Robust optimization revisited via robust vector Farkas lemmas

This paper provides characterizations of the weakly minimal elements of vector optimization problems and the global minima of scalar optimization problems posed on locally convex spaces whose objective functions are deterministic while the uncertain constraints are treated under the robust (or risk-averse) approach, i.e. requiring the feasibility of the decisions to be taken for any possible scenario. To get these optimality conditions we provide Farkas-type results characterizing the inclusion of the robust feasible set into the solution set of some system involving the objective function and possibly uncertain parameters. In the particular case of scalar convex optimization problems, we characterize the optimality conditions in terms of the convexity and closedness of an associated set regarding a suitable point.