In this paper, we introduce a second-order strong subdifferential of set-valued maps, and discuss some properties, such as convexity, sum rule and so on. By the new subdifferential and its properties, we establish a necessary and sufficient optimality condition of set-based robust efficient solutions for the uncertain set-valued optimization problem. We also introduce a Wolfe type dual problem of the uncertain set-valued optimization problem. Finally, we establish the robust weak duality theorem and the robust strong duality theorem between the uncertain set-valued optimization problem and its robust dual problem. Several main results extend to the corresponding ones in the literature.