Decomposition strategies are used in a variety of practical design optimization applications. Such decompositions are valid, if the solution of the decomposed problem is in fact also the solution to the original one. Conditions for such validity are not always obvious. In the present article, we develop conditions for two-level parametric decomposition under which: (1) isolated minima at the two levels imply an isolated minimum for the original problem; (2) necessary conditions at the two-levels are equivalent to the necessary conditions for the original problem; and, (3) a descent algorithm for computing Karush-Kuhn-Tucker points in decomposition formulations is globally convergent. Since no special problem structure is assumed, the results are general and could be used to evaluate the suitability of a variety of approaches and algorithms for decomposition strategies.