Outer space branch and bound algorithm for solving linear multiplicative programming problems

In this paper, we consider a linear multiplicative programming problem (LMP) that is known to be NP-hard even with one product term. We first introduce the auxiliary variables to obtain an equivalent problem of problem LMP. An outer space branch and bound algorithm is then designed, which integrates some basic operations such as the linear relaxation technique, branching rule and region reduction technique. The global convergence of the proposed algorithm is established by means of the subsequent solutions of a series of linear programming problems, and its computational complexity is estimated on the basis of the branching rule. Also, we discuss the relationship between the proposed linear relaxation and existing relaxations for LMP. Finally, preliminary numerical results demonstrate the proposed algorithm can efficiently find the globally optimal solutions for test instances.