This paper mainly studies the sum-of-linear-ratios problems, which have important applications in finance, economy and computational vision. In this process, we first propose a new method to re-represent the original problem as an equivalent problem (EP). Secondly, by relaxing these constraints, a nonlinear relaxation subproblem is constructed for EP. In view of the special structure of the relaxation, it is reconstructed as a second-order cone programming (SOCP) problem, which is essentially a SOCP relaxation of EP. Thirdly, through the structural characteristics of the objective function of EP, a region reduction technique is designed to accelerate the termination of the algorithm as much as possible. By integrating the SOCP relaxation and acceleration strategy into the branch and bound framework, a new global optimization algorithm is developed. Further, the theoretical convergence and computational complexity of the algorithm are analyzed. Numerical experiment results reveal that the algorithm is effective and feasible.

This paper investigated the linear ratio sum problem, a complex non-convex optimization problem with extensive applications in finance, economics, computer vision, and other fields. We proposed a novel global optimization approach that reformulated the original problem into an equivalent one with nonlinear constraints. The approach constructed linear fractional relaxation subproblems via constraint relaxation and leveraged the structural properties of the relaxations to transform these subproblems into linear programming formulations, thereby ensuring efficient computation. Furthermore, rectangular branching rules were designed based on the relaxed nonlinear constraints. These rules, complemented by region elimination techniques, accelerated convergence by exploiting the structure of the objective function. By integrating these components into a branch-and-bound framework, a novel global optimization algorithm was devised. Theoretical analysis confirmed the convergence and computational complexity of the proposed algorithm, while numerical tests validated its effectiveness and feasibility.