金江紅，李 煒，李好好

(1.杭州電子科技大學(xué)運(yùn)籌與控制研究所，浙江 杭州 310018；2.浙江財(cái)經(jīng)大學(xué)數(shù)據(jù)科學(xué)學(xué)院，浙江 杭州 310018)

一般區(qū)間線(xiàn)性系統(tǒng)的(Z，z)解

金江紅1，李 煒1，李好好2

(1.杭州電子科技大學(xué)運(yùn)籌與控制研究所，浙江 杭州 310018；2.浙江財(cái)經(jīng)大學(xué)數(shù)據(jù)科學(xué)學(xué)院，浙江 杭州 310018)

0 引 言

1 預(yù)備知識(shí)

全體m×n維區(qū)間矩陣的集合記為IRm×n，n維區(qū)間向量的全體記為IRn.

對(duì)于任意向量x∈Rn，其符號(hào)向量sgnx定義為

2 兩個(gè)引理

著名的Oettli-Prager定理給出了區(qū)間線(xiàn)性方程組AIx=bI弱解的特征[1]，為得到本文主要結(jié)果，首先給出更為一般的區(qū)間線(xiàn)性方程組AIx-bI=cI弱解的特征.眾所周知，IRn按照區(qū)間加法運(yùn)算只構(gòu)成一個(gè)交換幺半群(而不能構(gòu)成群)，所以AIx-bI=cI弱解的特征不可能直接利用Oettli-Prager定理通過(guò)簡(jiǎn)單的移項(xiàng)得到.

引理1x∈Rn是AIx-bI=cI的弱解當(dāng)且僅當(dāng)x滿(mǎn)足

由文獻(xiàn)[1]中命題2.27可得如下引理：

引理2 設(shè)AI∈IRm×n，bI∈IRm，x∈Rn，則有

3 主要結(jié)果

有Ax≤b成立.

(2)

(3)

證明 給定Z，z.首先考慮

AIx+BIy=bI,x≥0.

(4)

(5)

又根據(jù)引理2，易知

從而由式(5)知

簡(jiǎn)化得到

對(duì)于CIx+DIy≤dI證明方法類(lèi)似，不再贅述.因此得到式(3)，證畢.

4 結(jié)束語(yǔ)

[1]FIEDLER M, ROHN J, NEDOMA J, et al. Linear optimization problems with inexact date[M]. New York: Springer,2006:35-66.

[2]SHARY S P. A New Technique in Systems Analysis Under Interval Uncertainty and Ambiguity[J]. Reliable Computing, 2002,8(5):321-418.

[3]HLADIK M. AE solutions and AE solvability to general interval linear systems[J].Linear Algebra & its Applications, 2015,465:221-238.

[4]LI W, WANG H P, WANG Q. Localized solutions to interval linear equations[J]. Journal of Computational and Applied Mathematics, 2013,238(15):29-38.

[5]LI W, LUO J, WANG Q, et al. Checking weak optimality of the solution to linear programming with interval right-hand side[J]. Optimization Letters, 2014,8(4):1287-1299.

[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.

[7]LI W, XIA M, LI H. New method for computing the upper bound of optimal value in interval quadratic program[J]. Journal of Computational and Applied Mathematics, 2015,288:70-80.

[8] ROHN J. A Manual of Results on Interval Linear Problems[EB/OL].(2012-05-07) [2016-05-26]. http://uivtx.cs.cas.cz/～rohn/!manual.pdf.

(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

國(guó)家自然科學(xué)基金資助項(xiàng)目(11526184)；浙江省大學(xué)生科技創(chuàng)新活動(dòng)計(jì)劃(新苗計(jì)劃)資助項(xiàng)目(2016R407079)

金江紅(1992-)，女，河南商丘人，碩士研究生，數(shù)學(xué)規(guī)劃.通信作者：李煒教授，E-mail:weilihz@126.com.

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1001-9146(2017)03-0087-04