© 1994-2009 China Academic Journal Electronic Publishing House. All rights reserved. http://www.cnki.net
第 17卷 第 2期
2008年 4月
云南民族大学学报
(
自然科学版
)
Journal of Yunnan Nationalities University
(
Natural Sciences Edition
)
Vol. 17 No. 2
Ap r. 2008
3 收稿日期 : 2007 - 12 - 03.
基金项目 :国家自然科学基金资助项目
(
10671057
)
;河南科技学院自然科学基础计划研究项目
(
06055, 06054
)
.
作者简介 :焦红伟
(
1978～
)
,男 ,硕士 ,讲师 ,主要研究方向 :最优化理论与算法、全局优化.
A Determ in istic Algorithm for Solving a Series of
Fractional Programm ing Problem s
Jiao Hongwei
1
Guo Yunrui
1
Chen Yongqiang
2
(
1. Department ofMathematics, Heπnan Institute of Science and Technology, Xinxiang 453003, China;
2. College ofMathematics and Information Science, Heπnan Normal University, Xinxiang 453007, China
)
Abstract: In this paper a determ inistic algorithm is p roposed for a global solving of a series of fractional p ro2
gramming p roblem s
(
P
)
. By utilizing linearization technique the relaxation linear programm ing
(
RLP
)
about the
(
P
)
is established. The proposed algorithm is convergent to the global m inimum of
(
P
)
through the successive re2
finement of linear relaxation of the feasible region of objective function and solutions of a series of RLP. And finally
the numerical experiment is given to illustrate the feasibility of the presented algorithm.
Key words: fractional p rogramm ing; global op tim ization; linearization technique; branch2and2bound
求解一类比式规划问题的确定性算法
焦红伟
1
郭运瑞
1
陈永强
2
(
1. 河南科技学院 数学系 , 河南 新乡 453003; 2. 河南师范大学 数学与信息科学学院 , 河南 新乡 453007
)
摘 要 对一类比式规划问题
(
P
)
提出一确定性全局优化算法. 利用线性化技术建立了问题
(
P
)
的松弛线性规划
(
RLP
)
,
通过对目标函数线性松弛可行域的逐次细分以及一系列
(
RLP
)
的求解过程 ,提出的算法收敛问题
(
P
)
全局最优解. 最终数值
实验表明了提出方法的可行性.
关键词 比式规划 ; 全局优化 ; 线性化技术 ; 分枝定界
【中图分类号 】O22112 【文献标识码 】A 【文章编号 】1672—8513
(
2008
)
02 - 0106 - 03
0 Introduction
Consider the following a class of fractional p ro2
gramming p roblem s:
(
P
)
:
m in max
1Φ jΦ p
n
1
(
x
)
d
1
(
x
)
, …,
n
p
(
x
)
d
p
(
x
)
s1t1 AxΦ b
where pΕ 2, n
j
(
x
)
, d
j
(
x
)
are all linear affine functions
on R
n
, and n
j
(
x
)
> 0, d
j
(
x
)
≠0 for all x: { x |AxΦ b}.
Problem
(
P
)
have broad app lication in economy,
transaction, and so on, but since denom inators and nu2
merators of the ratios are all linear affine functions,
and there possibly existmany local op timal solutions for
problem
(
P
)
, so that it is more difficult to be solved.
In last decades, many algorithm s have been proposed
for solving fractional programm ing problem
(
P
)
with
assumption that n
j
(
x
)
Ε 0, d
j
(
x
)
> 0
(
Ref. [ 1 - 5 ]
)
.
But to our knowledge, there exist few algorithms for
solving fractional programm ing problem
(
P
)
which sat2
isfying that d
j
(
x
)
≠0 and n
j
(
x
)
may be arbitrary val2
ues. In this paper a deterministic algorithm is p roposed
for globally solving problem
(
P
)
. By utilizing new lin2
earization technique, the p roposed algorithm is conver2
gent to the global minimum through the successive re2
finement of the linear relaxation of feasible region of
the objective function and solutions of a series of
601
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