Quasi-Newton methods for multiobjective optimization problems: A systematic review
Unknown authors
Abstract
Quasi-Newton method is one of the most popular methods for solving unconstrained single and multiobjective optimization problems.In a quasi-Newton method, the search direction is computed based on a quadratic model of the objective function, where some approximations replace the true Hessian at each iteration.Several Hessian approximation schemes with an adequate line search technique provided higher-order accuracies in approximating the curvature and made the methods more effective in terms of convergence to solution.Considering the aforementioned reasons, we write a survey on quasi-Newton methods for multiobjective optimization problems.We discuss the development of all the variants of the quasi-Newton method for multiobjective optimization problems, along with some of the advantages and disadvantages of the existing methods.We give a brief discussion about the line search for these variants too.Two cases have been considered for BFGS, Huang BFGS, and self-scaling BFGS multiobjective versions of quasi-Newton methods: one is in the presence of the Armijo line search, and the other is in the absence of any line search.Subsequently, a nonmonotone line search version is also explained for multiobjective optimization problems.Commentary is given on the convergence properties of these methods.The rate of convergence of all these methods is highlighted.To prove the convergence of every method, it is reported that every sequence of points generated from the method converges to a critical point of the multiobjective optimization problem under some mild assumptions.Based on the available numerical data, we provide an unbiased opinion on the effectiveness of quasi-Newton methods for multiobjective optimization problems.