An algorithm based on continuation techniques for minimization problems with highly non-linear equality constraints

TL;DR

This paper introduces a continuation-based algorithm for solving complex minimization problems with numerous local minima, successfully computing high-order optimized integrators with minimal norm.

Contribution

The paper presents a novel continuation technique algorithm capable of effectively solving highly non-linear constrained minimization problems, especially for underdetermined polynomial systems.

Findings

Successfully computed 10th order time-symmetric integrators with minimal 1-norm.

Outperformed existing methods in finding optimized differential equation solvers.

Demonstrated effectiveness on problems with many local minima.

Abstract

We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local minimization algorithms with random starting guesses. We are particularly interested in the computation of minimal norm solutions of underdetermined systems of polynomial equations. Such systems arise, for instance, in the context of the construction of high order optimized differential equation solvers. By applying our algorithm, we are able to obtain 10th order time-symmetric composition integrators with smaller 1-norm than any other integrator found in the literature up to now.