Volume 2010, Article ID 939537,12pages doi:10.1155/2010/939537
Research Article
Optimality Conditions and Duality in
Nonsmooth Multiobjective Programs
Do Sang Kim and Hyo Jung Lee
Division of Mathematical Sciences, Pukyong National University, Busan 608-737, South Korea
Correspondence should be addressed to Do Sang Kim,dskim@pknu.ac.kr
Received 29 October 2009; Accepted 14 March 2010
Academic Editor: Jong Kyu Kim
Copyrightq2010 D. S. Kim and H. J. Lee. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions. Two types of Karush-Kuhn-Tucker optimality conditions with support functions are introduced. Sufficient optimality conditions are presented by using generalized convexity and certain regularity conditions. We formulate Wolfe-type dual and Mond-Weir-type dual problems for our nonsmooth multiobjective problems and establish duality theorems forweak Pareto-optimal solutions under generalized convexity assumptions and regularity conditions.
1. Introduction
Multiobjective programming problems arise when more than one objective function is to be optimized over a given feasible region. Pareto optimum is the optimality concept that appears to be the natural extension of the optimization of a single objective to the consideration of multiple objectives.
In 1961, Wolfe1obtained a duality theorem for differentiable convex programming. Afterwards, a number of different duals distinct from the Wolfe dual are proposed for the nonlinear programs by Mond and Weir 2. Duality relations for multiobjective programming problems with generalized convexity conditions were given by several authors
nonsmooth multiobjective programming problems involving locally Lipschitz functions for inequality and equality constraints. They extended sufficient optimality conditions in Kim et al.13to the nonsmooth case and established duality theorems for nonsmooth multiobjective programming problems involving locally Lipschitz functions.
In this paper, we apply the results in Kim and Schaible 6 for this problem to nonsmooth multiobjective programming problem involving support functions. We introduce nonsmooth multiobjective programming problems involving locally Lipschitz functions and support functions for inequality and equality constraints. Two kinds of sufficient optimality conditions under various convexity assumptions and certain regularity conditions are presented. We propose a Wolfe-type dual and a Mond-Weir-type dual for the primal problem and establish duality results between the primal problem and its dual problems under generalized convexity and regularity conditions.
2. Notation and Definitions
LetRnbe then-dimensional Euclidean space andRn
its nonnegative orthant.
We consider the following nonsmooth multiobjective programming problem involv-ing locally Lipschitz functions:
minimize fx sx|C f1x sxC1, . . . , fmx sx|Cm
subject to gjx0, j ∈P, hkx 0, k∈Q, x∈Rn,
MP
wherefi:Rn → R,i∈M {1,2, . . . , m},gj :Rn → R,j ∈P {1,2, . . . , p}, andhk:Rn → R,
k ∈Q {1,2, . . . , q}, are locally Lipschitz functions. Here,Ci,i∈M, is compact convex sets
inRn. We accept the formal writingC C
1, C2, . . . , Cmtwith the convention thatsx|C
sx|C1, . . . , sx|Cm, wheresx|Ciis the support function ofCiseeDefinition 2.2.
Throughout the article the following notation for order relations inRnwill be used:
xu⇐⇒u−x∈Rn,
x≤u⇐⇒u−x∈Rn\ {0},
x < u⇐⇒u−x∈intRn,
x /≤uis the negation ofx≤u,
x /< uis the negation ofx < u.
2.1
Definition 2.1. iA real-valued functionF :Rn → Ris said to be locally Lipschitz if for each
z ∈ Rn, there exist a positive constant K and a neighborhoodN of zsuch that, for every
x, y∈N,
Fx−FyKx−y, 2.2
iiThe Clarke generalized directional derivative14of a locally Lipschitz functionF atxin the directiond∈Rn, denoted byF0x;d, is defined as follows:
F0x;d lim sup
y→x, t→0t
−1Fytd−Fy, 2.3
whereyis a vector inRn.
iiiThe Clarke generalized subgradient14ofFatxis denoted by
∂Fx ξ∈Rn:F0x;dξtd,∀d∈Rn. 2.4
ivF is said to be regular atxif for alld ∈ Rn the one-sided directional derivative
Fx;dexists andFx;d F0x;d.
Definition 2.2see10. LetCbe a compact convex set inRn. The support functionsx|C
ofCis defined by
sx|C: maxxty:y∈C . 2.5
The support functionsx|C, being convex and everywhere finite, has a subdifferential, that is, for everyx∈Rnthere existszsuch that
sy|C≥sx|C zty−x, ∀y∈C. 2.6
Equivalently,
ztx sx|C. 2.7
The subdifferential ofsx|Cis given by
∂sx|C: z∈C:ztx sx|C . 2.8
For any setS⊂Rn, the normal cone toSat a pointx∈Sis defined by
NSx:
y∈Rn:ytz−x≤0,∀z∈S . 2.9
It is readily verified that for a compact convex setC,yis inNCxif and only ifsy|C xty,
or equivalently,xis in the subdifferential ofsaty.
In the notation of the problem MP, we recall the definitions of convexity, affine, pseudoconvexity, and quasiconvexity for locally Lipschitz functions.
Definition 2.3. if f1, f2, . . . , fmis convexstrictly convexatx0 ∈ X if for eachx ∈ X
iigA is convex atx0 ∈ X if for eachx ∈ Xand anyζj ∈ ∂gjx0,gjx−gjx0
ζt
jx−x0, wherej∈A, andgAdenotes the active constraints atx0.
iiih h1, h2, . . . , hkis convex at x0 ∈X if for eachx ∈ X and anyαk ∈∂hkx0,
hkx−hkx0αtkx−x0, for allk∈Q.
ivhis affine atx0 ∈ X if for eachx ∈ X and anyα
k ∈ ∂hkx0,hkx−hkx0
αt
kx−x0, for allk∈Q.
vf is pseudoconvex atx0 ∈Xif for eachx∈Xand anyξ
i ∈∂fix0,ξitx−x00
impliesfixfix0, for alli∈M.
vif is strictly pseudoconvex atx0 ∈ X if for eachx ∈ X withx /x0 and anyξ
i ∈
∂fix0,ξitx−x00 impliesfix> fix0, for alli∈M.
viif is quasiconvex at x0 ∈ X if for eachx ∈ X withx /x0 and anyξ
i ∈ ∂fix0,
fixfix0impliesξitx−x00, for alli∈M.
Finally, we recall the definition of Pareto-optimalefficient, nondominatedand weak Pareto-optimal solutions ofMP.
Definition 2.4. iA pointx0∈Xis said to be a Pareto-optimal solution ofMPif there exists no otherx∈Xwithfx≤fx0.
iiA pointx0∈Xis said to be a weak Pareto-optimal solution ofMPif there exists no otherx∈Xwithfx< fx0.
3. Sufficient Optimality Conditions
In this section, we introduce the following two types ofKKTconditions which differ only in the nonnegativity of the multipliers for the equality constraint and neither of which includes a complementary slackness condition, common in necessary optimality conditions14.
∃u0≥0, v0 0, w00u0∈Rm, v0 ∈Rp, w0∈Rqsuch that
0∈
i∈M
u0i∂fi
x0zi
j∈A
v0j∂gj
x0
k∈Q
w0k∂hk
x0,
gx00, hx0 0, ztix0i sx0|Ci
, i∈M,
whereA j∈P :gj
x0 0.
KKT
∃u0≥0, v0 0, w0 u0∈Rm, v0∈Rp, w0 ∈Rqsuch that
0∈
i∈M
u0i∂fi
x0zi
j∈A
v0j∂gj
x0
k∈Q
w0k∂hk
x0,
gx00, hx0 0, ztix0i sx0|Ci
, i∈M,
whereA j∈P : gj
x0 0.
In Theorems3.1and3.2and Corollaries3.3and3.4below we present new versions including support functions of the results by Kim and Schaible in 6 for smooth problems MP
involvingKKT.
Theorem 3.1. Letx0, u0, v0, w0satisfyKKT. Iff· zt·is pseudoconvex atx0,g
Aandhare
quasiconvex atx0, andfis regular atx0, thenx0is a weak Pareto-optimal solution of MP.
Proof. Letx0, u0, v0, w0satisfyKKT. Thengx00,hx0 0,
0∈
i∈M
u0i∂fi
x0zi
j∈A
v0j∂gj
x0
k∈Q
w0k∂hk
x0. 3.1
From3.1, there existξi∈∂fix0,ζj∈∂gjx0, andαk∈∂hkx0such that
i∈M
u0iξizi
j∈A
vj0ζj
k∈Q
wk0αk 0. 3.2
Suppose thatx0 ∈Xis not a weak Pareto-optimal solution ofMP. Then there existsx∈X such thatfx sx|C< fx0 sx0|Cthat impliesfx ztx < fx0 ztx0because of ztxsx|Cand the assumptionztx0 sx0|Cwhich means that this functionsx|Cis subdifferentiable and regular atx0. By pseudoconvexity off· zt·atx0, we have
ξizit
x−x0<0 for anyξi∈∂fi
x0, i∈M. 3.3
Since gjx 0 gjx0, j ∈ A, we obtain the following inequality with the help of
quasiconvexity ofgAatx0:
ζtjx−x00 for anyζj ∈∂gj
x0, j ∈A. 3.4
Also, sincehkx 0 hkx0,k∈Q, it follows from quasiconvexity ofhatx0that
αt k
x−x00 for anyα
k∈∂hk
x0, k∈Q. 3.5
From3.3–3.5, we obtain
⎡ ⎣
i∈M
u0iξizi
j∈A
v0jζj
k∈Q
w0kαk
⎤ ⎦
t
x−x0<0, 3.6
which contradicts3.2. Hencex0is a weak Pareto-optimal solution ofMP.
Theorem 3.2. Letx0, u0, v0, w0satisfyKKT. Iff· zt·is strictly pseudoconvex atx0,g
A
Corollary 3.3. Letx0, u0, v0, w0satisfyKKT. Iff· zt·,g
Aandhare convex atx0, andfis
regular atx0, thenx0is a weak Pareto-optimal solution of MP.
Corollary 3.4. Letx0, u0, v0, w0satisfyKKT. Iff· zt·is strictly convex atx0,g
A andh
are convex atx0, andfis regular atx0, thenx0is a Pareto-optimal solution ofMP.
Theorem 3.5. Let x0, u0, v0, w0satisfy KKT. If f· zt· is quasiconvex atx0, v0tg A
w0this strictly pseudoconvex atx0, andf,g
A, andhare regular atx0, thenx0is a Pareto-optimal
solution of MP.
Proof. Suppose thatx0 ∈Xis not a Pareto-optimal solution ofMP. Then there existsx∈X such thatfx sx|C≤fx0 sx0|C, that impliesfx ztx≤fx0 ztx0because of ztxsx|Cand the assumptionztx0 sx0|C. By quasiconvexity off· zt·atx0, we have
ξizit
x−x00 for anyξi∈∂fi
x0, i∈M. 3.7
Sincex0, u0, v0, w0satisfyKKT, we obtainj∈Av0jζjk∈Qwk0αk
tx−x00 for some ζj ∈ ∂gjx0,j ∈ A, andαk ∈ ∂hkx0,k ∈ Q. By regularity ofgA andhatx0, there exists
β ∈∂v0tg
A w0thsuch thatβtx−x0 0. With the help of a strict pseudoconvexity of
v0tgA w0th, we have
v0tg
Ax
w0thx>v0tg
A
x0w0thx0. 3.8
Sincex∈X, we obtain
v0tg
Ax
w0thx0. 3.9
SincegAx0 hx0 0, we obtain
v0tgA
x0w0thx0 0. 3.10
Substituting 3.9 and 3.10for 3.8, we arrive at a contradiction. Hence x0 is a Pareto-optimal solution ofMP.
Now we present a result for nonsmooth problemsMPwhich in the smooth case is similar to Singh’s earlier result in12under generalized convexity.
Theorem 3.6. Let x0, u0, v0, w0 satisfy KKT. If u0tf· zt· v0tg
A w0th is
pseudoconvex atx0, andf,g
A, and hare regular at x0, thenx0 is a weak Pareto-optimal solution
of MP.
gAx0 gAx0andhx 0 hx0. Therefore,fx sx|C−fx0 sx0 |C<0,
gAx−gAx00, andhx−hx0 0. Henceu0tfx ztx v0tgAx w0thx<
u0tfx0 ztx0 v0tg
Ax0 w0thx0. Sincef,gA, andhare regular atx0, we obtain
∂ ⎛ ⎝
i∈M
u0
i
fi
x0sx0|C
i
j∈A
v0
jgj
x0
k∈Q
w0
khk
x0
⎞ ⎠
i∈M
u0
i
∂fi
x0∂sx0 |C
i
j∈A
v0
j∂gj
x0
k∈Q
w0
k∂hk
x0.
3.11
By pseudoconvexity ofu0tf· zt· v0tg
A w0th, we haveβtx−x0<0 for anyβ∈
∂i∈Mu0
ifix0 ztix0
j∈Avj0gjx0
k∈Qw0khkx0. We easily see that this contradicts
0 ∈ i∈Mu0i∂fix0 zi j∈Av0j∂gjx0
k∈Qw0k∂hkx0. Hence x0 is a weak
Pareto-optimal solution ofMP.
Theorem 3.7. Letx0, u0, v0, w0satisfyKKT. Ifu0tf· zt· v0tgA w0this strictly
pseudoconvex atx0, andf,g
A, andhare regular atx0, thenx0is a Pareto-optimal solution of MP.
The proof is similar to the one used for the previous theorem.
4. Duality
Following Mond and Weir2, in this section we formulate a Wolfe-type dual problemWD
and a Mond-Weir-type dual problemMDof the nonsmooth problemMP and establish duality theorems. We begin with a Wolfe-type dual problem:
maximize fyztyvtgyewthye
subject to 0∈
i∈M
ui
∂fi
yzi
j∈P
vj∂gj
y
k∈Q
wk∂hk
y,
y∈Rn, u≥0 withute 1, v0, w0.
WD
Heree 1, . . . ,1t∈Rm.
We now derive duality relations.
Theorem 4.1. Letxbe feasible forMPandy, u, v, wfeasible forWD. Iff·zt·,gandwth
are convex, andfis a regular function, thenfx sx|C/< fy ztyvtgyewthye.
Proof. Letxbe feasible forMPandy, u, v, wfeasible forWD. Thengx0,hx 0,
0∈
i∈M
ui
∂fi
yzi
j∈P
vj∂gj
y
k∈Q
wk∂hk
According to4.1, there existξi∈∂fiy,ζj∈∂gjy, andαk∈∂hkysuch that
i∈M
uiξizi
j∈P
vjζj
k∈Q
wkαk 0. 4.2
Assume that
fx sx|C< fyztyvtgyewthye. 4.3
Multiplying4.3byuand using the equalityztx sx|C, we have
utfx ztx−utfyzty−vtgy−wthy<0 4.4
sinceu≥0 andute 1. Now by convexity off· zt·,gandwth, we obtain
utfx ztx−utfyzty
i∈M
uiξizit
x−y, ∀ξi∈∂fi
y,
vtgx−vtgy
j∈P
vjζtj
x−y, ∀ζj ∈∂gj
y,
wthx−wthy
k∈Q
wkαtk
x−y, ∀αk∈∂hk
y.
4.5
Sincevtgx0 andwthx 0, we obtain the following inequality from4.2,4.5:
utfx ztx−utfyzty−vtgy−wthy0, 4.6
which contradicts4.4.
Hence the weak duality theorem holds.
Now we derive a strong duality theorem.
Theorem 4.2. Letxbe a weak Pareto-optimal solution of MPat which a constraint qualification holds [14]. Then there existu∈Rm,v∈Rp, andw ∈Rqsuch thatx, u, v, wis feasible forWD
andztx sx|C. In addition, iff·zt·,gandwthare convex, andfis a regular function, then
x, u, v, wis a weak Pareto-optimal solution of WDand the optimal values of MPandWD
are equal.
Proof. From theKKTnecessary optimality theorem14, there existu ∈ Rm,v ∈ Rp, and
w∈Rqsuch that
0∈
i∈M
ui
∂fix zi
j∈P
vj∂gjx
k∈Q
wk∂hkx,
vtgx 0, u≥0, v0.
Sinceu≥0, we can scale theuis,vjsandwksas follows:
ui ui i∈Mui
, vj
vj
i∈Mui
, wk wk i∈Mui
. 4.8
Thenx, u, v, wis feasible forWD. Sincexis feasible forMP, it follows fromTheorem 4.1
that
fx sx|C fx sx|C vtgxewthxe
/
<fyztyvtgyewthye 4.9
for any feasible solution x, u, v, w of WD. Hence x, u, v, w is a weak Pareto-optimal solution ofWDand the optimal values ofMPandWDare equal.
Remark 4.3. If we replace the convexity hypothesis of f· zt· by strict convexity in
Theorems4.1and4.2, then these theorems hold for the case of a Pareto-optimal solution.
Remark 4.4. If we replace the convexity hypothesis ofwthby affineness ofhin Theorems4.1
and4.2, then these theorems are also valid.
Theorem 4.5. Letx be feasible forMPand y, u, v, wfeasible for WD. Ifutf· zt·
vtgwthis pseudoconvex andf,g, andhare regular functions, thenfx sx|C/≤fy zty
vtgyewthye.
Proof. Suppose to the contrary thatfx sx | C ≤ fy ztyvtgyewthye. By
feasibility ofx, we obtain
utfx ztxvtgx wthx< utfyztyvtgywthy. 4.10
Sincef,g, and hare regular functions, we have βtx−y < 0 by the pseudoconvexity of
utf· zt· vtgwthfor anyβ∈∂
i∈Muifiy ztiy j∈Pvjgjy k∈Qwkhky.
This contradicts the feasibility ofy, u, v, w. Hence the weak duality theorem holds.
Theorem 4.6. Letxbe a weak Pareto-optimal solution of MPat which a constraint qualification holds [14]. Then there existu∈Rm,v∈Rp, andw ∈Rqsuch thatx, u, v, wis feasible forWD
andztx sx|C. If in additionutf· zt· vtgwthis pseudoconvex andf,g, andhare regular functions, thenx, u, v, wis a weak Pareto-optimal solution ofWDand the optimal values of MPandWDare equal.
The proof is similar to the one used forTheorem 4.2.
Remark 4.7. If we replace the pseudoconvexity hypothesis ofutf· zt· vtgwthby
We now prove duality relations between MP and the following Mond-Weir-type dual problem:
maximize fyzty
subject to 0∈
i∈M
ui
∂fi
yzi
j∈P
vj∂gj
y
k∈Q
wk∂hk
y,
vtgywthy0,
y∈Rn, u≥0 withute 1, v0, w0.
MD
Theorem 4.8. Letxbe feasible forMPandy, u, v, wfeasible forMD. Iff·zt·,gandwth
are convex, andfis a regular function, thenfx sx|C/<fy zty.
Proof. Letxbe feasible forMPandy, u, v, wfeasible forMD. Thengx0,hx 0,
0∈
i∈M
ui
∂fi
yzi
j∈P
vj∂gj
y
k∈Q
wk∂hk
y, 4.11
vtgywthy0. 4.12
According to4.11, there existξi∈∂fiy,ζj∈∂gjy, andαk∈∂hkysuch that
i∈M
uiξizi
j∈P
vjζj
k∈Q
wkαk 0. 4.13
Assume that
fx sx|C< fyzty. 4.14
Multiplying4.14byuand using the inequalityztxsx|C, we have
utfx ztx< utfyzty. 4.15
By convexity off· zt·,gandwth, we obtain
utfx ztxvtgx wthxutfyztyvtgywthy. 4.16
From4.12and4.16, we obtain
utfx ztx−utfyzty0, 4.17
since vtgx 0 andwthx 0. However, 4.17contradicts 4.15. Hence the proof is
Theorem 4.9. Letxbe a weak Pareto-optimal solution of MPat which a constraint qualification holds. Then there existu∈Rm,v∈Rp, andw ∈Rq such thatx, u, v, wis feasible forMDand
ztx sx|C. If in additionf· zt·,g andwthare convex, andf is a regular function, then
x, u, v, wis a weak Pareto-optimal solution of MDand the optimal values ofMPandMDare equal.
Proof. Letxbe a weak Pareto-optimal solution ofMPsuch that a constraint qualification is satisfied atx. According to the KKTnecessary optimality theorem, there existu ∈ Rm,
v∈Rp, andw∈Rqsuch that
0∈
i∈M
ui
∂fix zi
j∈P
vj∂gjx
k∈Q
wk∂hkx,
vtgx 0, u≥0, v0.
4.18
Since u ≥ 0, we can scale theuis, vjs, andwks as in the proof of Theorem 4.2such that
x, u, v, wis feasible forMD, it follows fromTheorem 4.8thatfxsx|C/<fyztyfor
any feasible solutiony, u, v, wofMD. Hencex, u, v, wis a weak Pareto-optimal solution ofMDand the optimal values ofMPandMDare equal.
Remark 4.10. If we replace the convexity hypothesis of f· zt· by strict convexity in
Theorems4.8and4.9, then these theorems hold in the sense of a Pareto-optimal solution.
Remark 4.11. If we replace the convexity hypothesis ofwthby affineness ofhin Theorems4.8
and4.9, then these theorems are also valid.
Theorem 4.12. Letxbe feasible forMPand y, u, v, wfeasible forMD. Ifutf· zt·
vtgwthis pseudoconvex andf,g, andhare regular functions, thenfx sx|C/≤fy zty.
Proof. Suppose thatfx sx | C ≤ fy zty. By using the feasibility assumptions and
ztxsx|C, we obtain
utfx ztxvtgx wthx< utfyztyvtgywthy. 4.19
By the same argument as in the proof ofTheorem 4.5, we arrive at a contradiction.
Theorem 4.13. Letxbe a weak Pareto-optimal solution of MPat which a constraint qualification holds. Then there existu ∈ Rm,v ∈ Rp, andw ∈ Rq such thatx, u, v, wis feasible forMD
andztx sx|C. If in additionutf· zt· vtgwthis pseudoconvex andf,g, andhare regular functions, thenx, u, v, wis a weak Pareto-optimal solution of MDand the optimal values of MPandMDare equal.
The proof is similar to the one used for the previous theorem.
Remark 4.14. If we replace the pseudoconvexity hypothesis ofutf· zt· vtgwthby
Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technologyno. 2009-0074482.
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