This paper describes applications of global and local monotonicity analysis within a decomposition framework. We present a general formulation and solution procedure, based on a bottom-level global monotoncity analysis, for a design optimization problem which is decomposed into three levels of subproblems. We then perform an optimality test to prove that the optimality conditions for the decomposed subproblems will form the optimality conditions of the overall problem. Furthermore, applications of a two-level decomposition method is presented in which an overall global monotonicity analysis or first-level local monotonicity analysis is performed. Well-known examples illustrate applications of the methods.