AN IMPROVED NONMONOTONE QUASI-NEWTON METHOD FOR MULTIOBJECTIVE OPTIMIZATION PROBLEMS

In this paper, we discuss one of the most popular methods for multiobjective optimization problems, namely quasi-Newton method. We start with a brief survey on quasi-Newton methods for multiobjective optimization problems. In this, we focus on all three components, namely Hessian approximation, quasi-Newton direction, and step length of quasi-Newton algorithms for multiobjective optimization problems. It is commonly observed that the BFGS update formula is used to approximate the Hessian matrix in the quasi-Newton methods. However, we also mention the case in which self-scaling-BFGS and Huang-BFGS update formulae are also operated. It is also highlighted that the quasi-Newton direction has been calculated by solving the subproblems involving the Hessian approximation of the objectives. We mention an algorithm using a nonmonotone Armijo line search instead of a monotone Armijo line search. After the survey, an improved nonmonotone quasi-Newton method has been proposed in this paper. In this method, we use the conventional BFGS update formula to approximate the Hessian matrix of each component of the objective function of multiobjective optimization problems. The well-definedness of the proposed algorithm is also provided. Subsequently, global convergence has been established under some mild assumptions. Finally, the proposed algorithm is performed on some test problems to demonstrate the numerical performance of the proposed algorithm.