Newton method for set optimization problems with set-valued mapping of finitely many vector-valued functions

In this paper, we propose a Newton method for unconstrained set optimization problems to find their 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 converges locally to a point satisfying the necessary optimality condition for weakly minimal points. To find this sequence of iterates, we formulate a family of vector optimization problems using the partition set concept. Then, we find a descent direction for the family of vector optimization problems obtained to progress from the current iterate to the next iterate. As the chosen vector optimization problem differed across the iterations, 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 analysed. 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 a 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. We provide examples to illustrate the performance of the proposed method. The performance of the proposed method is also compared with that of the existing steepest-descent method.