A criterion-space branch-reduction-bound algorithm for solving generalized multiplicative problems

In this paper, we investigate a generalized multiplicative problem (GMP) that is known to be NP-hard even with one linear product term. We first introduce some criterion-space variables to obtain an equivalent problem of the GMP. A criterion-space branch-reduction-bound algorithm is then designed, which integrates some basic operations such as the two-level linear relaxation technique, rectangle branching rule and criterion-space region reduction technologies. The global convergence of the presented algorithm is proved by means of the subsequent solutions of a series of linear relaxation problems, and its maximum number of iterations is estimated on the basis of exhaustiveness of branching rule. Finally, numerical results demonstrate the presented algorithm can efficiently find the global optimum solutions for some test instances with the robustness.