This paper proposes an efficient method for acquiring the global solution of the sum of affine ratios problem (SARP) in the reduced outer space. Using equivalence conversions, the original problem was transformed into an equivalent problem. Then, an affine relaxation problem of the equivalent problem was constructed by exploiting linearization techniques. Subsequently, an outcome space branch-and-bound algorithm was proposed, the convergence of the algorithm was proved and the computational complexity was estimated. Finally, numerical examples were presented to demonstrate the effectiveness and feasibility of the presented algorithm.

This paper proposed an algorithm based on the branch-and-bound framework for globally solving the sum of linear ratios problem (SLRP) with a large number of ratios and a small number of variables. First, we introduced new variables to construct an equivalent problem of the problem (SLRP). Then, using a new linear relaxation technique, we obtained the linear relaxation problem for the equivalent problem. By utilizing the separable nature of the linear relaxation problem, we computed the linear relaxation problem by solving its $p$ linear programming subproblems, thereby the lower bound for the problem (SLRP) could be obtained. Additionally, we conducted a theoretical analysis of the proposed algorithm and validated its feasibility and effectiveness through numerical experiments.