An Efficient Algorithm for Solving a Class of Fractional Programming Problems

Yongjian Qiu,Jingben Yin,Yuming Zhu,Leandro F F Miguel

doi:10.1155/2021/5556130

Abstract

This paper presents an efficient branch-and-bound algorithm for globally solving a class of fractional programming problems, which are widely used in communication engineering, financial engineering, portfolio optimization, and other fields. Since the kind of fractional programming problems is nonconvex, in which multiple locally optimal solutions generally exist that are not globally optimal, so there are some vital theoretical and computational difficulties. In this paper, first of all, for constructing this algorithm, we propose a novel linearizing method so that the initial fractional programming problem can be converted into a linear relaxation programming problem by utilizing the linearizing method. Secondly, based on the linear relaxation programming problem, a novel branch-and-bound algorithm is designed for the kind of fractional programming problems, the global convergence of the algorithm is proved, and the computational complexity of the algorithm is analysed. Finally, numerical results are reported to indicate the feasibility and effectiveness of the algorithm.

Highlights

IntroductionWe consider the following a class of fractional programming problems:
In this paper, we consider the following a class of fractional programming problems:⎧⎪⎪⎪⎪⎪⎨ (FP): ⎪⎪⎪⎪⎪⎩ max s.t. g(x) pm 􏽘􏽙 fjt(x) j 1 t 1 hjt(x)x ∈ D {x|Ax ≤ b}, (1)where p and m are all any arbitrary natural numbers and x is an N−dimensional variable, and for any x ∈ D ⊂ RN, fjt(x) and hjt(x)(j 1, . . . , p and t 1, . . . , m) are all affine functions such that fjt(x) > 0 and hjt(x) > 0
As special cases of the (FP), the linear sum-of-ratios problem and linear multiplicative programming problem have attracted a huge attention of practitioners and researchers for many years. is is because the linear sum-of-ratios problem and linear multiplicative programming problem exist in very important applications such as chance optimization, portfolio optimization, engineering optimization, and data envelopment analysis [1]

Summary

Introduction

We consider the following a class of fractional programming problems:. Several differential evolution algorithms [38,39,40], a novel gate resource allocation method using improved PSO-based QEA [41], and an enhanced MSIQDE algorithm with novel multiple strategies [42] are proposed for solving the global optimization problems including the problem (FP). Some researchers have proposed some algorithms for solving the linear sum-of-ratios problem, the linear multiplicative programming problem, or the special form of the problem (FP), to our knowledge, little work has been still done for globally the general form of the problem (FP) considered in this paper. Based on the branch-and-bound scheme and the constructed problem (LRP), a global optimization algorithm is established, and its convergence is derived.

Novel Linearizing Technique

Algorithm and Its Global Convergence

Computational Complexity of the Algorithm

Numerical Experiments

Conclusions