Untitled

A Linearization to the Sum of Linear Ratios Programming Problem

“…Motivated by Kuno, Benson [24,25] presented branch and bound based algorithms to reach global optimal solutions for S-LFP. According to the theory of monotonic optimization introduced by Tuy [26], Phuong and Tuy [27] presented an iterative efficient unified method to address a wide category of generalized LFPPs. In [28], Benson presented and validated a simplicial branch and bound duality-bounds algorithm to find the global optimal solution for S-LFP.…”

Inner approximation algorithm for generalized linear multiplicative programming problems

“…They are mainly classified as primal-based algorithms that directly solve the primal problem, dual-based algorithms that solve the dual problem, and adapted general nonlinear programming methods [13–15]. Recently, many works aimed at globally solving special forms of (GLMP) are presented, for example, global algorithms for signomial geometric programming problems, branch and bound algorithms for multiplicative programming with linear constraints, branch and reduction methods for quadratic programming problems, and sum of ratios problems are all in this category [16–21]. Despite these various contributions to their special forms, however, optimization algorithms for solving the general case of (GLMP) are still scarce.…”

Effective algorithm and computational complexity for solving sum of linear ratios problem

“…For instance, Konno et al [22] proposed a parametric simplex algorithm for solving the problem (SLRP) with only two linear ratios terms; based on the idea of the approximation algorithm, Shen and Lu [30] designed a practical regional division and reduction method for minimizing the sum of linear fractional functions over a polyhedron; based on the general purpose algorithm for generalized convex multiplicative programming problem, Konno and Yamashita [23] presented an outer approximation algorithm for solving the sum of linear ratios problem. Moreover, based on linear relaxation of the objective function, Carlsson and Shi [6] proposed a linear relaxation algorithm for the sum of linear ratios problem with lower dimension; based on the image space analysis, Falk and Palocsay [11] designed an image space branch-and-bound approach for globally solving the generalized fractional programming problem; based on the theory of monotonic optimization, Phuong and Tuy [27] put forward an unified monotonic optimization algorithm for the generalized affine fractional programming problem which includes the sum of linear ratios problem. Recently, Nesterov and Nemirovskii [25] presented an interior point method for generalized linear fractional programming problem; Shen et al [32,29] designed two polynomial time approximation algorithms for generalized fractional programming problem, respectively; Jiao et al [13] proposed an outcome space range reduction method for the sum of linear ratios problem; Kuno [20] presented a new method for the maximization of sum of linear ratios, which used a concave function to overestimate the optimal value of the original problem; Benson [2] designed a global optimization method by using simplicial branch and bound duality-bounds for the sum of linear ratios problem with linear constraints; Jiao et al [14] and Jiao & Shang [17] proposed two different branch-and-bound algorithms for the sum of linear ratios problem based on outer space partition; Huang and Shen [10] proposed a convex relaxation algorithm for the sum of linear ratios problem; Zhang et al [37] proposed an outer space branch-and-bound algorithm for the sum of linear ratios problem; Jiao et al [18] proposed an image space direct relaxation method for the minimax linear fractional programming problem.…”