processes, extending finite-dimensional algorithms.

Findings

Effective in solving nonlinear boundary value problems with constraints

Demonstrated feasibility for nonlinear optimal control problems

Numerical results validate the algorithm's performance

Abstract

We propose an algorithm to approximate solutions of global optimization problems in Sobolev spaces that follows the spirit of Consensus-based algorithms in finite dimensions. The main ingredient are Gaussian processes. In fact, we exploit their rich toolbox in order to draw sample functions from Sobolev spaces that satisfy initial values, boundary conditions or state constraints. Well-known marginalization properties of Gaussian processes help us to discretize the algorithm, that is stated in infinite dimensions, appropriately. We illustrate the performance of the algorithm and show its feasibility for nonlinear boundary value problems with state constraints as well as nonlinear optimal control problems constrained by a system of ordinary differential equations with several numerical results.