A New Global Optimization Algorithm for a Class of Linear Fractional Programming

X. Liu,B. Zhang,Y.L. Gao,F.P. Tian

doi:10.3390/math7090867

Abstract

In this paper, we propose a new global optimization algorithm, which can better solve a class of linear fractional programming problems on a large scale. First, the original problem is equivalent to a nonlinear programming problem: It introduces p auxiliary variables. At the same time, p new nonlinear equality constraints are added to the original problem. By classifying the coefficient symbols of all linear functions in the objective function of the original problem, four sets are obtained, which are     I  i  +    ,     I  i  −    ,     J  i  +    and     J  i  −    . Combined with the multiplication rule of real number operation, the objective function and constraint conditions of the equivalent problem are linearized into a lower bound linear relaxation programming problem. Our lower bound determination method only needs      e i T  x +  f i  ≠ 0    , and there is no need to convert molecules to non-negative forms in advance for some special problems. A output-space branch and bound algorithm based on solving the linear programming problem is proposed and the convergence of the algorithm is proved. Finally, in order to illustrate the feasibility and effectiveness of the algorithm, we have done a series of numerical experiments, and show the advantages and disadvantages of our algorithm by the numerical results.

Highlights

Fractional programming is an important branch of nonlinear optimization and it has attracted interest from researchers for several decades
Assume that when the algorithm iterates to step k, we only need to solve problem LRPk, whose optimal value v( LRPk ) is a lower bound of the global optimum value v( EOPk )of problem equivalent optimization problem (EOP) on rectangle H k ⊆ H
The optimal value v( LRPk ) is an effective lower bound of the global optimum value v( LFPk ) of the original problem linear fractional programming (LFP) on H k, i.e., v( LRPk ) ≤ v( EOPk ) =

Summary

Introduction

Fractional programming is an important branch of nonlinear optimization and it has attracted interest from researchers for several decades. The primary challenges in solving linear fractional programming (LFP) arise from a lack of useful properties (convexity or otherwise) and from the number of ratios and the dimension of decision space. Jiao et al gave a new interval reduced branch-and-bound algorithm for solving the global problem of linear ratio and denominator outcome space [14]. Hu et al proposed a new branch-and-bound algorithm for solving the low-dimensional linear fractional programming [16]. By adopting the exponent transformation technique, Jiao et al proposed a branch and bound algorithm of three-level linear relaxation to solve the generalized polynomial ratios problem with coefficients [19]. A new branch-and-bound algorithm based on the branch of output-space is proposed for globally solving the LFP problem. The method of this paper is briefly reviewed, and the extension of this method to multi-objective fractional programming is prospected

The Equivalence Problem of LFP

A New Linear Relaxation Technique

Branching Process

Output-Space Branch-and-Bound Algorithm and Its Convergence

Numerical Examples

Conclusions