The decomposition of a structuring element for a morphological operation reduces the amount of the computation required for executing the operation. In this paper, we present a new technique for the decomposition of convex structuring elements for morphological operations. We formulated the linear constraints for the decomposition of a convex polygon in discrete space, then the constraints are applied to the decomposition of a convex structuring element. Also, a cost function is introduced to represent the optimal criteria for decomposition. We use linear integer programming technique to find the combination of basis structuring elements which minimizes the amount of the computation required for executing the morphological operation. Formulating different cost functions for different implementation methods and computer architectures, we can determine the optimal decompositions which guarantee the minimal amounts of computation on different computing environment.