An efficient image space branch-reduction-bound algorithm to globally solve generalized fractional programming problems for large-scale real applications

This study focuses on finding a global optimal solution to a generalized fractional programming problem (GFP) in the perspective of large-scale applications. However, few algorithms exist to solve the GFP due to its nonconvexity and NP-hard. Therefore, it is significant to propose an efficient algorithm for the GFP. For this purpose, first of all, using logarithmic and exponential properties, we transform the GFP into an equivalent problem (EP) by introducing some image space variables. Next, the relaxation problem (RP) is constructed using a two-phase relaxation method and bilinear relaxation technique based on the structure of the EP. Furthermore, through the comprehensive utilization of the unique structures of EP and RP, along with the attributes of branch and bound algorithms, some novel region-reduction techniques have been introduced to eliminate regions devoid of the global optimal solution. Subsequently, the integration of the two-phase relaxation method and region-reduction techniques within the branch-and-bound framework has led to the design of the image space branch-reduction-bound algorithm (ISBRBA). Moreover, the convergence and computational complexity of the ISBRBA are analyzed by considering the infinite approximation property between the objective functions of the EP and RP. Finally, the effectiveness and robustness of the ISBRBA are demonstrated through numerous numerical experiments.