Solution existence for a class of nonsmooth robust optimization problems

The main purpose of this paper is to investigate the existence of global optimal solutions for nonsmooth and nonconvex robust optimization problems. To do this, we first introduce a concept called extended tangency variety and show how a robust optimization problem can be transformed into a minimizing problem of the corresponding tangency variety. We utilize this concept together with a constraint qualification condition and the boundedness of the objective function to provide relationships among the concepts of robust properness, robust M-tamesness and robust Palais-Smale condition related to the considered problem. The obtained results are also employed to derive necessary and sufficient conditions for the existence of global optimal solutions to the underlying robust optimization problem.