Sensitivity-based Heuristic for Guaranteed Global Optimization with Nonlinear Ordinary Differential Equations

TL;DR

This paper introduces a sensitivity-based heuristic for branch and bound algorithms to efficiently solve global optimization problems constrained by nonlinear ODEs, reducing computational effort by up to 30%.

Contribution

It proposes a novel smear-like sensitivity heuristic for branching, improving efficiency in guaranteed global optimization with nonlinear differential constraints.

Findings

Up to 30% reduction in number of branches.

Effective heuristic for nonlinear ODE constrained optimization.

Implementation within branch and bound framework.

Abstract

We focus on interval algorithms for computing guaranteed enclosures of the solutions of constrained global optimization problems where differential constraints occur. To solve such a problem of global optimization with nonlinear ordinary differential equations, a branch and bound algorithm can be used based on guaranteed numerical integration methods. Nevertheless, this kind of algorithms is expensive in term of computation. Defining new methods to reduce the number of branches is still a challenge. Bisection based on the smear value is known to be often the most efficient heuristic for branching algorithms. This heuristic consists in bisecting in the coordinate direction for which the values of the considered function change the most "rapidly". We propose to define a smear-like function using the sensitivity function obtained from the differentiation of ordinary differential equation with respect to parameters. The sensitivity has been already used in validated simulation for local optimization but not as a bisection heuristic. We implement this heuristic in a branch and bound algorithm to solve a problem of global optimization with nonlinear ordinary differential equations. Experiments show that the gain in term of number of branches could be up to 30%.