R E S E A R C H
Open Access
On minimax programming problems
involving right upper-Dini-derivative
functions
Anurag Jayswal
1, Izhar Ahmad
2, Krishna Kummari
1and Suliman Al-Homidan
2**Correspondence: [email protected] 2Department of Mathematics and
Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia
Full list of author information is available at the end of the article
Abstract
In this paper, we derive necessary and sufficient optimality conditions for a general minimax programming problem involving some classes of generalized convexities with the tool-right upper-Dini-derivative. Moreover, using the concept of optimality conditions, Mond-Weir type duality theory has been developed for such a minimax programming problem.
MSC: 26A51; 49J35; 90C32
Keywords: minimax programming; upper-Dini-derivative; optimality; duality
1 Introduction
The minimax approach to optimization theory certainly is not new. It takes its origins in von Neumann’s game theory. The broad spectrum of existing results and applications of minimax theory in the field of optimization is captured in the book edited by Duet al.
[]. Starting with work of Schmittendorf [], minimax programming problems have been studied by several authors; for example, see [–] and the references cited therein.
Convexity is a sufficient but not necessary condition for many important results of math-ematical programming, since there are diverse extensions of the notion of convexity bear-ing the same properties. Moreover, it is well known that a function is convex iff its re-striction to each line segment in its domain is convex. This property inspired Ortega and Rheinboldt [] to introduce an important generalization of convex functions by replacing a line segment joining two points by a continuous arc and called them arcwise connected functions defined on arcwise connected sets.
Following the idea of arcwise convexity, Avriel and Zang [] introducedQ-connected (QCN) functions andP-connected (PCN) functions and also they have discussed neces-sary and sufficient local-global minimum properties of these functions. Some elementary properties of these functions in terms of their directional derivatives have been studied by Bhatia and Mehra []. Bhatia and Mehra [] also established optimality conditions for scalar-valued nonlinear programming problems involving these functions.
To relax the definition of arcwise convexity in terms of directional derivative recently Yuan and Liu [] introduced the concept of (α,ρ)-right upper-Dini-derivative locally ar-cwise connected with respect to the arcHand established optimality and duality results for a nonlinear multiobjective programming problem. In this paper, we use generalized
convex functions, in terms of the right upper-Dini-derivative to derive necessary and suf-ficient optimality conditions for a general minimax programming problem and duality results for its Mond-Weir type dual model.
This paper is structured as follows: Some preliminary concepts and properties regarding generalized convex functions are given in Section . In Section , we establish necessary and sufficient optimality conditions for a general minimax programming problem involv-ing generalized convex functions. In Section , we establish appropriate duality theorems for a Mond-Weir type dual problem. Finally, in Section we summarize our main results and also point out some further research opportunities.
2 Preliminaries
LetRndenote then-dimensional Euclidean space,Rn
+its nonnegative orthant andX⊂Rn.
For a nonempty setQin a topological vector paceE,Q¯ denote the closure ofQand
Q∗=ν∈E∗|ν(q)≥,∀q∈Q
denotes the dual cone ofQ, whereE∗is the dual space ofE.
For some nonempty subsetY, letRY=YRdenote the product space in a product topol-ogy. Then the topological dual space ofRYis the generalized finite sequence space consist-ing of all the functionsu:Y→Rwith finite support []. The setRY
+=YR+denote the
convex cone of all nonnegative functions onY. Then the topological dual ofRY
+ is given
by
RY+∗=∧
=λ= (λy)y∈Y|∃a finite setY⊆Ysuch thatλy= ,∀y∈Y\Y
andλy≥,∀y∈Y
.
Now, we recall some well-known results and concepts which will be used in the sequel.
Definition .[] A setX⊂Rnis said to be an arcwise connected set if, for everyx
∈X,
x∈X, there exists a continuous vector-valued functionHx,x: [, ]→X, called an arc,
such that
Hx,x() =x, Hx,x() =x.
Definition .[] Letϕbe a real-valued function defined on an arcwise connected set
X⊂Rn. Letx
,x∈X, andHx,x be the arc connectingx andxinX. The right
upper-Dini-derivative ofϕwith respect toHx,x(t) att= is defined as follows:
(dϕ)+Hx,x
+= lim t→+sup
ϕ(Hx,x(t)) –ϕ(x)
t . ()
Definition .[] A setX⊂Rnis said to be locally arcwise connected atx¯if for any
x∈Xandx=x¯there exist a positive numbera(x,x¯), with <a(x,x¯)≤, and a continuous arcH¯x,xsuch thatHx¯,x(t)∈Xfor anyt∈(,a(x,x¯)).
The setXis locally arcwise connected onXifXis locally arcwise connected at anyx∈X.
Definition .[] LetX⊂Rnbe a locally arcwise connected set andϕ:X→Rbe a real function defined onX. The functionϕis said to be (α,ρ)-right upper-Dini-derivative locally arcwise connected with respect toHatx¯, if there exist real functionsα:X×X→R, ρ:X×X→Rsuch that
ϕ(x) –ϕ(x¯)≥α(x,x¯)(dϕ)+H¯x,x
++ρ(x,x¯), ∀x∈X.
Ifϕis (α,ρ)-right upper-Dini-derivative locally arcwise connected with respect toHatx¯
for anyx¯∈X, thenϕis called (α,ρ)-right upper-Dini-derivative locally arcwise connected with respect toHonX.
Remark . It revealed by an example given in [] that there exists a function, which is (α,ρ)-right upper-Dini-derivative locally arcwise connected but neitherd-ρ-(η,θ)-invex [] nord-invex [] nor directional differentially B-arcwise connected [].
Now we define the notions ofρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected, strictlyρ-generalized-pseudo-right upper-Dini-derivative locally arc-wise connected andρ-generalized-quasi-right upper-Dini-derivative locally arcwise con-nected functions.
Definition . The functionϕ:X→Ris said to beρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect toH) atx¯, if there exists a real functionρ:X×X→Rsuch that
(dϕ)+H¯x,x
+≥–ρ(x,x¯) ⇒ ϕ(x)≥ϕ(x¯), ∀x∈X, equivalently
ϕ(x) <ϕ(x¯) ⇒ (dϕ)+H¯x,x
+< –ρ(x,x¯), ∀x∈X.
The functionϕ:X→Ris said to beρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect to H) onX if it isρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect toH) at anyx¯∈X.
The following example shows that there exists a function which is
ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected but not (α,ρ)-right upper-Dini-derivative locally arcwise connected with respect to the arcH.
Example . LetX= (–, ) and the functionϕ:X→Rbe defined by
ϕ(x) =
⎧ ⎨ ⎩
|x|sin
x; ifx∈(–, )∪(, ),
For any,x,y∈R, defining the arcH: [, ]→Rby
Hy,x(t) =tx+ ( –t)y, t∈[, ].
Note that, by the definition of right upper-Dini-derivative by (), forx∈(–, )∪(, ) we have
(dϕ)+H,x
+=|x|.
Letρ:X×X→Rbe defined by
ρ(x,y) =
⎧ ⎨ ⎩
|x|; ifx∈(–, )∪(, ), ; ifx= .
Now, forx¯= , it follows that
(dϕ)+H¯x,x
+≥–ρ(x,x¯) ⇒ ϕ(x)≥ϕ(x¯), ∀x∈X.
This means thatϕ isρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect toH) atx¯= . Butϕ is not a (α,ρ)-right upper-Dini-derivative locally arcwise connected with respect to same arc H and ρ at x¯= because for x∈
(–, )∪(, ) andα(x,x¯) = , we can see that
ϕ(x) –ϕ(x¯) –α(x,x¯)(dϕ)+Hx¯,x
+–ρ(x,x¯) < .
Definition . The functionϕ:X→Ris said to be strictlyρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect toH) atx¯, if there exists a real functionρ:X×X→Rsuch that
(dϕ)+H¯x,x
+≥–ρ(x,x¯) ⇒ ϕ(x) >ϕ(x¯), ∀x∈X,x=x¯,
equivalently
ϕ(x)≤ϕ(x¯) ⇒ (dϕ)+H¯x,x
+< –ρ(x,x¯), ∀x∈X,x=x¯.
The functionϕ:X→Ris said to be strictlyρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect toH) onXif it is strictly ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected (with respect toH) at any
¯ x∈X.
Definition . The function ϕ:X→Ris said to be ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected with respect toHatx¯, if there exists a real func-tionρ:X×X→Rsuch that
ϕ(x)≤ϕ(x¯) ⇒ (dϕ)+H¯x,x
equivalently
(dϕ)+H
¯
x,x
+> –ρ(¯ x,x¯) ⇒ ϕ(x) >ϕ(x¯), ∀x∈X.
The functionϕ:X→Ris said to beρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect toH) onXif it isρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect toH) at anyx¯∈X.
The next example shows that there exists a function which is ρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected but neither (α,ρ)-right upper-Dini-derivative locally arcwise connected nor ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected with respect to the arcH.
Example . LetX= (–π
,
π
) and the functionϕ:X→Rbe defined by
ϕ(x) =
⎧ ⎨ ⎩
π|x|sinπx – π|x|; ifx∈(–π, )∪(,π),
; ifx= .
For any,x,y∈R, defining the arcH: [, ]→Rby
Hy,x(t) =tx+ ( –t)y, t∈[, ].
Clearly, forx∈(–π, )∪(,π) we have
(dϕ)+H,x
+= –π|x|.
Letρ:X×X→Rbe defined by
ρ(x,y) =
⎧ ⎨ ⎩
π|x|; ifx∈(–π, )∪(,π),
; ifx= .
Now, we can easily verify thatϕisρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with respect toH) atx¯= . However, forx∈(–π, )∪(,π),α(x,x¯) = andx¯= , we can deduce that
ϕ(x) –ϕ(x¯) –α(x,x¯)(dϕ)+Hx¯,x
+–ρ(x,x¯) < ,
and
ϕ(x) –ϕ(x¯) < implies (dϕ)+Hx¯,x
++ρ(¯ x,x¯) = .
Definition .[] A functionf :X→Ris called preinvex (with respect toη:X×X→ Rn) onXif there exists a vector-valued functionηsuch that
fu+tη(x,u)≤tf(x) + ( –t)f(u)
holds for allx,u∈Xand anyt∈[, ].
Definition .[] A functionf :X→Ris said to be convexlike if for anyx,y∈Xand ≤θ≤, there isz∈Xsuch that
f(z)≤θf(x) + ( –θ)f(y).
Remark . The convex and the preinvex functions are convexlike functions.
In the next section we will use the following version of Theorem . from [].
Lemma . Let G:X×Y→R andψ:X→R,where X and Y are arbitrary nonempty sets.Let the pair(G,ψ)be convexlike on X.Assume that for some neighborhood U ofin RY and a constantν> ,the set
∩ ¯U×(–∞,ν]is a nonempty closed subset of RY×R,
where
=
(u,r)|u:Y→R and∃x∈X such thatψ(x)≤r,G(x,y)≤u(y),∀y∈Y.
Then exactly one of the following systems is solvable: (I) G(x,y)≤,∀y∈Y,ψ(x) < ,
(II) ∃an integers> ,scalarsλi≥,≤i≤s,μ≥and vectorsyi∈Y,≤i≤s,such
that(λ, . . . ,λs,μ)= and si=λiG(x,yi) +μψ(x)≥.
3 Optimality conditions
Consider the following general minimax programming problem:
Minimize max
y∈Y f(x,y)
subject tog(x)≤,x∈X,
(P)
wheref :X×Y→R,g= (g,g, . . . ,gm) :X→Rm,Xis an open arcwise connected subset ofRn,Yis a compact subset ofRmandf(x,·) is continuous onYfor everyx∈X.X={x∈
X|gj(x)≤,∀≤j≤m}denote the set of feasible solutions of (P). Forx∈X, we define
I(x) =j|gj(x) = ,
J(x) ={, , . . . ,m} \I(x),
Y(x) =
y∈Yf(x,y) =sup z∈Y
f(x,z)
.
upper-Dini-derivatives of the functionsf(·,y),gj(·),j∈ {, , . . . ,m}with respect to an arc
Hx¯,xatt= exist∀¯x,x∈X,∀y∈Y, and (df)+(Hx¯,x(+),·) is continuous onY,∀¯x,x∈X.
Also assume thatgj(·), ≤j≤mis continuous onX.
The following lemma can be proved without difficulty on the same lines as in Lemma . (Mehra and Bhatia []).
Lemma . Letx be an optimal solution of¯ (P).Then the system ⎧
⎨ ⎩
(df)+(H¯
x,x(+),y) < , ∀y∈Y(x¯),
(dgj)+(Hx¯,x(+)) < , ∀j∈I(x¯)
()
has no solution x∈X.
We now prove the following theorem by using Lemmas . and ., which gives the necessary optimality conditions for an optimal solution of problem (P).
Theorem .(Necessary optimality conditions) Letx be an optimal solution of¯ (P). Fur-ther, let(df)+(H
¯
x,x(+)), (dgj)+(H¯x,x(+)), j∈I(x¯)be convexlike functions of x on X and
let there exist a neighborhood U of in RY(¯x) and a constant ν= (νj)j
∈I(x¯) such that
(x¯)∩ ¯U×j∈I(¯x)(–∞,νj]is a nonempty closed set,where
(x¯) =(u,r)|r= (rj)j∈I(x¯),u:Y→R and∃x∈X such that f(x,y)≤u(y),∀y∈Y(x¯),
gj(x)≤rj,j∈I(x¯).
Then there exist an integer s> ,scalarsλi≥, ≤i≤s,μj≥, ≤j≤m,and vectors
yi∈Y(x¯), ≤i≤s,such that
s
i=
λi(df)+H¯x,x
+,yi+ m
j=
μj(dgj)+Hx¯,x
+≥, ∀x∈X,
μjgj(x¯) = , ∀≤j≤m,
s
i=
λi+ m
j=
μj= .
Proof Ifx¯is an optimal solution of (P) then, by Lemma ., the system () has no solution
x∈X. But the assumption of Lemma . also holds and since the system () has no solution
x∈X, we obtain the result that there exist an integers> , scalarsλi≥, ≤i≤s,μj≥,
j∈I(x¯), and vectorsyi∈Y(x¯), ≤i≤s, such that
(λ,λ, . . . ,λs,μj)j∈I(x¯)= , ()
and
s
i=
λi(df)+
H¯x,x
+,yi+ j∈I(¯x)
μj(dgj)+
Hx¯,x
+≥, ∀x∈X. ()
Now, we prove the following sufficient optimality conditions for the considered minimax problem (P) under generalized convexity with upper-Dini-derivative concept.
Theorem .(Sufficient optimality conditions) Letx¯∈Xand there exist an integer s> ,
scalarsλi≥, ≤i≤s, si=λi= ,μj≥, ≤j≤m,and vectors yi∈Y(x¯), ≤i≤s,such
thatα(x,x¯) > ,
s
i=
λi(df)+H¯x,x
+,yi+ m
j=
μj(dgj)+Hx¯,x
+≥, ∀x∈X, ()
μjgj(x¯) = , ∀≤j≤m. ()
Also,assume that
(i) for≤i≤s,f(·,yi)is(α,ρi)¯ -right upper-Dini-derivative locally arcwise connected (with respect toH)atx¯,
(ii) for≤j≤m,gj(·)is(α,ρ˘j)-right upper-Dini-derivative locally arcwise connected
(with respect toH)atx¯,
(iii) si=λiρ¯i(x,x¯) + mj=μjρ˘j(x,x¯)≥.
Thenx is an optimal solution of¯ (P).
Proof Suppose to the contrary thatx¯is not an optimal solution of (P). Then there exists anx˜∈Xsuch that
sup z∈Y
f(x˜,z) <sup z∈Y
f(x¯,z).
Further, asyi∈Y(x¯), we have
sup z∈Y
f(x¯,z) =fx¯,yi, ≤i≤s.
Also,yi∈Y, ≤i≤s, we have
fx˜,yi≤sup z∈Y
f(x˜,z), ≤i≤s.
Thus, from the above three inequalities, we get
fx˜,yi<fx¯,yi, ≤i≤s.
Usingλi≥, ≤i≤sand si=λi= , we obtain s
i=
λif
˜ x,yi<
s
i=
λif
¯
x,yi. ()
Forx˜∈X,μj≥, ≤j≤m, we haveμjgj(x˜)≤, which in view of () implies that m
j=
μjgj(x˜)≤ m
j=
Now, by () and () we obtain
s
i=
λif
˜ x,yi+
m
j=
μjgj(x˜) < s
i=
λif
¯ x,yi+
m
j=
μjgj(x¯). ()
On the other hand, from the assumptions thatf(·,yi), ≤i≤sandg
j(·), ≤j≤mare (α,ρ¯i) and (α,ρ˘j)-right upper-Dini-derivative locally arcwise connected (with respect to
H) atx¯, we have
fx˜,yi–fx¯,yi≥α(x˜,x¯)(df)+Hx¯,x˜
+,yi+ρ¯i(x˜,x¯), ≤i≤s, ()
gj(x˜) –gj(x¯)≥α(x˜,x¯)(dgj)+Hx¯,x˜
++ρj(˘ x˜,x¯), ≤j≤m. ()
From () and () together withλi≥, ≤i≤s, andμj≥, ≤j≤m, we get
s
i=
λif
˜ x,yi+
m
j=
μjgj(x˜)
–
s
i=
λif
¯ x,yi+
m
j=
μjgj(x¯)
≥α(x˜,x¯)
s
i=
λi(df)+Hx¯,x˜
+,yi+ m
j=
μj(dgj)+Hx¯,x˜ + + s i=
λiρ¯i(x˜,x¯) + m
j=
μjρ˘j(x˜,x¯).
By () and usingα(x˜,x¯) > , si=λiρ¯i(x˜,x¯) + mj=μjρ˘j(x˜,x¯)≥, it follows that
s
i=
λif
˜ x,yi+
m
j=
μjgj(x˜)
–
s
i=
λif
¯ x,yi+
m
j=
μjgj(x¯)
≥,
which is a contradiction to (). Hencex¯is an optimum solution for (P) and the theorem is
proved.
Theorem .(Sufficient optimality conditions) Letx¯∈Xand there exist an integer s> ,
scalarsλi≥, ≤i≤s, si=λi= ,μj≥, ≤j≤m and vectors yi∈Y(x¯), ≤i≤s,such
that the conditions()and()of Theorem.hold.Also,assume that
(i) si=λif(·,yi)isρ¯-generalized-pseudo-right upper-Dini-derivative locally arcwise
connected(with respect toH)atx¯,
(ii) mj=μjgj(·)isρ˘-generalized-quasi-right upper-Dini-derivative locally arcwise
connected(with respect toH)atx¯, (iii) ρ(¯x,x¯) +ρ(˘x,x¯)≥.
Thenx is an optimal solution of¯ (P).
Proof Suppose to the contrary thatx¯ is not an optimal solution of (P) and following the proof of Theorem ., we have
s
i=
λif
˜ x,yi<
s
i=
λif
which by ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected¯ (with respect toH) of si=λif(·,yi) atx¯, we have
s
i=
λi(df)+H¯x,x˜
+,yi< –ρ(¯x˜,x¯). ()
Forx˜∈X,μj≥, ≤j≤m, we haveμjgj(x˜)≤, which in view of () implies that m
j=
μjgj(x˜)≤ m
j=
μjgj(x¯),
which byρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with˘ respect toH) of mj=μjgj(·) atx¯, we have
m
j=
μj(dgj)+Hx¯,x˜
+≤–ρ(˘x˜,x¯). ()
By () and (), we get
s
i=
λi(df)+
H¯x,x˜
+,yi+ m
j=
μj(dgj)+
Hx¯,x˜
+< –ρ(¯x˜,x¯) –ρ(˘ x˜,x¯)≤,
where the last inequality is according toρ(¯x˜,x¯) +ρ(˘x˜,x¯)≥. Therefore,
s
i=
λi(df)+H¯x,x˜
+,yi+ m
j=
μj(dgj)+Hx¯,x˜
+< ,
which is a contradiction to (). Hencex¯is an optimum solution for (P) and the theorem is
proved.
Theorem .(Sufficient optimality conditions) Letx¯∈Xand there exist an integer s> ,
scalarsλi≥, ≤i≤s, si=λi= ,μj≥, ≤j≤m,and vectors yi∈Y(x¯), ≤i≤s,such
that the conditions()and()of Theorem.hold.Also,assume that
(i) si=λif(·,yi)is strictlyρ¯-generalized-pseudo-right upper-Dini-derivative locally
arcwise connected(with respect toH)atx¯,
(ii) mj=μjgj(·)isρ˘-generalized-quasi-right upper-Dini-derivative locally arcwise
connected(with respect toH)atx¯, (iii) ρ(¯x,x¯) +ρ(˘x,x¯)≥.
Thenx is an optimal solution of¯ (P).
Proof The proof follows along similar lines as the proof of Theorem . and hence is
omit-ted.
Theorem .(Sufficient optimality conditions) Letx¯∈Xand there exist an integer s> ,
scalarsλi≥, ≤i≤s, si=λi= ,μj≥, ≤j≤m,and vectors yi∈Y(x¯), ≤i≤s,such
(i) si=λif(·,yi)isρ¯-generalized-quasi-right upper-Dini-derivative locally arcwise
connected with respect toHatx¯,
(ii) forj∈I(x¯),j=l,gj(·)isρ˘j-generalized-quasi-right upper-Dini-derivative locally
arcwise connected andgl(·)is strictlyρl˘-generalized-pseudo-right
upper-Dini-derivative locally arcwise connected with respect toHatx¯,withμl> , (iii) ρ(¯x,x¯) + mj=μjρ˘j(x,x¯)≥.
Thenx is an optimal solution of¯ (P).
Proof Suppose to the contrary thatx¯ is not an optimal solution of (P) and following the proof of Theorem ., we have
s
i=
λif
˜ x,yi<
s
i=
λif
¯ x,yi,
which byρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected of¯ s
i=λif(·,yi) with respect toHatx¯, we have s
i=
λi(df)+
H¯x,x˜
+,yi≤–ρ(¯x˜,x¯). ()
Since forx˜∈Xand forj∈I(x¯), we have
gj(x˜)≤ =gj(x¯),
which by ρ˘j-generalized-quasi-right upper-Dini-derivative locally arcwise connected of
gj(·),j∈I(x¯),j=land strictlyρ˘l-generalized-pseudo-right upper-Dini-derivative locally arcwise connected ofgl(·) with respect toHatx¯, we have
(dgj)+
Hx¯,x˜
+≤–ρ˘j(x˜,x¯), forj∈I(x¯),j=l, ()
(dgl)+Hx¯,x˜
+< –ρ˘l(x˜,x¯). ()
Sinceμj≥,∀j∈I(x¯),μl> , andμj= forj∈J(x¯), from () and (), we get
m
j=
μj(dgj)+
Hx¯,x˜
+< – m
j=
μjρ˘j(x˜,x¯). ()
By () and (), we get
s
i=
λi(df)+H¯x,x˜
+,yi+ m
j=
μj(dgj)+Hx¯,x˜
+< –ρ(¯x˜,x¯) – m
j=
μjρ˘j(x,x¯)≤,
where the last inequality is according toρ(¯x˜,x¯) + mj=μjρ˘j(x,x¯)≥. Therefore, s
i=
λi(df)+H¯x,x˜
+,yi+ m
j=
μj(dgj)+Hx¯,x˜
+< ,
which is a contradiction to (). Hencex¯is an optimum solution for (P) and the theorem is
4 Duality
This section deals with the duality theorems for the following Mond-Weir type dual (D) of minimax problem (P):
max
(s,λ,y)∈K(z,μ)sup∈H(s,λ,y)
s
i=
λif
z,yi, (D)
whereK={(s,λ,y)|sis an integer,λ∈Rs
+,
s
i=λi= ,y= (y,y, . . . ,ys),yi∈Y(x) for some
x∈X, ≤i≤s}, andH(s,λ,y) denotes the set of all (z,μ)∈X×Rm
+ satisfying
s
i=
λi(dfi)+Hz,x
+,yi+ m
j=
μj(dgj)+Hz,x
+≥, for allx∈X, ()
m
j=
μjgj(z)≥. ()
If for a triplet (s,λ,y) inKthe setH(s,λ,y) is empty then we define the supremum over it to be –∞.
Theorem .(Weak duality) Let x and(z,μ,s,λ,y)be feasible solutions of(P)and(D),
respectively.Assume that
(i) si=λif(·,yi)isρ¯-generalized-pseudo-right upper-Dini-derivative locally arcwise
connected(with respect toH)atz,
(ii) mj=μjgj(·)isρ˘-generalized-quasi-right upper-Dini-derivative locally arcwise
connected(with respect toH)atz, (iii) ρ(¯x,z) +ρ(˘ x,z)≥.
Then
sup y∈Y
f(x,y)≥ s
i=
λif
z,yi.
Proof Suppose to the contrary that
sup y∈Y
f(x,y) < s
i=
λifz,yi.
Thus, we have
fx,yi< s
i=
λif
z,yi, for allyi∈Y(x), ≤i≤s.
It follows fromλi≥, ≤i≤s, and si=λi= , that s
i=
λif
x,yi<
s
i=
λif
which by ρ-generalized-pseudo-right upper-Dini-derivative locally arcwise connected¯ (with respect toH) of si=λif(·,yi) atz, we have
s
i=
λi(df)+Hz,x
+,yi< –ρ(¯ x,z). ()
Forx∈X,μj≥, ≤j≤m, we haveμjgj(x)≤, which in view of () implies that m
j=
μjgj(x)≤ m
j=
μjgj(z),
which byρ-generalized-quasi-right upper-Dini-derivative locally arcwise connected (with˘ respect toH) of mj=μjgj(·) atz, we have
m
j=
μj(dgj)+Hz,x
+≤–ρ(˘ x,z). ()
By () and (), we get
s
i=
λi(df)+Hz,x
+,yi+ m
j=
μj(dgj)+Hz,x
+< –ρ(¯x,z) –ρ(˘ x,z)≤,
where the last inequality is according toρ(¯x,z) +ρ(˘ x,z)≥. Therefore,
s
i=
λi(df)+Hz,x
+,yi+ m
j=
μj(dgj)+Hz,x
+< ,
which is a contradiction to (). Hence the theorem is proved.
Theorem . (Strong duality) Let x∗ be an optimal solution of (P). Assume that the conditions of Theorem . are satisfied. Then there exist(s∗,λ∗,y∗)∈K, and(x∗,μ∗)∈ H(s∗,λ∗,y∗)such that(x∗,μ∗,s∗,λ∗,y∗)is a feasible solution of(D)and the two objectives have same values.If,in addition,the assumption of weak duality Theorem.hold for all feasible solutions of(D),then(x∗,μ∗,s∗,λ∗,y∗)is an optimal solution of(D).
Proof Sincex∗ is an optimal solution for (P) and all the conditions of Theorem . are satisfied, there exist (s∗,λ∗,y∗)∈K, and (x∗,μ∗)∈H(s∗,λ∗,y∗) such that (x∗,μ∗,s∗,λ∗,y∗) is a feasible solution of (D) and the two objective values are equal. The optimality of
(x∗,μ∗,s∗,λ∗,y∗) for (D) thus follows from Theorem ..
Theorem .(Strict converse duality) Let x∗and(z∗,μ∗,s∗,λ∗,y∗)be optimal solutions of(P)and(D),respectively.Assume that the hypothesis of Theorem.is fulfilled.Also,
assume that
(i) si=∗ λ∗if(·,y∗i)is strictlyρ¯-generalized-pseudo-right upper-Dini-derivative locally arcwise connected(with respect toH)atz∗,
(iii) ρ(¯x∗,z∗) +ρ(˘ x∗,z∗)≥.
Then z∗=x∗.
Proof Suppose to the contrary thatz∗=x∗. According to Theorem ., we know that there exist (s∗,λ∗,y∗)∈K, and (x∗,μ∗)∈H(s∗,λ∗,y∗) such that (x∗,μ∗,s∗,λ∗,y∗) is a feasible so-lution of (D) and
sup y∗∈Y
fx∗,y∗= s∗
i=
λ∗ifz∗,y∗i.
Thus, we have
fx∗,y∗i≤
s∗
i=
λ∗ifz∗,y∗i, for ally∗i∈Yx∗, ≤i≤s∗.
It follows fromλ∗i ≥, ≤i≤s∗, and si∗=λ∗i = , that s∗
i=
λ∗ifx∗,y∗i≤
s∗
i=
λ∗ifz∗,y∗i. ()
Now proceeding on the same lines as in Theorem ., we get
s∗
i=
λ∗i(df)+Hz∗,x∗
+,y∗i+ m
j=
μ∗j(dgj)+
Hz∗,x∗
+< ,
which is a contradiction to (). Hence the theorem is proved.
5 Conclusion
In this study we have established necessary and sufficient optimality conditions under generalized convexity using the tool-right upper-Dini-derivative for a general minimax programming problem. Mond-Weir type duality theory is also obtained. These results can be extended to the following semiinfinite minimax programming problem (SIP) with the tool-right upper-Dini-derivative:
Minimize max
y∈Y f(x,y)
subject to
Gj(x,t)≤, for allt∈Tj,j∈q={, , . . . ,q},
Hk(x,s) = , for alls∈Sk,k∈r={, , . . . ,r},
x∈X,
(SIP)
whereX⊂Rnis a nonempty open arcwise connected set,Yis a compact metrizable topo-logical space,f(·,y) is a real-valued function defined onX.TjandSkare compact subsets of complete metric spaces, for eachj∈q,Gj(·,t) is a real-valued function defined onX
for eachj∈qandk∈r,Gj(x,·), andHk(x,·) are continuous real-valued functions defined, respectively, onTjandSkfor allx∈X. We shall investigate this semiinfinite programming problem in subsequent papers.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors carried out the proof. All authors conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
Author details
1Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826 004, India.2Department of
Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia.
Acknowledgements
The research of the second and fourth author is financially supported by King Fahd University of Petroleum and Minerals, Saudi Arabia under the Internal Research Project No. IN131026.
Received: 26 May 2014 Accepted: 29 July 2014 Published:26 Aug 2014 References
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