Solving multiplicative programs by binary-encoding the multiplication operation

Multiplicative programs in the form of maximization and/or minimization have numerous applications in conservation planning, game theory, and multi-objective optimization settings. In practice, multiplicative programs are challenging to solve because of their multiplicative objective function (a product of continuous or integer variables). These challenges are twofold: 1. As the number of factors in the objective increases, so does the solution time, and the problems become computationally expensive to solve. 2. If all factors are in $(0, 1)$ or in $(1, \infty)$, the objective may cause ill-conditioning and numerical instability. The solution methods proposed in this paper help overcome both of these challenges. The main idea is to binary-encode the multiplication operation analogously to how a computer conducts it internally. This not only solves the aforementioned numerical issues but also allows us to develop a new family of solution methods for multiplicative programs. One such method is to solve the multiplicative programs bit-by-bit, i.e., iteratively computing the optimal value of each bit of the objective function. In an extensive computational study, we explore a number of solution methods that solve multiplicative programs faster and more accurately.