一般区间线性系统的(Z，z)解

金江红，李 炜，李好好

(1.杭州电子科技大学运筹与控制研究所，浙江 杭州 310018；2.浙江财经大学数据科学学院，浙江 杭州 310018)

一般区间线性系统的(Z，z)解

金江红1，李 炜1，李好好2

(1.杭州电子科技大学运筹与控制研究所，浙江 杭州 310018；2.浙江财经大学数据科学学院，浙江 杭州 310018)

0 引 言

1 预备知识

全体m×n维区间矩阵的集合记为IRm×n，n维区间向量的全体记为IRn.

对于任意向量x∈Rn，其符号向量sgnx定义为

2 两个引理

著名的Oettli-Prager定理给出了区间线性方程组AIx=bI弱解的特征[1]，为得到本文主要结果，首先给出更为一般的区间线性方程组AIx-bI=cI弱解的特征.众所周知，IRn按照区间加法运算只构成一个交换幺半群(而不能构成群)，所以AIx-bI=cI弱解的特征不可能直接利用Oettli-Prager定理通过简单的移项得到.

引理1x∈Rn是AIx-bI=cI的弱解当且仅当x满足

由文献[1]中命题2.27可得如下引理：

引理2 设AI∈IRm×n，bI∈IRm，x∈Rn，则有

3 主要结果

有Ax≤b成立.

(2)

(3)

证明 给定Z，z.首先考虑

AIx+BIy=bI,x≥0.

(4)

(5)

又根据引理2，易知

从而由式(5)知

简化得到

对于CIx+DIy≤dI证明方法类似，不再赘述.因此得到式(3)，证毕.

4 结束语

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[6]LI W, LIU P Z, LI H H. Checking weak optimality of the solution to interval linear program in the general form[J]. Optimization Letters, 2016,10(1):77-88.

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(Z,z)-Solutions to General Interval Linear Systems

JIN Jianghong1, LI Wei1, LI Haohao2

(1.InstituteofOperationalResearchandCybernetic,HangzhouDianziUniversity,HangzhouZhejiang310018,China; 2.SchoolofDataSciences,ZhejiangUniversityofFinanceandEconomics,HangzhouZhejiang310018,China)

How to present the characterization of the various solution sets is an important research subject in the field of interval analysis and interval optimization. Jiri Rohn proposed the characterization of (Z,z)-solutions of interval linear equations. However, the characteristics of (Z,z)-solutions of interval linear inequalities and more general interval linear systems have not been studied. In this paper, it first generalizes the far-reaching Oettli-Prager theorem, and then establishes the characterization of a new interval linear system, in the end, necessary and sufficient conditions of the (Z,z)-solutions for interval linear inequalities and general interval linear systems are given.

interval linear systems; interval matrix; (Z,z)-solutions; Hadamard product

10.13954/j.cnki.hdu.2017.03.018

2016-06-27

国家自然科学基金资助项目(11526184)；浙江省大学生科技创新活动计划(新苗计划)资助项目(2016R407079)

金江红(1992-)，女，河南商丘人，硕士研究生，数学规划.通信作者：李炜教授，E-mail:weilihz@126.com.

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1001-9146(2017)03-0087-04