In this paper, we present co-processor selection problem for minimum energy consumption in hw/sw co-design on FPGAs with dual power mode. We provide theoretical analysis for the problem under no constraint, resource constraint, and timing constraint. We prove that the complexity of the problem in each case is NP-Hard and we provide a generalized ILP formulation. We compared the result of our approach in minimizing energy to the result of other approaches that had not considered both static and dynamic power during optimization and we showed that we can reduce energy by 63% in some cases.