Newton Method for Set Optimization Problems with Set-Valued Mapping of Finitely Many Vector-Valued Functions

Abstract: In this paper, we propose a Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The objective function of the problem under consideration is given by finitely many strongly convex twice continuously differentiable vector-valued functions. At first, with the help of a family of vector optimization problems and the Gerstewitz scalarizing function, we identify a necessary optimality condition for weakly minimal solutions of the considered problem. In the proposed Newton method, we derive a sequence of iterative points that exhibits local convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. To find this sequence of iterates, we formulate a family of vector optimization problems with the help of a partition set concept. Then, we find a descent direction for this obtained family of vector optimization problems to progress from the current iterate to the next iterate. As the chosen vector optimization problem differed across the iterates, the proposed Newton method for set optimization problems is not a straight extension of that for vector optimization problems. A step-wise algorithm of the entire process is provided. The well-definedness and convergence of the proposed method are analyzed. To establish the convergence of the proposed algorithm under some regularity condition of the stationary points, we derive three key relations: a condition of nonstationarity, the boundedness of the norm of Newton direction, and the existence of step length that satisfies the Armijo condition. We obtain the local superlinear convergence of the proposed method under uniform continuity of the Hessian and local quadratic convergence under Lipschitz continuity of the Hessian.