A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

Abstract

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex-convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.

A practicable branch and bound algorithm for sum of linear ratios problem

“…algorithms can be used to solve special form of sum of linear ratios problem, as far as we know, only Shen and Wang (2006) present a branch and bound algorithm for maximizing the sum of linear ratios with coefficients. In addition, several algorithms (Benson, 2002a(Benson, , 2002bDai, Shi, & Wang, 2005;Fang, Gao, Sheu, & Xing, 2009;Gao, Mishra, & Shi, 2012;Jaberipour & Khorram, 2010;Jiao, Wang, & Chen, 2013;Pei & Zhu, 2013;Shen & Jin, 2010;Shen, Duan, & Pei, 2009;Wang & Zhang, 2004) have been developed for globally solving sum of nonlinear ratios problems.…”

Branch and bound computational method for multi-objective linear fractional optimization problem

“…Banson [22] developed a branch-and-bound outer approximation algorithm for globally solving a sum-ofratios fractional programming problem. Shen et al [23] concerned a simplicial branch and duality bound algorithm for globally solving the sum of convex-convex ratios problem with non convex feasible region. Zhou and Cao [24] presented a simplicial branch and bound dualitybounds algorithm to globally solving the linear multiplicative programming to convert the problem into an equivalent problem by introducing p-auxiliary variables.…”

“…Multi-objective linear fractional (MOLF) optimization problem is not yet solved by using branch and bound technique as far as authors aware. The propose method is developed with help of branch-andbound technique proposed by Shen et al [23].…”

Global Optimization for the Sum of Concave-Convex Ratios Problem

“…Freund and Jarre [10] present a suitable interior-point approach for the solution of much more general problems with convex-concave ratios and convex constraints. Shen et al [11] present a simplicial branch and duality bound algorithm for globally solving the sum of convex-convex ratios problem with nonconvex feasible region.…”