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In this paper, with the unique hypothesis that the denominator is not equal to 0, an efficient outer space rectangle branch-and-bound algorithm is presented to globally solve the sum of general affine ratios problem. By using equivalent transformation and the characteristics of general single ratio function, a linear program relaxation problem is constructed for computing the lower bound of the global minimum value of the original problem. Moreover, to enhance the running speed of the presented algorithm, we design an outer space accelerating technic for deleting the entire outer space rectangle or a part of the entire examined outer space rectangle, in which there contains no the global minimum point of the equivalent problem. Furthermore, through the complexity analysis of the presented algorithm, we estimate its maximum iteration times. In addition, we prove the global convergence of the presented algorithm, report and analyze the numerical computational results for indicating the validity of the presented algorithm. Finally, two practical problems from power transportation and production planning are solved to verify the usefulness of the presented algorithm. • We present a novel outer space branch-and-bound algorithm for the SGAR. • The algorithm economizes the required computations by dividing space. • The main calculation involves solving a series of linear programming problems. • We give the complexity analysis of the algorithm for the first time. • The computing efficiency is significantly higher than the existing algorithms. [ABSTRACT FROM AUTHOR]

Based on the image space branch-and-bound scheme, this paper presents a novel algorithm for globally solving the min–max linear fractional programs (MMLFP), which has many applications in management optimization, engineering optimization, economic investment and so on. For finding the global optimal solution of the MMLFP, by leading into new parameter, we firstly transform the MMLFP into an equivalent fractional problem (EFP). Next, by using convex hull and concave hull approximation of bilinear function, we construct the linear relaxation problem (LRP) for computing the lower bound of the global minimum value of the EFP in the image space branch-and-bound algorithm. By subsequently solving a series of linear relaxation problems and refining the initial image space rectangle, the proposed algorithm is globally convergent to the optimum solution of the EFP. In addition, we give the computational complexity analysis of the algorithm based on the exhaustive branching rule. Finally, computational comparisons show better computational performance of the algorithm. [ABSTRACT FROM AUTHOR]