Continuation Newton methods with deflation techniques for global optimization problems

TL;DR

This paper introduces a memetic algorithm combining continuation Newton methods with deflation techniques and automatic differentiation to efficiently find global minima in large-scale nonconvex optimization problems, outperforming existing methods.

Contribution

The paper proposes a novel hybrid optimization algorithm that integrates continuation Newton methods with deflation and automatic differentiation, enhancing global minimum detection.

Findings

The new algorithm effectively finds global minima in large-scale nonconvex problems.

It outperforms existing global optimization methods in numerical experiments.

The method is applicable to unconstrained optimization problems with improved efficiency.

Abstract

The global minimum point of an optimization problem is of interest in engineering fields and it is difficult to be found, especially for a nonconvex large-scale optimization problem. In this article, we consider a new memetic algorithm for this problem. That is to say, we use the continuation Newton method with the deflation technique to find multiple stationary points of the objective function and use those found stationary points as the initial seeds of the evolutionary algorithm, other than the random initial seeds of the known evolutionary algorithms. Meanwhile, in order to retain the usability of the derivative-free method and the fast convergence of the gradient-based method, we use the automatic differentiation technique to compute the gradient and replace the Hessian matrix with its finite difference approximation. According to our numerical experiments, this new algorithm works well for unconstrained optimization problems and finds their global minima efficiently, in comparison to the other representative global optimization methods such as the multi-start methods (the built-in subroutine GlobalSearch.m of MATLAB R2021b, GLODS and VRBBO), the branch-and-bound method (Couenne, a state-of-the-art open-source solver for mixed integer nonlinear programming problems), and the derivative-free algorithms (CMA-ES and MCS).