Regional division and reduction algorithm for minimizing the sum of linear fractional functions

Pei-Ping Shen,Ting Lu

doi:10.1186/s13660-018-1651-9

Pei-Ping Shen, Ting Lu

Open Access

https://doi.org/10.1186/s13660-018-1651-9

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Abstract

This paper presents a practicable regional division and cut algorithm for minimizing the sum of linear fractional functions over a polyhedron. In the algorithm, by using an equivalent problem (P) of the original problem, the proposed division operation generalizes the usual standard bisection, and the deleting and reduction operations can cut away a large part of the current investigated region in which the global optimal solution of (P) does not exist. The main computation involves solving a sequence of univariate equations with strict monotonicity. The proposed algorithm is convergent to the global minimum through the successive refinement of the solutions of a series of univariate equations. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.

Highlights

Consider the following class of fractional programming:⎧ ⎨min (FP) : ⎩s.t.N ci y+c0i i=1 di y+d0iAy ≤ b, y ≥ 0, where A =m×n is a real matrix, ci =1×n and di =1×n are real vectors for each i = 1, . . . , N, b =m×1, c0i, d0i ∈ R
A new division and reduction algorithm is proposed for globally solving problem (FP)
2 Equivalent problem For solving problem (FP), we first convert the primary problem (FP) into an equivalent optimization problem (P), in which the objective function is a single variable and the constraint functions are the difference of two increasing functions

Summary

Introduction

Ay ≤ b, y ≥ 0, where A = (arj)m×n is a real matrix, ci = (cij)1×n and di = (dij)1×n are real vectors for each i = 1, . The proposed adaptive division operation both generalizes and is superior to the usual standard bisection in BB methods according to the numerical computational result in Sect.

Objectives

Results

Conclusion