A linear relaxation algorithm for solving the sum-of-linear-ratios problem with lower dimension

In this paper, we present an algorithm for solving the sum-of-linear-ratios problem based on a linear relaxation of the objective function. Though there already exist linear relaxation algorithms for solving this problem, they all work on a space whose dimension increases with the number of ratios. When the number of ratios becomes large, these algorithms are unable to solve the problem efficiently. Our numerical experiments indicate that the proposed algorithm in this paper is superior to these existing algorithms when the number of ratios is large.