An efficient outer space branch-and-bound algorithm for globally minimizing linear multiplicative problems

We propose an efficient outer space branch-and-bound algorithm for minimizing linear multiplicative problems (LMP). First, by introducing auxiliary variables, LMP is transformed into an equivalent problem (ELMP), where the number of auxiliary variables is equal to the number of linear functions. Subsequently, based on the properties of exponential and logarithmic functions, further equivalent transformation of ELMP is performed. Next, a novel linear relaxation technique is used to obtain the linear relaxation problem, which provides a reliable lower bound for the global optimal value of LMP. Once more, branching operation takes place in the outer space of the linear function while embedding compression technique to remove infeasible regions to the maximum extent possible, which significantly reduces the computational cost. Therefore, an outer space branch-and-bound algorithm is proposed. In addition, we conduct convergence analysis and complexity proof for the algorithm. Finally, the computational performance of the algorithm is demonstrated based on the experimental results obtained by testing a series of problems.