The nondominated solutions and weakly nondominated solutions for the interval-valued optimization problems are proposed in this paper. Based on these solutions concepts, the Karush-Kuhn-Tucker optimality conditions for interval-valued optimization problems are derived, and the different concepts of solvability are also proposed. Moreover, we introduce the Wolfe's type of primal-dual pair problems for the interval-valued optimization problems, and derive the strong duality theorems in weak and strong sense.