A Revised Algorithm for Solving the Sum of Linear Ratios Problem with Lower Dimension using Linear Relaxation *

 In this paper, we propose a revision of the linear relaxation algorithm [Carlsson and Shi (2013): A linear relaxation algorithm for solving the sum-of-linear-ratios problem with lower dimension. OR Letters, 41(4): 381-389] for solving the Sum-of-Linear-Ratios (SOLR) problem. Carlsson and Shi casted the SOLR problem into an equivalent problem with linear objective and a set of linear and nonconvex quadratic constraints. By dropping out the nonconvex quadratic constraints, they proposed a linear relaxation for the SOLR problem and designed a branch-and-bound algorithm to solve the SOLR problem with lower dimension. To circumvent the nonconvex quadratic constraints, we do not drop them out but make a linear relaxation for the nonconvex constraints with some extra variables. Therefore, this linear relaxation is generally tighter than the previous one. With the help of the new relaxation, we propose an algorithm for solving the SOLR problem and prove the convergence of the algorithm. The numerical experiments are conducted and the results indicate that our method is more efficient than the previous.

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