擾動(dòng)變分不等式的序下半連續(xù)性

胡志軍，趙素平

(1.廣西師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，桂林 541004;

2.重慶大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，重慶 401331)

擾動(dòng)變分不等式的序下半連續(xù)性

胡志軍1，趙素平2

(1.廣西師范大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，桂林 541004;

2.重慶大學(xué)數(shù)學(xué)與統(tǒng)計(jì)學(xué)院，重慶 401331)

對(duì)向量?jī)?yōu)化準(zhǔn)則下集值映射(SMVOR)半連續(xù)性和集優(yōu)化準(zhǔn)則下集值映射(SMSOR)序半連續(xù)性進(jìn)行了研究。通過(guò)舉例指出了SMVOR下半連續(xù)與SMSOR序下半連續(xù)沒(méi)有關(guān)系，在一定條件下由SMVOR上半連續(xù)性得到了SMSOR序下半連續(xù)。重點(diǎn)討論了集序關(guān)系意義下廣義向量擬變分不等式(GVQVI)的序下半連續(xù)性。通過(guò)對(duì)集序關(guān)系意義下序下半連續(xù)的進(jìn)一步研究，得到了擾動(dòng)變分不等式問(wèn)題的解集在空間上序下半連續(xù)的充分條件，并用不同方法證明了該充分條件。

集序關(guān)系;序下半連續(xù);擾動(dòng)變分不等式;半連續(xù)

Lions等建立了初期變分不等式理論，Giannessi在有限維空間中引入了向量變分不等式?；谙蛄?jī)?yōu)化的發(fā)展，向量變分不等式問(wèn)題在理論和應(yīng)用上日益完善［1-2］，被廣泛應(yīng)用于工程、經(jīng)濟(jì)等領(lǐng)域。向量變分不等式的穩(wěn)定性、適定性和半連續(xù)性等被廣泛研究［3-5］。集優(yōu)化問(wèn)題是以函數(shù)的每一個(gè)像集作為一個(gè)整體來(lái)比較像集間的優(yōu)劣關(guān)系。Kuroiwa在文獻(xiàn)［6-7］中介紹了多種集序關(guān)系及其性質(zhì)。Jahn在文獻(xiàn)［8］中引入了新的集序關(guān)系，并給出了序下半連續(xù)的概念。Madeda在文獻(xiàn)［9］中給出了集值映射在整個(gè)空間序半連續(xù)的條件。

1 預(yù)備知識(shí)

2 SMVOR半連續(xù)與SMSOR≤l序下半連續(xù)的關(guān)系

3 集序關(guān)系意義下的變分不等式問(wèn)題

3.1 擾動(dòng)廣義向量變分不等式模型

3.2 下面討論≤l序關(guān)系意義下一種特殊的變分不等式問(wèn)題

4 結(jié)束語(yǔ)

本文給出了變分不等式在集序關(guān)系意義下的表示，并用2種方法證明了擾動(dòng)變分不等式在空間上≤l下半連續(xù)的充分條件，為繼續(xù)研究集序關(guān)系意義下的變分不等式開(kāi)拓了思路。

［1］Chen G Y，Li S J.Existence of Solutions for a Generalized Vector Quasivariational Inequality［J］.Optim Theory Appl，1996，90 (1):321-334.

［2］Yang X Q，Goh C J.On Vector Variational Inequalities:Application to Vector Equilibria［J］.Optim.Theory Appl，1997，95(2): 431-443.

［3］Li S J，Zhang W Y.Hadamard Well-posed Vector Optimization Problems［J］.Optim Theory Appl，2010，46:383-393.

［4］曾靜.向量?jī)?yōu)化及相關(guān)問(wèn)題解的存在性和適定性研究［D］.重慶:重慶大學(xué)，2011.

［5］陳光亞.向量?jī)?yōu)化問(wèn)題某些基礎(chǔ)理論及其發(fā)展［J］.重慶師范大學(xué)學(xué)報(bào):自然科學(xué)版，2005，22(3):6-9.

［6］Kuroiwa D.On Set-valued Optimization［J］.Nonlinear Anal，2001，47:1395-1400.

［7］Kuroiwa D.The natural Criteria in Set-valued Optimization［J］.RIMS Kokyuroku，1998，1031:85-90.

［8］Jahn J，Ha T X D.New Order Relations in Set Optimization［J］.Optim.Theory Appl，2011，148:209-236.

［9］Maeda T.On Optimization Problems with Set-valued Objective Maps:Existence and Optimality［Z］.Optim Theory Appl，2011.

［10］Aubin J P，Ekland I.Applied Nonlinear Analysis［M］.New York:NY，1984:518.

［11］Hogan W W.Point-to-set Maps in Mathematical Programming［J］.SI AM Rev，1973，15:591-603.

(責(zé)任編輯 劉舸)

Lower Semicontinuity with Set-relations of Perturbed Generalized Vector Quasivational Inequality Problems

HU Zhi-jun1，ZHAO Su-ping2
(1.College of Mathematics and Statistics，Guangxi Normal University，Guilin 541004，China; 2.College of Mathematics and Statistics，Chongqing University，Chongqing 401331，China)

The semicontinuity of set-valued mapping under set-valued optimization rules and the lower semicontinuity with set-relations under set optimization rules have been studied.By some examples，one can find the semicontinuity of set-valued mapping under set-valued optimization rules has no relations with the lower semicontinuity with set-relations under set optimization rules.It is easy to get the relation between the upper semicontinuity with the lower semicontinuity with set-relations.In addtition，the key point is discussing the lower semicontinuity with set-relations of GVQVI.By analyzing the lower semicontinuity with set-relations，one can conclude the sufficient conditions of the lower semicontinuity with set-relations of perturbed GVQVI and prove the sufficient condition by two different ways.

set-relations;lower semicontinuity with set-relations;perturbed vector vational inequality;semicontinuous

O175.8

A

1674-8425(2014)06-0122-04

10.3969/j.issn.1674-8425(z).2014.06.024

2013-12-21

廣西教育廳科研立項(xiàng)項(xiàng)目(201106LX047)

胡志軍(1981—)，男，四川成都人，碩士研究生，講師，主要從事最優(yōu)化理論和分形圖形壓縮方面的研究。

胡志軍，趙素平.擾動(dòng)變分不等式的序下半連續(xù)性［J］.重慶理工大學(xué)學(xué)報(bào):自然科學(xué)版，2014(6):122 -125.

format:HU Zhi-jun，ZHAO Su-ping.Lower Semicontinuity with Set-relations of Perturbed Generalized Vector Quasivational Inequality Problems［J］.Journal of Chongqing University of Technology:Natural Science，2014 (6):122-125.