A logarithmic method for eliminating binary variables and constraints for the product of free-sign discrete functions

In this paper, a logarithmic method was developed to solve optimization problems containing the product of free-sign discrete functions (PFDF). The current deterministic methods used to handle these problems are based on the concept of continuous variables; therefore, the methods always transform the original model into another programming model (e.g., DC programming, convex programming) and solve them with a commercial solver. As the nature of a discrete variable is quite different from that of a continuous one, developing a novel method to address the above mentioned problems is necessary. This study proposes a concise and efficient method that linearizes PFDF term into a set of linear inequalities directly without redundant transformation. Further, the proposed method only requires the logarithmic numbers of binary variables and constraints. Numerical examples demonstrate that the proposed formulation significantly outperforms current approaches.