A projected gradient method for constrained set optimization problems with set-valued mappings of finite cardinality

In this article, a projected gradient method is proposed for constrained set optimization problems, where the objective set-valued mapping is given by a finite number of continuously differentiable vector-valued functions. The proposed method is a set approach to identify the weakly minimal solutions with respect to the lower set less order relation. At first, a necessary condition for weakly minimal solutions of the considered set optimization problem is derived, and then this necessary condition is exploited to find weakly minimal solutions of the considered problem. In deriving the necessary optimality condition, a vectorization of the constrained set optimization problem is used based on the concept of the partition set at a point. Two particular types of projected gradient methods are proposed and analysed: with constant step size and with variable step size along the negative gradient before taking the projection on the constraint set. The well-definedness and global convergence of the proposed methods are provided without any convexity assumption on the objective function. In what follows, it is shown that the sequence of points generated by the methods is feasible and bounded. Further, the bounds of the sequence of descent directions generated by the methods are found. Lastly, some numerical examples are illustrated to exhibit the performance of the proposed method with respect to both the usual standard cone and the general ordering cone.