We introduce a new notion of a vector-based robust minimal solution for a vector-valued uncertain optimization problem, which is defined by means of some open cone. We present necessary and sufficient conditions for this kind of solution, which are stated in terms of some directional derivatives of vector-valued functions. To prove these results, we apply the methods of set-valued analysis. We also study relations between our definition and three other known optimality concepts. Finally, for the case of scalar optimization, we present two general algorithm models for computing vector-based robust minimal solutions.