The objective of this paper is twofold. First, an optimality test is presented to show that the optimality conditions for a two- level design optimization problem before and after its decomposition are the same. Second, based on identification of active constraints and exploitation of problem structure, a simple approach for calculating the gradient of a "second-level" problem is presented. This gradient is an important piece of information which is needed for solution of two-level design optimization problems. Three examples are given to demonstrate applications of the approach.