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In this article, we study the nonlinear and nonsmooth interval-valued optimization problems in the face of data uncertainty, which are called interval-valued robust optimization problems (IVROPs). We introduce the concept of nondominated solutions for the IVROP. If the interval-valued objective function f and constraint functions gi{g}_{i} are nonsmooth on Banach space E, we establish a nonsmooth and robust Karush-Kuhn-Tucker optimality theorem.