R E S E A R C H
Open Access
Optimality and duality for nonsmooth
multiobjective optimization problems
Kwan Deok Bae and Do Sang Kim
**Correspondence:
[email protected] Department of Applied Mathematics, Pukyong National University, Busan, 608-737, Korea
Abstract
In this paper, we consider a nonsmooth multiobjective programming problems including support functions with inequality and equality constraints. Necessary and sufficient optimality conditions are obtained by using higher-order strong convexity for Lipschitz functions. Mond-Weir type dual problem and duality theorems for a strict minimizer of ordermare given.
Keywords: nonsmooth multiobjective programming; strict minimizers; optimality conditions; duality
1 Introduction
Nonlinear analysis problems are a new and vital area of optimization theory, mathematical physics, economics, engineering and functional analysis. Moreover, nonsmooth problems occur naturally and frequently in optimization.
In , Rockafellar wrote in his book that practical applications are not necessarily differentiable in applied mathematics (see []). So, dealing with nondifferentiable mathe-matical programming problems was very important. Vial [] studied strongly and weakly convex sets andρ-convex functions.
Auslender [] introduced the notion of lower second-order directional derivative and obtained necessary and sufficient conditions for a strict local minimizer. Based on Auslen-der’s results, Studniarski [] proved necessary and sufficient conditions for the problem of the feasible set defined by an arbitrary set. Moreover, Ward [] derived necessary and sufficient conditions for strict minimizer of ordermin nondifferentiable scalar programs. Jimenez [] introduced the notion of super-strict efficiency for vector problems and gave necessary conditions for strict minimality. Jimenez and Novo [, ] obtained first- and second-order optimality conditions for vector optimization problems. Bhatia [] gave the higher-order strong convexity for Lipschitz functions and established optimality condi-tions for the new concept of strict minimizer of higher order for a multiobjective opti-mization problem.
Kim and Bae [] formulated nondifferentiable multiobjective programs with the sup-port functions. Also, Baeet al.[] established duality theorems for nondifferentiable mul-tiobjective programming problems under generalized convexity assumptions. Also, Kim and Lee [] introduced the nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. They introduced Karush-Kuhn-Tucker type optimality conditions and established duality theorems for (weak) Pareto-optimal
lutions. Recently, Bae and Kim [] established optimality conditions and duality theorems for a nondifferentiable multiobjective programming problem with support functions.
In this paper, we consider nonsmooth multiobjective programming with inequality and equality constraints. In Section , we introduce the concept of a strict minimizer of order
mand higher-order strong convexity for this problem. In Section , necessary and suffi-cient optimality theorems are established for a strict minimizer of ordermunder gener-alized strong convexity assumptions. In Section , we formulate a Mond-Weir type dual problem and obtain weak and strong duality theorems.
2 Preliminaries
Letx,y∈Rn. The following notation will be used for vectors inRn:
x<y ⇐⇒ xi<yi, i= , , . . . ,n;
xy ⇐⇒ xiyi, i= , , . . . ,n;
x≤y ⇐⇒ xiyi, i= , , . . . ,nbutx=y;
x≮yis the negation ofx<y;
xyis the negation ofx≤y.
Forx,u∈R,xuandx<uhave the usual meaning. LetRnbe then-dimensional
Eu-clidean space, and letRn
+be its nonnegative orthant.
Definition .[] LetDbe a compact convex set inRn. The support functions(·|D) is
defined by
s(x|D) :=maxxTy:y∈D.
The support function s(·|D) has a subdifferential. The subdifferential of s(·|D) atxis given by
∂s(x|D) :=z∈D:zTx=s(x|D).
The support functions(·|D) is convex and everywhere finite, that is, there existsz∈Dsuch that
s(y|D)≥s(x|D) +zT(y–x) for ally∈D.
Equivalently,
zTx=s(x|D).
We consider the following multiobjective programming problem.
(MOP) Minimize f(x) +s(x|D) =f(x) +s(x|D), . . . ,fp(x) +s(x|Dp)
wheref :X→Rp,g:X→Rqandh:X→Rrare locally Lipschitz functions, respectively,
andXis the convex set ofRn. For eachi∈P={, , . . . ,p},D
iis a compact convex subset
ofRn.
Further, letS:={x∈X|gj(x),j= , . . . ,q,hl(x) = ,l= , . . . ,r}be the feasible set of
(MOP),B(x,) ={x∈Rn| x–x <}be an open ball with centerxand radiusand
I(x) :={j∈ {, . . . ,q} |g
j(x) = }be the index set of active constraints atx.
We introduce the following definitions due to Jimenez [].
Definition . A pointx∈Xis called a strict local minimizer for (MOP) if there exists
> such that
f(x) +s(x|D)≮fx+sx|D, ∀x∈Bx,∩X.
Definition . Letm be an integer. A pointx∈Xis called a strict local minimizer
of ordermfor (MOP) if there exist> andc∈intRp+such that
f(x) +s(x|D)≮fx+sx|D+x–xmc, ∀x∈Bx,∩X.
Definition . Letm be an integer. A pointx∈Xis called a strict minimizer of order
mfor (MOP) if there existsc∈intRp+such that
f(x) +s(x|D)≮fx+sx|D+x–xmc, ∀x∈X.
Definition .[] Suppose thatf :X→Ris Lipschitz on X. Clarke’s generalized
di-rectional derivative off atx∈Xin the directiond∈Rn, denoted byf(x,d), is defined as
f(x,d) =lim sup y→x t↓
f(y+td) –f(y)
t .
Definition .[] Clarke’s generalized gradient offatx∈X, denoted by∂f(x), is defined as
∂f(x) =ξ∈Rn:f(x,d)≥ ξ,d ∀d∈Rn.
Definition . For a nonempty subsetXofRn, we denoteX∗, the dual cone ofX, defined
by
X∗=u∈Rn|uTx,∀x∈X.
Further, forx∈X,N
X(x) denotes the normal cone toXatxdefined by
NX
x=d∈Rn|d,x–x ,∀x∈X.
It is clear that (X–x)∗= –N
X(x).
Definition . A functionf :X→Ris said to be strongly convex of ordermon a convex setXif there existsc> such that forx,x∈Xandt∈[, ],
ftx+ ( –t)x
tf(x) + ( –t)f(x) –ct( –t) x–x m.
Proposition .[] If fi,i= , . . . ,p,are strongly convex of order m on a convex set X,then p
i=tifiandmax≤i≤pfiare also strongly convex of order m on X,where ti≥,i= , . . . ,p.
Definition . A locally Lipschitz functionf is said to be strongly quasiconvex of order
mon X if there exists a constantc> such that forx,x∈X,
f(x)f(x) ⇒ ξ,x–x+ x–x mc, ∀ξ∈∂f(x).
For each k∈ {, . . . ,p} and x∈X, we consider the following scalarizing problem of (MOP) due to the one in [].
(Pk(x)) Minimize fk(x) +s(x|Dk)
subject to fi(x) +s(x|Di)fi
x+sx|D
i
, k∈P,i=k,
gj(x), j= , . . . ,q, hl(x) = , l= , . . . ,r.
The following definition is due to the one in [].
Definition . Letxbe a feasible solution for (MOP). We say that the basic regularity
condition (BRC) is satisfied atxif there exist no non-zero scalarsλi ,wi∈Di,i=
, . . . ,p,i=k,k∈P,μ
j ,j∈I(x),μj = ,j∈/I(x), andνl,l= , . . . ,r, such that
∈
p
i=,i=k
λi∂fi
x+wi
+
q
j=
μj∂gj(x) +
r
l=
νl∂hl
x+NX
x.
3 Optimality conditions
In this section, we establish Fritz John necessary optimality conditions, Karush-Kuhn-Tucker necessary optimality conditions and Karush-Kuhn-Karush-Kuhn-Tucker sufficient optimality condition for a strict minimizer of (MOP).
Theorem .(Fritz John necessary optimality conditions) Suppose that xis a strict min-imizer of order m for(MOP)and fi,i= , . . . ,p,gj,j= , . . . ,q,and hl,l= , . . . ,r,are locally Lipschitz functions at x.Then there existλi ,wi ∈Di,i= , . . . ,p,μj ,j= , . . . ,q, andνl,l= , . . . ,r,not all zero such that
∈
p
i=
λi∂fi
x+wi+
q
j=
μj∂gj
x+
r
l=
νl∂hl
x+NX
x,
wi,x=sx|Di
, i= , . . . ,p,
μjgj
Proof Sincex is a strict minimizer of ordermfor (MOP), it is a strict minimizer for
(MOP). It can be shown thatxsolves the following problem:
minimize F(x)
subject to g(x), h(x) = ,
where
F(x) =maxf(x) +s(x|D)
–f
x+sx|D
, . . . ,
fp(x) +s(x|Dp)
–fp
x+sx|Dp
.
If it is not so, then there exitsx∈Rnsuch thatF(x) <F(x),g(x),h(x) = . Since
F(x) = , we haveF(x) < . This contradicts the fact thatx is a strict minimizer for
(MOP). SincexminimizesF(x), from Theorem .. in Clarke [], there exists (λ,μ,ν)∈ (Rp,Rq,Rr) not all zero such that
∈
p
i=
λi∂F
x+
j∈I(x)
μj∂gj
x+
r
l=
νl∂hl
x+NX
x.
Lettingμj= , forj∈/I(x), we have
∈
p
i=
λi∂F
x+
q
j=
μj∂gj
x+
r
l=
νl∂hl
x+NX
x.
SinceF(x) =max{(f(x) +s(x|D)) – (f(x) +s(x|D))}for anyx∈Xands(x|D
i) = (x)Twi, i= , . . . ,p, we have
∂F(x)⊂co∂fi
x+sx|Di
=co∂fi
x+w
i
,
whereco{∂(fi(x) +s(x|Di))}denotes the convex hull of{∂(fi(x) +s(x|Di))}. Hence, there
existλi ,wi ∈Di,i= , . . . ,p,μj ,j= , . . . ,q, andνl,l= , . . . ,r, not all zero such
that
∈
p
i=
λi
∂fi
x+wi+
q
j=
μj∂gj
x+
r
l=
νl∂hl
x+NX
x,
wi,x=sx|Di
, i= , . . . ,p,
μjgj
x= , j= , . . . ,q.
Theorem .(Karush-Kuhn-Tucker necessary optimality conditions) Suppose that xis
a strict minimizer of order m for(MOP)and fi,i= , . . . ,p,gj,j= , . . . ,q,and hl,l= , . . . ,r, are locally Lipschitz functions at x.If the basic regularity condition(BRC)holds at x,then
∈
p
i=
λi∂fi
x+wi+
q
j=
μj∂gj
x+
r
l=
νl∂hl
x+NX
x,
wi,x=sx|Di
, i= , . . . ,p,
μjgj
x= , j= , . . . ,q,
λ, . . . ,λp= (, . . . , ).
Proof Sincex is a strict minimizer of ordermfor (MOP), by Theorem ., there exist
wi ∈Di,λi ,i= , . . . ,p,μj ,j= , . . . ,q, andνl,l= , . . . ,r, not all zero such that
∈
p
i=
λi∂fi
x+wi+
q
j=
μj∂gj
x+
r
l=
νl∂hl
x+NX
x,
wi,x=sx|Di
, i= , . . . ,p,
μjgj
x= , j= , . . . ,q.
It can be shown that (λ, . . . ,λp)= (, . . . , ). Ifλi = ,i= , . . . ,p, then we have
∈
p
i=,i=k
λi∂fi
x+wi
+
q
j=
μj∂gj(x) +
r
l=
νl∂hl
x+NX
x
for eachk∈P={, . . . ,p}. Since the basic regularity condition (BRC) holds atx, we have
λk= ,k∈P,k=i={, . . . ,p},μj= ,j∈I(x), andνl= ,l= , . . . ,r. This contradicts the
fact thatλi,λk,k∈P,k=i,μj,j∈I(x) andνl,l= , . . . ,r, are not all simultaneously zero.
Hence, (λ, . . . ,λp)= (, . . . , ).
Theorem .(Karush-Kuhn-Tucker sufficient optimality conditions) Assume that there
existλi ,wi ∈Di,i= , . . . ,p,μj ,j= , . . . ,q,andνl,l= , . . . ,r,such that for x∈X,
∈
p
i=
λi∂fi
x+wi+
q
j=
μj∂gj
x+
r
l=
νl∂hl
x+NX
x,
wi,x=sx|Di
, i= , . . . ,p,
μjgj
x= , j= , . . . ,q,
λ, . . . ,λp= (, . . . , ).
Assume further that fi,i= , . . . ,p, are strongly convex of order m on X,gj,j∈I(x)are strongly quasiconvex of order m on X and νTh is strongly quasiconvex of order m on X.
Then xis a strict minimizer of order m for(MOP).
Proof Sincefi,i= , . . . ,p, are strongly convex of ordermonXand (·)Twi,i= , . . . ,p, are
convex, there exists ci> ,i= , . . . ,p, such that for all x∈X, ξi∈∂fi(x) andwi∈Di, i= , . . . ,p,
fi(x) –fi
xξi,x–x
+x–xmci, xTwi–
xTwi
xi,x–x
So, we obtain
fi(x) +xTwi
–fi
x+xTxi
ξi+wi,x–x
+x–xmci. (.)
Forλi ,i= , . . . ,p, (.) implies
p
i=
λifi(x) +xTwi
–
p
i=
λifi
x+xTwi
p
i=
λiξi+wi,x–x
+
p
i=
λix–xmci. (.)
Forx∈X, we have
gj(x)gj
x, j∈Ix,
νTh(x) =νThx.
Sincegj,j∈I(x) are strongly quasiconvex of ordermonXandνThis strongly quasiconvex
of order monX, it follows that there exist cj> ,ηj∈∂gj(x),j∈I(x),c> , andζ ∈
∂νTh(x) such that
ηj,x–x
+x–xmcj,
ζ,x–x+x–xmc.
(.)
Forμj ,j∈I(x), we obtain
j∈I(x)
μjηj,x–x
+
j∈I(x)
μjx–xmcj. (.)
Sinceμj = forj∈/I(x), (.) implies
q
j=
μjηj,x–x
+
q
j=
μjx–xmcj. (.)
By (.), (.) and (.), we get
p
i=
λifi(x) +xTwi
–
p
i=
λifi
x+xTwi
x–xma,
wherea= pi=λici+ q
j=μjcj+ ml=νlcl. This implies that p
i=
λifi(x) +xTwi
–fi
x+xTwi
–x–xmdi
, (.)
Suppose thatxis not a strict minimizer of ordermfor (MOP). Then there existx,x∈X
andc∈Rp+such that
f(x) +s(x|D) <fx+sx|D+x–xmc, ∀x∈X.
SincexTws(x|D) and (x)Tw=s(x|D), we have
f(x) +xTwf(x) +s(x|D)
<fx+sx|D+x–xmc
=fx+xTw+x–xmc.
Forλi , we obtain
p
i=
λifi(x) +xTwi
<
p
i=
λifi
x+xTwi
+
p
i=
λix–xmci.
This is a contradiction to (.).
Remark . Suppose thatgj,j∈I(x) are strongly convex of ordermonXand thatνTh
is strongly convex of ordermonX. Then the conclusion of Theorem . also holds.
Proof It follows on the lines of Theorem ..
4 Duality theorems
Now we propose the following Mond-Weir type dual (MOD) to (MOP):
(MOD) Maximize f(u) +uTw
subject to ∈
p
i=
λi
∂fi(u) +wi
+
q
j=
μj∂gj(u)
+
r
l=
νl∂hl(u) +NX(u), (.)
q
j=
μjgj(u) + r
l=
νlhl(u), (.)
λi, wi∈Di, i= , . . . ,p,λTe= , (.)
μj, j= , . . . ,q, νl, l= , . . . ,r,u∈X. (.)
Theorem .(Weak duality) Let x and(u,w,λ,μ,ν)be feasible solutions of(MOP)and
(MOD),respectively.If fi,i= , . . . ,p,are strongly convex of order m at u and q
j=μjgj(·) + r
Proof Sincexis a feasible solution of (MOP) and (u,w,λ,μ,ν) is a feasible solution of (MOD), we have
q
j=
μjgj(x) + r
l=
μlhl(x) q
j=
μjgj(u) + r
l=
μlhl(u).
Since qj=μjgj(·) + lr=νlhl(·) is strongly quasiconvex of ordermatu, it follows that there
existcj> ,ηj∈∂gj(u),j= , . . . ,q,cl> , andζl∈∂hl(u) such that q
j=
μjηj+ r
l=
νlζl,x–u
+
q
j=
x–u mcj+ r
l=
x–u mcl. (.)
Now, suppose contrary to the result that (.) holds. SincexTwis(x|Di),i= , . . . ,p, we
obtain
fi(x) +xTwi<fi(u) +uTwi, i= , . . . ,p.
Forc∈intRp+, we obtain
fi(x) +xTwi<fi(u) +uTwi+ x–u mci, i= , . . . ,p. (.)
Forλi, we obtain p
i=
λi
fi(x) +xTwi < p i= λi
fi(u) +uTwi
+
p
i=
λi x–u mci. (.)
Sincefi,i= , . . . ,p, are strongly convex of ordermatuand (·)Twi,i= , . . . ,p, are convex at u, there existsci> ,i= , . . . ,p, such that for allx∈X,ξi∈∂fi(x) andwi∈Di,i= , . . . ,p,
fi(x) –fi(u)ξi,x–u+ x–u mci, xTwi–uTwiwi,x–u.
So, we obtain
fi(x) +xTwi
–fi(u) +uTwi
ξi+wi,x–u+ x–u mci. (.)
Forλi,i= , . . . ,p, we obtain p
i=
λi
fi(x) +xTwi – p i= λi
fi(u) +uTwi
p
i=
λi(ξi+wi),x–u
+
p
i=
λi x–u mci. (.)
By (.) and (.), we get
p
i=
λi
fi(x) +xTwi – p i= λi
fi(u) +uTwi
wherea= pi=λici+ q
j=μjcj+ rl=νlcl. This implies that p
i=
λi
fi(x) +xTwi
–fi(u) +uTwi
– x–u mdi
, (.)
whered=ae, sinceλTe= . This is a contradiction to (.).
Lemma . If gj(·),j= , . . . ,m,are strongly convex of order m on X andνTh is strongly convex of order m on X,then the same conclusion of Theorem.also holds.
Proof It follows on the lines of Theorem ..
Definition . Letm be an integer. A pointx∈Xis called a strict maximizer of order
mfor (MOD) if there existsc∈intRp+such that
fx+xTw+x–xmc≮f(x) +xTw, ∀x∈X.
Theorem .(Strong duality) If xis a strict minimizer of order m for(MOP)and the
basic regularity condition(BRC)holds at x,then there existλ
i ,wi ∈Di,i= , . . . ,p,
μj ,j= , . . . ,q,andνl,l= , . . . ,r,such that(x,w,λ,μ,ν)is a feasible solution of
(MOD)and(x)Twi =s(x|Di),i= , . . . ,p.Moreover,if the assumptions of Theorem.are satisfied,then(x,w,λ,μ,ν)is a strict maximizer of order m for(MOD).
Proof By Theorem ., there existsλ
i ,wi ∈Di,i= , . . . ,p,μj ,j= , . . . ,q, andνl, l= , . . . ,r, such that
∈
p
i=
λi∂fi
x+wi+
q
j=
μj∂gj
x+
r
l=
νl∂hl
x+NX
x,
wi,x=sx|Di
, i= , . . . ,p,
μjgj
x= , j= , . . . ,q,
λ, . . . ,λp= (, . . . , ).
Thus (x,w,λ,μ,ν) is a feasible solution of (MOD) and (x)Twi =s(x|Di),i= , . . . ,p.
By Theorem ., we obtain that the following holds:
fi
x+xTwi =fi
x+sx|Di
≮fi(u) +uTwi, i= , . . . ,p,
for a given feasible solution (u,w,λ,μ,ν) of (MOD). Forx,u∈Xandc∈intRp, we have fx+xTw+u–xmc
<f(u) +uTw.
Remark . Theorem . and Theorem . reduce to [, Theorem . and Theorem .] in an inequality constraint case. More exactly,fi(·) + (·)Twi,i= , . . . ,p, andgj(·),j∈I(u) at
the considered point in the framework of [, Theorem . and Theorem .] are strongly convex of ordermand strongly quasiconvex of orderm, respectively.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
DSK obtained necessary and sufficient optimality conditions by using higher-order strong convexity of Lipschitz functions, formulated a Mond-Weir type dual problem and established weak and strong duality theorems for a strict minimizer of orderm. KDB carried out the duality studies and participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2A10008908).
Received: 24 July 2013 Accepted: 21 October 2013 Published:22 Nov 2013 References
1. Rockafellar, RT: Convex Analysis. Princeton University Press, Princeton (1970)
2. Vial, JP: Strong and weak convexity of sets and functions. Math. Oper. Res.8, 231-259 (1983)
3. Auslender, A: Stability in mathematical programming with non-differentiable data. SIAM J. Control Optim.22, 239-254 (1984)
4. Studniarski, M: Necessary and sufficient conditions for isolated local minima of nonsmooth functions. SIAM J. Control Optim.24, 1044-1049 (1986)
5. Ward, DE: Characterizations of strict local minima and necessary conditions for weak sharp minima. J. Optim. Theory Appl.80, 551-571 (1994)
6. Jimenez, B: Strictly efficiency in vector optimization. J. Math. Anal. Appl.265, 264-284 (2002)
7. Jimenez, B, Novo, V: First and second order sufficient conditions for strict minimality in multiobjective programming. Numer. Funct. Anal. Optim.23, 303-322 (2002)
8. Jimenez, B, Novo, V: First and second order sufficient conditions for strict minimality in nonsmooth vector optimization. J. Math. Anal. Appl.284, 496-510 (2003)
9. Bhatia, G: Optimality and mixed saddle point criteria in multiobjective optimization. J. Math. Anal. Appl.342, 135-145 (2008)
10. Kim, DS, Bae, KD: Optimality conditions and duality for a class of nondifferentiable multiobjective programming problems. Taiwan. J. Math.13(2B), 789-804 (2009)
11. Bae, KD, Kang, YM, Kim, DS: Efficiency and generalized convex duality for nondifferentiable multiobjective programs. J. Inequal. Appl.2010, Article ID 930457 (2010)
12. Kim, DS, Lee, HJ: Optimality conditions and duality in nonsmooth multiobjective programs. J. Inequal. Appl.2010, Article ID 939537 (2010)
13. Bae, KD, Kim, DS: Optimality and duality theorems in nonsmooth multiobjective optimization. Fixed Point Theory Appl.2011, Article ID 42 (2011)
14. Clarke, FH: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)
15. Lin, GH, Fukushima, M: Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints. J. Optim. Theory Appl.118, 67-80 (2003)
16. Chankong, V, Haimes, YY: Multiobjective Decision Making: Theory and Methodology. North-Holland, New York (1983) 17. Chandra, S, Dutta, J, Lalitha, CS: Regularity conditions and optimality in vector optimization. Numer. Funct. Anal.
Optim.25, 479-501 (2004) 10.1186/1029-242X-2013-554
Cite this article as:Bae and Kim:Optimality and duality for nonsmooth multiobjective optimization problems.
