[PDF] Approximate-Karush-Kuhn-Tucker Conditions and Interval Valued

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ValuedVectorVariationalInequalities

KINKEUNGLAI

CollegeofEconomics

ShenzhenUniversity

Shenzhen,518060

CHINA mskklai@outlook.com

SANJEEVKUMARSINGH

DepartmentofMathematics

InstituteofScience

BanarasHinduUniversity

Varanasi,221005

INDIA sanjeevk.singh1@bhu.ac.in

SHASHIKANTMISHRA

DepartmentofMathematics

InstituteofScience

BanarasHinduUniversity

Varanasi,221005

INDIA shashikant.mishra@bhu.ac.in

1Introduction

whichisconvergingtosomexwiththecondition thatxkisaKarush-Kuhn-Tucker(KKT)pointfor xkconvergestozero.

TheKKTconditions[12]playavitalrole

straintqualification[14].

LionsandStampacchia[13]introducedthecon-

affinityassumptions. valvaluedobjectivefunction. functions.

Inthispaper,weintroduceApproximateKKT

associatedwiththevectorvariationalinequalityprob-WSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.28Kin Keung Lai, Sanjeev Kumar Singh,

Shashi Kant Mishra

E-ISSN: 2224-2880

280

Volume 19, 2020

lem. In addition to that, we establish relationship be- tween vector variational inequality problems and mul- tiobjective interval valued Optimization problems un- der the assumption ofLUconvex smooth and nons- mooth objective functions.

Motivated by the work of Wu [3], Andreaniet al.

[16], Haeser and Schuverdt [10], Mastroeni [11] and

Giorgiet al.[9], we introduce Approximate-Karush-

Kuhn-Tucker optimality conditions for interval val- ued objective function and discuss the sufficiency of

AKKTconditions for the interval valued problems

and generalize its definition to the structure of vector variational inequality problems.

The organization of this paper is as follows: In

Section 2, we collected some basic definitions and re- sults. In Section 3, we develop sequential optimality conditions asAKKTconditions for interval valued vector variational inequality problem and proved suf- ficiency withLUconvex and affine conditions.

2 Preliminaries

2.1 Interval Analysis

We collect some basic concepts and essential defini- tions related to interval valued functions.

We denote byIthe class of all closed intervals in

R :LetU= [uL;uU];whereuLanduUdenotes the lower and upper bounds ofU;respectively. Let

U= [uL;uU]andV= [vL;vU]be inI;then, we

have (i)U+V=fu+v:u2U; v2Vg= [uL+ v

L;uU+vU];

(ii)U=fu:u2Ug= [uU;uL]; (iii)UV=U+ (V) = [uLvU;uUvL]; (iv)tU=ftu:u2Ug=[tuL;tuU]ift0 tu

U;tuL]fort <0

wheretis a real number. we refer to Moore [5], for further details on interval analysis .

Suppose thatURnandVRn;then the Haus-

dorff metric betweenUandVis denoted and defined by d

H(U;V) = maxn

sup u 2

Uinfv2Vkuvk;sup

v2Vinfu2Ukuvko wherek:kis an Euclidean norm.

LetU= [uL;uU]andB= [vL;vU]be two closed

intervals, then it is easy to prove that d H(U;V) = maxfjuLvLj;juUvUjg:LetfUn= [uLn;uUn]gandUbe closed intervals inR; then the sequence of closed intervalfUngconverges toU;if for every >0;there existsN >0such that, forn > N;we havedH(Un;U)< :Wu [2] proved that lim n !1Un=Uif and only iflimn!1uLn=uLand lim n !1uUn=uU:

The function:Rn! Iis called interval val-

ued function, this means(x) =(x1;;xn)is a closed interval inRfor eachx2Rn: can be written as(x) = [L(x);U(x)];whereLandU are two real valued functions defined onRnsuch that

L(x)U(x);8x2Rn:

Wu [2] discussed limit and continuity of interval val- ued functions. Letbe an interval valued function defined onRnandU= [uL;uU]be an interval inR; we say lim x a(x) =U;if and only iflimx!aL(x) =uLand lim x aU(x) =uU:

The interval valued functionfdefined onRnis said

to be continuous ata2Rnif lim x a(x) =(a):

Proposition 2.1[3] Supposeis an interval valued

function dened onRn;thenis continuous ata2 R nif and only ifLandUare continuous ata:

Denition 2.1[3] SupposeKis an open set inRThe

interval valued function:K! Iwith(x) =

L(x);U(x)]is called weakly differentiable atx0if

the real valued functionsLandUare differentiable atx0(in the ordinary sense).

ForU;V2 I;if there exists aW2 Isuch thatU=

V+W;thenWis called the Hukuhara difference of

UandV. Also,Wcan be written asW=UV;

considering the Hukuhara differenceWexists, which means thatuLvLuUvUandW= [uL v

L;uUvU]:

Proposition 2.2[3] SupposeU= [uL;uU]andV=

v

L;vU]are two closed intervals inR:IfuLvL

u

UvU;then the Hukuhara differenceWexists and

W= [uLvL;uUvU]:

Denition 2.2[3] SupposeKis an open set inR:

The interval valued function:K! Iis calledWSEAS TRANSACTIONS on MATHEMATICS DOI: 10.37394/23206.2020.19.28Kin Keung Lai, Sanjeev Kumar Singh, Shashi Kant MishraE-ISSN: 2224-2880281Volume 19, 2020 Hdifferentiable atx0if there exists a closed interval

U(x0)2 Isuch that the limits

lim h 0 +(x

0+h)(x0)h

and lim h 0 +(x

0)(x0h)h

both exist and equal toU(x0);which is called theH derivative ofatx0:

2.2 Solution Concepts

SupposeU= [uL;uU]andV= [vL;vU]are two

closed intervals inR:We writeULUVif and only ifuLvLanduUvU:

Consider multiobjective programming problem with

multiple interval valued objective functions

MIV P) min(x) = (1(x);;p(x))

subject tox= (x1;;xn)2KRn; where each k(x) = [Lk(x);Uk(x)]is an interval valued func- tion fork= 1;;p:

We writeULUVif and only ifULUVand

U6=V:We sayU= (U1;;Up)is an inter-

val valued vector if each componentUk= [uLk;uUk] is closed interval fork= 1;;p:SupposeU= U

1;;Up)andV= (V1;;Vp)be two interval

valued vectors. We writeULUVif and only if U kLUVk8k= 1;;p;andULUVif and only ifUkLUVk;8k= 1;;pandUqLUVqfor at least oneq:Supposexis a feasible solution of(MIV P);then(x)is an interval valued vector. The concepts of Pareto optimal (efficient) solution is given below.

Denition 2.3[2] Supposex0is a feasible solution

to the problem(MIV P): (i)x0is said to be an efcient solution to the prob- lem(MIV P)if there exists noxsuch that (x)LU(x0). (ii)x0is said to be a strong efcient solution to the problem(MIV P)if there exists noxsuch that (x)LU(x0). (iii)x0is said to be a weak efcient solution to the problem(MIV P)if there exists noxsuch that k(x)LUk(x0)8k= 1;;p:

Denition 2.4[2] Supposex0is feasible solution of

the problem(MIV P): x0is said to be local weak ef- cient solution of the problem(MIV P);if there existsa neighborhoodNofx0such that for allx2K\N; then the following cannot satisfy for anyk= 1;;p k(x)LUk(x0): Zhanget al.[19] defined the concepts of local quasi efficient and local weak quasi efficient solutions for the problem(MIV P):

Denition 2.5Supposex0is feasible solution of the

problem(MIV P): x0is said to be local quasi ef- cient solution of the problem(MIV P);if there exist

2int(Rp

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