A new proof of the dual optimization problem and its application to the optimal material distribution of SiC/graphene composite

This paper presents a simple and direct proof of the dual optimization problem. The stationary conditions of the original and the dual problems are exactly equivalent, and the duality gap can be completely eliminated in the dual problem, where the maximal or minimal value is solved together with the stationary conditions of the dual problem and the original constraints. As an illustration, optimization of SiC/graphene composite is addressed with an objective of maximizing certain material properties under the constraint of a given strength.