Outer-space branch-and-bound algorithm for generalized linear multiplicative programs

This paper introduces a new global optimization algorithm for solving the generalized linear multiplicative problem (GLMP). The algorithm starts by introducing $\bar{p}$ new variables and applying a logarithmic transformation to convert the problem into an equivalent problem (EP). By using the strong duality of linear program, a new convex relaxation subproblem is formulated to obtain the lower bounds for the optimal value of EP. This relaxation subproblem, combined with a simplicial branching process, forms the foundation of a simplicial branch-and-bound algorithm that can globally solve the problem. The paper also includes an analysis of the theoretical convergence and computational complexity of the algorithm. Additionally, numerical experiments are conducted to demonstrate the effectiveness of the proposed algorithm in various test instances.