Robust alternative theorem for linear inequalities with applications to robust multiobjective optimization

We first establish a new alternative theorem for a robust linear inequality system, where the dual statement is expressed in terms of linear matrix inequalities and thus, it can be verified by solving a semidefinite linear program. We then apply the established alternative theorem to derive a characterization of optimality for weakly Pareto solutions of a robust linear multiobjective optimization problem, and to examine weak, strong and converse duality relations in robust linear multiobjective optimization.