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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

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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, jP, hkx 0, kQ, x∈Rn,

MP

wherefi:Rn → R,iM {1,2, . . . , m},gj :Rn → R,jP {1,2, . . . , p}, andhk:Rn → R,

kQ {1,2, . . . , q}, are locally Lipschitz functions. Here,Ci,iM, 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⇐⇒ux∈Rn,

xu⇐⇒ux∈Rn\ {0},

x < u⇐⇒ux∈intRn,

x /uis the negation ofxu,

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, yN,

FxFyKxy, 2.2

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iiThe Clarke generalized directional derivative14of a locally Lipschitz functionF atxin the directiond∈Rn, denoted byF0x;d, is defined as follows:

F0x;d lim sup

yx, t→0t

−1FytdFy, 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:yC . 2.5

The support functionsx|C, being convex and everywhere finite, has a subdifferential, that is, for everyx∈Rnthere existszsuch that

sy|Csx|C ztyx,yC. 2.6

Equivalently,

ztx sx|C. 2.7

The subdifferential ofsx|Cis given by

∂sx|C: zC:ztx sx|C . 2.8

For any setS⊂Rn, the normal cone toSat a pointxSis defined by

NSx:

y∈Rn:ytzx≤0,∀zS . 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 eachxX

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iigA is convex atx0 ∈ X if for eachxXand anyζj∂gjx0,gjxgjx0

ζt

jxx0, wherejA, andgAdenotes the active constraints atx0.

iiih h1, h2, . . . , hkis convex at x0 ∈X if for eachxX and anyαk∂hkx0,

hkxhkx0αtkxx0, for allkQ.

ivhis affine atx0 X if for eachx X and anyα

k∂hkx0,hkxhkx0

αt

kxx0, for allkQ.

vf is pseudoconvex atx0 Xif for eachxXand anyξ

i∂fix0,ξitxx00

impliesfixfix0, for alliM.

vif is strictly pseudoconvex atx0 X if for eachx X withx /x0 and anyξ

i

∂fix0,ξitxx00 impliesfix> fix0, for alliM.

viif is quasiconvex at x0 X if for eachx X withx /x0 and anyξ

i∂fix0,

fixfix0impliesξitxx00, for alliM.

Finally, we recall the definition of Pareto-optimalefficient, nondominatedand weak Pareto-optimal solutions ofMP.

Definition 2.4. iA pointx0Xis said to be a Pareto-optimal solution ofMPif there exists no otherxXwithfxfx0.

iiA pointx0Xis said to be a weak Pareto-optimal solution ofMPif there exists no otherxXwithfx< 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.

u00, v0 0, w00u0Rm, v0 Rp, w0Rqsuch that

0∈

iM

u0i∂fi

x0zi

jA

v0j∂gj

x0

kQ

w0k∂hk

x0,

gx00, hx0 0, ztix0i sx0|Ci

, iM,

whereA jP :gj

x0 0.

KKT

u0≥0, v0 0, w0 u0∈Rm, v0∈Rp, w0 ∈Rqsuch that

0∈

iM

u0i∂fi

x0zi

jA

v0j∂gj

x0

kQ

w0k∂hk

x0,

gx00, hx0 0, ztix0i sx0|Ci

, iM,

whereA jP : gj

x0 0.

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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∈

iM

u0i∂fi

x0zi

jA

v0j∂gj

x0

kQ

w0k∂hk

x0. 3.1

From3.1, there existξi∂fix0,ζj∂gjx0, andαk∂hkx0such that

iM

u0iξizi

jA

vj0ζj

kQ

wk0αk 0. 3.2

Suppose thatx0 Xis not a weak Pareto-optimal solution ofMP. Then there existsxX 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

xx0<0 for anyξi∂fi

x0, iM. 3.3

Since gjx 0 gjx0, jA, we obtain the following inequality with the help of

quasiconvexity ofgAatx0:

ζtjxx00 for anyζj∂gj

x0, jA. 3.4

Also, sincehkx 0 hkx0,kQ, it follows from quasiconvexity ofhatx0that

αt k

xx00 for anyα

k∂hk

x0, kQ. 3.5

From3.3–3.5, we obtain

⎡ ⎣

iM

u0iξizi

jA

v0jζj

kQ

w0kαk

⎤ ⎦

t

xx0<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

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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 existsxX such thatfx sx|Cfx0 sx0|C, that impliesfx ztxfx0 ztx0because of ztxsx|Cand the assumptionztx0 sx0|C. By quasiconvexity off· zt·atx0, we have

ξizit

xx00 for anyξi∂fi

x0, iM. 3.7

Sincex0, u0, v0, w0satisfyKKT, we obtainjAv0jζjkQwk0αk

txx00 for some ζj∂gjx0,jA, andαk∂hkx0,kQ. By regularity ofgA andhatx0, there exists

β∂v0tg

A w0thsuch thatβtxx0 0. With the help of a strict pseudoconvexity of

v0tgA w0th, we have

v0tg

Ax

w0thx>v0tg

A

x0w0thx0. 3.8

SincexX, 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.

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gAx0 gAx0andhx 0 hx0. Therefore,fx sx|Cfx0 sx0 |C<0,

gAxgAx00, andhxhx0 0. Henceu0tfx ztx v0tgAx w0thx<

u0tfx0 ztx0 v0tg

Ax0 w0thx0. Sincef,gA, andhare regular atx0, we obtain

⎛ ⎝

iM

u0

i

fi

x0sx0|C

i

jA

v0

jgj

x0

kQ

w0

khk

x0

⎞ ⎠

iM

u0

i

∂fi

x0∂sx0 |C

i

jA

v0

j∂gj

x0

kQ

w0

k∂hk

x0.

3.11

By pseudoconvexity ofu0tf· zt· v0tg

A w0th, we haveβtxx0<0 for anyβ

iMu0

ifix0 ztix0

jAvj0gjx0

kQw0khkx0. We easily see that this contradicts

0 ∈ iMu0i∂fix0 zi jAv0j∂gjx0

kQw0k∂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∈

iM

ui

∂fi

yzi

jP

vj∂gj

y

kQ

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∈

iM

ui

∂fi

yzi

jP

vj∂gj

y

kQ

wk∂hk

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According to4.1, there existξi∂fiy,ζj∂gjy, andαk∂hkysuch that

iM

uiξizi

jP

vjζj

kQ

wkαk 0. 4.2

Assume that

fx sx|C< fyztyvtgyewthye. 4.3

Multiplying4.3byuand using the equalityztx sx|C, we have

utfx ztxutfyztyvtgywthy<0 4.4

sinceu≥0 andute 1. Now by convexity off· zt·,gandwth, we obtain

utfx ztxutfyzty

iM

uiξizit

xy,ξi∂fi

y,

vtgxvtgy

jP

vjζtj

xy,ζj∂gj

y,

wthxwthy

kQ

wkαtk

xy,αk∂hk

y.

4.5

Sincevtgx0 andwthx 0, we obtain the following inequality from4.2,4.5:

utfx ztxutfyztyvtgywthy0, 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,vRp, 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∈

iM

ui

∂fix zi

jP

vj∂gjx

kQ

wk∂hkx,

vtgx 0, u≥0, v0.

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Sinceu≥0, we can scale theuis,vjsandwksas follows:

ui ui iMui

, vj

vj

iMui

, wk wk iMui

. 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 aneness 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 | Cfy ztyvtgyewthye. By

feasibility ofx, we obtain

utfx ztxvtgx wthx< utfyztyvtgywthy. 4.10

Sincef,g, and hare regular functions, we have βtxy < 0 by the pseudoconvexity of

utf· zt· vtgwthfor anyβ

iMuifiy ztiy jPvjgjy kQwkhky.

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,vRp, 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

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We now prove duality relations between MP and the following Mond-Weir-type dual problem:

maximize fyzty

subject to 0∈

iM

ui

∂fi

yzi

jP

vj∂gj

y

kQ

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∈

iM

ui

∂fi

yzi

jP

vj∂gj

y

kQ

wk∂hk

y, 4.11

vtgywthy0. 4.12

According to4.11, there existξi∂fiy,ζj∂gjy, andαk∂hkysuch that

iM

uiξizi

jP

vjζj

kQ

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 ztxutfyzty0, 4.17

since vtgx 0 andwthx 0. However, 4.17contradicts 4.15. Hence the proof is

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Theorem 4.9. Letxbe a weak Pareto-optimal solution of MPat which a constraint qualification holds. Then there existu∈Rm,vRp, 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∈

iM

ui

∂fix zi

jP

vj∂gjx

kQ

wk∂hkx,

vtgx 0, u0, 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 aneness 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 | Cfy 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

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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.

References

1 P. Wolfe, “A duality theorem for non-linear programming,”Quarterly of Applied Mathematics, vol. 19, pp. 239–244, 1961.

2 B. Mond and T. Weir, “Generalized concavity and duality,” inGeneralized Concavity in Optimization and Economics, S. Schaible and W. T. Ziemba, Eds., pp. 263–279, Academic Press, New York, NY, USA, 1981.

3 Q. H. Ansari, S. Schaible, and J.-C. Yao, “η-pseudolinearity,” Rivista di Matematica per le Scienze Economiche e Sociali, vol. 22, no. 1-2, pp. 31–39, 1999.

4 R. R. Egudo and M. A. Hanson, “On sufficiency of Kuhn-Tucker conditions in nonsmooth multiobjective programming,” Technical Report M-888, Florida State University, Tallahassee, Fla, USA, 1993.

5 V. Jeyakumar and B. Mond, “On generalised convex mathematical programming,” Journal of the Australian Mathematical Society. Series B, vol. 34, no. 1, pp. 43–53, 1992.

6 D. S. Kim and S. Schaible, “Optimality and duality for invex nonsmooth multiobjective programming problems,”Optimization, vol. 53, no. 2, pp. 165–176, 2004.

7 J. C. Liu, “Optimality and duality for generalized fractional programming involving nonsmooth pseudoinvex functions,”Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp. 667–685, 1996.

8 O. L. Mangasarian,Nonlinear Programming, McGraw-Hill, New York, NY, USA, 1969.

9 S. K. Mishra and R. N. Mukherjee, “On generalised convex multi-objective nonsmooth program-ming,”Australian Mathematical Society, vol. 38, no. 1, pp. 140–148, 1996.

10 B. Mond and M. Schechter, “Nondifferentiable symmetric duality,” Bulletin of the Australian Mathematical Society, vol. 53, no. 2, pp. 177–188, 1996.

11 A. A. K. Majumdar, “Optimality conditions in differentiable multiobjective programming,”Journal of Optimization Theory and Applications, vol. 92, no. 2, pp. 419–427, 1997.

12 C. Singh, “Optimality conditions in multiobjective differentiable programming,”Journal of Optimiza-tion Theory and ApplicaOptimiza-tions, vol. 53, no. 1, pp. 115–123, 1987.

13 D. S. Kim, G. M. Lee, B. S. Lee, and S. J. Cho, “Counterexample and optimality conditions in differentiable multiobjective programming,”Journal of Optimization Theory and Applications, vol. 109, no. 1, pp. 187–192, 2001.

References

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