Continuation Newton methods with deflation techniques for global optimization problems
This addresses the challenge of global optimization in engineering fields, but it is incremental as it builds on existing methods like continuation Newton and evolutionary algorithms.
The authors tackled the problem of finding global minima for nonconvex large-scale optimization by proposing a new memetic algorithm that combines continuation Newton methods with deflation techniques to find multiple stationary points as initial seeds for an evolutionary algorithm, showing it works well and finds global minima efficiently in numerical experiments compared to other methods like multi-start, branch-and-bound, and derivative-free algorithms.
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).