A New Global Optimization Method Based on Separable Relaxation for Solving Sum of Linear Ratios Problem

This paper proposes an effective separable relaxation algorithm to address the challenges associated with the sum of linear ratios problem (SLRP), which is characterized by a large number of ratios and low-dimensional variables. First, by introducing auxiliary variables and applying equivalent transformations, the equivalent problem (EP) corresponding to the SLRP is derived. Subsequently, using an innovative linearization technique, the relaxation problem (LRP) of the (EP) is formulated, demonstrating separable properties. Within this framework, solving a sequence of linear programming problems is sufficient to obtain a globally optimal solution to the (SLRP), significantly reducing computational effort. Furthermore, the paper provides a rigorous proof of the convergence properties of the proposed algorithm and estimates the maximum number of iterations required. Numerical experiments are conducted to validate the practicality and efficiency of the proposed algorithm.