Output-space branch-and-bound reduction algorithm for solving generalized linear multiplicative programming programs

We consider a class of generalized linear multiplicative problems (GLMP), which have a wide range of applications and are known to be NP-hard. In this paper, we first transform it into an equivalent problem (EP) by introducing p new variables and applying logarithmic transformation. Secondly, in order to calculate the lower bound, we derived the linear relaxation problem (LRP) of EP by constructing a novel relaxation strategy. Additionally, a rectangular region reduction technique is proposed to accelerate the convergence speed of the algorithm. Based on the output-space search, we propose a new branch-and-bound algorithm for tackling the GLMP or EP. The global convergence of the algorithm is proved, and its computational complexity is analyzed to estimate the maximum number of iterations. Especially on the basis of LRP, we also propose another new convex relaxation based branch-and-bound algorithm for GLMP. Some experimental examples demonstrate the feasibility and effectiveness of these two algorithms.