An efficient branch-and-bound algorithm using an adaptive branching rule with quadratic convex relaxation for globally solving general linear multiplicative programs

This article proposes an efficient global algorithm for solving general linear multiplicative programming problem (GLMP). The new algorithm combines the quadratic convex relaxation problem, adaptive branching rule, region reduction technique and branch-and-bound scheme. Firstly, a transformation technique can transform GLMP into a non-convex quadratic program with quadratic and linear constraints. The non-convexity parts of the equivalent problem are addressed by employing secant lines, so that a quadratic convex relaxation problem is structured. Secondly, we introduce the adaptive branching rule to improve the upper bound of the optimal value. Thirdly, the convergence of the proposed algorithm is analyzed and its worst-case complexity is provided. Finally, numerical experiments demonstrate the efficiency and advantage of the proposed algorithm for obtaining the global ɛ-optimal solutions of test instances.