Multiobjective Optimization Involving V-Type I Invex Functions

Volume 2010, Article ID 898626,14pages doi:10.1155/2010/898626

Research Article

Optimality and Duality in Nonsmooth

Multiobjective Optimization Involving V-Type I Invex Functions

Ravi P. Agarwal,

I. Ahmad,

Z. Husain,

and A. Jayswal

1Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

2Department of Mathematical Sciences, Florida Institute of Techynology, Melbourne 32901, USA

3Department of Mathematics, Aligarh Muslim University, Aligarh-202 002, India

4Department of Mathematics, Faculty of Applied Sciences, Integral University, Lucknow 226026, India

5Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi 835 215, India

Correspondence should be addressed to Ravi P. Agarwal,[email protected] Received 4 June 2010; Accepted 26 September 2010

Academic Editor: R. N. Mohapatra

Copyrightq2010 Ravi P. Agarwal et al. 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.

A new class of generalized V-type I invex functions is introduced for nonsmooth multiobjective programming problem. Based upon these generalized invex functions, we establish sufficient optimality conditions for a feasible point to be an efficient or a weakly efficient solution. Weak, strong, and strict converse duality theorems are proved for Mond-Weir type dual program in order to relate the weakly efficient solutions of primal and dual programs.

1. Introduction

There is a vital role of convexity in many aspects of mathematical programming including optimality conditions, duality theorems, and alternative theorems, but, due to insufficiency of convexity notion in many mathematical models used in decision science, economics, engineering, and so forth, there has been an increasing interest in relaxing convexity assumptions in connection with sufficiency and duality theorems. One of the most lively generalizations of convexity is owed to Hanson1, which was named as invexity by Craven 2. Later, Hanson and Mond 3 defined two new classes of functions, called type I and type II functions, which have been further generalized by many researchers and applied to nonlinear programming problems in different settings. This concept was further generalized

to pseudo-type I and quasi-type I functions by Rueda and Hanson 4 and to pseudo- quasi-type I, quasi-pseudo-type I, and strictly quasi-pseudo-type I functions by Kaul et al.

5.

Since many practical problems encountered in economics, engineering design, and management science, and so forth can be described only by nonsmooth functions, consequently, the theory of nonsmooth optimization using locally Lipschitz functions was put forward by Clarke in 1980’s see6. He extended the properties of convex functions to the case of locally Lipschitz functions by suitably defining a generalized derivative and a subdifferential. Later on, the notion of invexity was extended to locally Lipschitz functions by Craven 7, by replacing the derivative with Clarke’s generalized gradient.

Reiland 8 pointed out that under the invexity assumption, the Kuhn-Tucker conditions also assure the optimality in nondifferentiable programming involving locally Lipschitz functions. Recent development of optimality conditions and duality relations for nonsmooth multiobjective programming problems involving locally Lipschitz functions can be seen in 9–17.

In order to resolve the difficulty of demanding same function η for objective and constraint functions in problems dealing with invexity, Jeyakumar and Mond 18 introduced the concept of V-invexity and its generalization for differentiable multiobjective programming problems. However, the extension of their studies to nonsmooth case was discussed by Egudo and Hanson 9. Further development in this direction can be found in 14,17. Zhao 19established optimality conditions and duality results in nonsmooth scalar programming assuming Clarke’s generalized subgradients under type I functions6.

Kuk and Tanino15obtained Karush-Kuhn-Tucker type necessary and sufficient optimality conditions and duality theorems for nonsmooth multiobjective programming problems involving generalized type I vector-valued functions.

In this paper, we are motivated by Kuk and Tanino 15 to introduce generalized type I invex functions, called generalized V-type I invex functions, an extension of V- type I functions introduced by Hanson et al. 20 to nonsmooth cases. By utilizing these new concepts, we obtain Karush-Kuhn-Tucker type sufficient optimality conditions and Mond-Weir type duality relations for nonsmooth multiobjective programming problems. Our results generalize a variety of previously known results in this area.

2. Notations and Preliminaries

Throughout the paper, we use the following conventions of vectors inRⁿ. For anyx, y∈Rⁿ, x y ⇔ xi yi, i 1,2, . . . , n, x ≥ y ⇔ x y,x /y, andx > y ⇔ xi > yi, i 1,2, . . . , n.

A functionf : Rⁿ → Ris said to be locally Lipschitz at a pointx ∈ Rⁿ if there exist scalarsζ >0 and >0 such that

f x¹

−f

x²ζx¹−x², ∀x¹, x²∈x B, 2.1

wherex Bis the open ball of radiusaroundxand · is any norm inRⁿ.

The Clarke generalized directional derivative6of a locally Lipschitz functionf:Rⁿ → Ratxin the directionv∈Rⁿ, denoted byf^◦x;v, is defined as

f^◦x;v lim sup

y→x

t↓0

f y tv

−f y

t , 2.2

whereyis a vector inRⁿ.

The Clarke generalized gradient6off :Rⁿ → Ratx, denoted by∂^cfx, is defined as

∂^cfx

ξ∈Rⁿ:f^◦x;vξv, ∀v∈Rⁿ . 2.3

It follows that for anyv∈Rⁿ,f^◦x;v max{ξ^Tv:ξ∈∂^cfx}.

We consider the following nonlinear multiobjective programming problem:

Minimize fx

f₁x, f2x, . . . , fkx , subject to x∈S

x∈X:gx0

, MP

where X ⊆ Rⁿ is an open set and the functions f f1, f₂, . . . , f_k : X → R^k and g g1, g₂, . . . , g_m:X → R^mare locally Lipschitz onX.

Since the objectives in multiobjective programming problems generally conflict with one another, an optimal solution is chosen from the set of efficientweakly efficientsolutions in the following sensesee21.

Definition 2.1. A pointx∈Sis said to be an efficient solution ofMPif there exists nox∈S such thatfx≤fx.

Definition 2.2. A pointx∈Sis said to be a weakly efficient solution ofMPif there exists no x∈Ssuch thatfx< fx.

LetK{1,2, . . . , k}, and letM{1,2, . . . , m}be any index set. Forx∈S,Jx {j ∈ M:g_jx 0}andg_Jdenotes the vector of active constraints atx.

We define the following generalized V-type I invex functions. Letf andg be locally Lipschitz functions at a given pointu∈X.

Definition 2.3. The pairf, gis said to be V-type I invex atu∈Xif for eachx∈Sand for any ξi∈∂^cfiu, ζj∈∂^cgju,there exist vectorsαiandβj, whereαi, βj :X×X → R \ {0}, and a functionη:S×X → Rⁿsuch that for alli∈K, j ∈M

f_ix−f_iuα_ix, uξiηx, u,

−gjuβjx, uζjηx, u. 2.4 Remark 2.4. Ifα_ix, u β_jx, u 1, fori ∈ K,j ∈ M, we obtain the definition of type I function given by Kuk and Tanino15.

Definition 2.5. The pairf, g is said to be V-pseudo-quasi-type I invex atu∈ Xif for each x ∈ S and for anyξ_i ∈ ∂^cf_iu, ζj ∈ ∂^cg_ju, there exist vectorsα_i and β_j, where α_i,β_j : X×X → R \ {0}, and a functionη:S×X → Rⁿsuch that

k i1

αifix<^k

i1

αifiu ⇒^k

i1

ξiηx, u<0,

−^m

j1

βjgju0⇒^m

j1

ζjηx, u0.

2.5

If in the above definition first inequality is satisfied as

k i1

αifix^k

i1

αifiu ⇒^k

i1

ξiηx, u<0, 2.6

then we say thatf, gis V-strictly pseudo-quasi-type I invex atu.

Definition 2.6. The pairf, gis said to be V-quasi-pseudo-type I invex atu ∈ X if for each x∈Sand for anyξi∈∂^cfiu,ζj∈∂^cgju, there exist vectorsαiandβj, whereαi,βj:X×X → R \ {0}, and a functionη: S×X → Rⁿsuch that

k i1

ξ_iηx, u>0⇒^k

i1

α_if_ix>^k

i1

α_if_iu,

−^m

j1

β_jg_ju<0⇒^m

j1

ζ_jηx, u<0.

2.7

If in the above definition second inequality is satisfied as

−^m

j1

βjgju0⇒^m

j1

ζjηx, u<0, 2.8

then we say thatf, gis V-quasistrictly pseudo-type I invex atu.

We will need the following result.

Theorem 2.7 see21, page 45. Let the functions f_i : Rⁿ → Ri 1,2,· · ·, k be locally Lipschitzian at a pointx^∗∈Rⁿ. Then, for weightsw_i∈R, one has

∂^c _k

i1

wifi

x^∗⊂^k

i1

wi∂^cfix^∗. 2.9

3. Karush-Kuhn-Tucker Type Sufficient Optimality Conditions

In this section, we derive some sufficient optimality conditions for a feasible solution to be an efficient or a weakly efficient solution forMP. Throughout this section, and inSection 4, f^λdenotes the vectorλ1f₁, λ₂f₂, . . . , λ_kf_kandg_J^μdenotes the vector whose components are μ_jgj, j ∈Jx.

Theorem 3.1. Suppose that there exist a feasible solutionxforMPand scalarsλi>0, i∈K, μ_j 0, j ∈Jxsuch that

i0∈_k

i1λ_i∂^cf_ix

j∈Jxμ_j∂^cg_jx, ii f, gJis V-type I invex atx.

Then,xis an efficient solution forMP.

Proof. Hypothesis iimplies that there existξi ∈ ∂^cfix, i ∈ K and ζj ∈ ∂^cgjx, j ∈ Jx satisfying

0^k

i1

λ_iξ_i

j∈Jx

μ_jζ_j. 3.1

Sincef, gJis V-type I invex atx, we have for allx∈S

f_ix−f_ixα_ix, xξiηx, x, for anyξ_i∈∂^cf_ix, i∈K,

0−gjxβjx, xζjηx, x, for anyζj∈∂^cgjx, j ∈Jx. 3.2

By usingα_ix, x>0, i∈Kandβ_jx, x>0, j ∈Jx, we get

1

α_ix, xfix− 1

α_ix, xfixξiηx, x, for anyξ_i∈∂^cfix, i∈K, 0ζ_jηx, x for any ζ_j∈∂^cg_jx, j ∈Jx.

3.3

Asλ_i>0, i∈Kandμ_j0, j∈Jx, using3.3, we obtain

k i1

λ_i

αix, xf_ix−^k

i1

λ_i

αix, xf_ix

⎛

⎝^k

i1

λ_iξ_i

j∈Jx

μ_jζ_j

⎞

⎠ηx, x, 3.4

which on using3.1yields k i1

λ_i

αix, xf_ix^k

i1

λ_i

αix, xf_ix. 3.5

Suppose thatxis not an efficient solution forMP. Then, there exist a feasible solutionxfor MPand an indexrsuch that

frx< frx,

f_ixf_ix, ∀i /r. 3.6

Becauseλi>0, andαix, x>0, i∈K,we have k

i1

λ_i

αix, xf_ix<^k

i1

λ_i

αix, xf_ix. 3.7

This contradicts inequality3.5, andxis thus an efficient solution forMP.

Theorem 3.2. Suppose that there exist a feasible solutionxforMPand scalarsλi>0, i∈K, μ_j 0, j ∈Jxsuch that

i0∈_k

i1λi∂^cfix

j∈Jxμ_j∂^cgjx, ii f^λ, g_J^μis V-pseudo-quasi-type I invex atx.

Then,xis an efficient solution forMP.

Proof. Suppose thatxis not an efficient solution forMP. Then, there exist a feasible solution xforMPand an indexrsuch that

f_rx< f_rx,

fixfix, ∀i /r. 3.8

Sinceλ_i>0 andα_ix, x>0,i∈K,above inequalities give k

i1

λiαix, xfix<^k

i1

λiαix, xfix. 3.9

Alsogjx 0,j∈Jxyields

j∈Jx

β_jx, xμ_jg_jx 0. 3.10

The hypothesisiiand inequalities3.9and3.10imply k

i1

ξ_iηx, x<0, for anyξ_i∈∂^c λ_if_i

x,

j∈Jx

ζ_jηx, x0, for anyζ_j∈∂^c μ_jgj

x.

3.11

Adding these inequalities, we obtain

⎛

⎝^k

i1

ξ_i

j∈Jx

ζ_j

⎞

⎠ηx, x<0, 3.12

but, byTheorem 2.7, for someξ_i ∈ ∂^cλif_ixandζ_j ∈ ∂^cμ_jg_jx, there existξ_i ∈ ∂^cf_ix andζj∈∂^cgjxsuch that

ξ_iλ_iξ_i, i∈K andζ_jμ_jζ_j, j ∈Jx. 3.13

Hence, the above inequality becomes

⎛

⎝^k

i1

λiξi

j∈Jx

μ_jζj

⎞

⎠ηx, x<0. 3.14

This contradictsi, as forξi∈∂^cfix, ζj∈∂^cgjx,_k

i1λiξi

j∈Jxμ_jζj0.Hence,xis an efficient solution forMP.

Remark 3.3. If we takeλ_i0,i∈K,_k

i1λ_i1,then the above theorem still holds under the assumption thatf^λ, g_J^μis V-strictly pseudo-quasi-type I invex atx.

Theorem 3.4. Suppose that there exist a feasible solution xforMPand scalarsλi 0,i ∈ K, _k

i1λ_i1, μ_j0, j ∈Jxsuch that i0∈_k

i1λ_i∂^cf_ix

j∈Jxμ_j∂^cg_jx, ii f, gJis V-type I invex atx.

Then,xis a weakly efficient solution forMP.

Proof. Following the proof ofTheorem 3.1, we obtain

k i1

λi

α_ix, xfix^k

i1

λi

α_ix, xfix. 3.15

Suppose thatxis not a weakly efficient solution forMP. Then, there exists a feasible solution xx /xforMPsuch that

f_ix< f_ix, i∈K. 3.16

Becauseλ_i0,i∈K,_k

i1λ_i 1, andα_ix, x>0,i∈K,we have k

i1

λ_i

αix, xf_ix<^k

i1

λ_i

αix, xf_ix. 3.17

This contradicts inequality3.15, andxis thus a weakly efficient solution forMP.

Theorem 3.5. Suppose that there exist a feasible solution x for MP and scalars λ_i 0, i ∈ K,_k

i1λi1 andμ_j0, j ∈Jxsuch that i0∈_k

i1λ_i∂^cf_ix

j∈Jxμ_j∂^cg_jx, ii f^λ, g_J^μis V-pseudo-quasi-type I invex atx.

Then,xis a weakly efficient solution forMP.

Proof. Suppose thatxis not a weakly efficient solution forMP. Then, there exists a feasible solutionxx /xforMPsuch that

fix< fix, i∈K. 3.18

Sinceλi0, i∈K,_k

i1λi1, andαix, x>0, i∈K,the above inequality gives k

i1

λiαix, xfix<^k

i1

λiαix, xfix. 3.19

The remaining part of the proof is similar to that ofTheorem 3.2.

Theorem 3.6. Suppose that there exist a feasible solution x for MP and scalars λ_i 0, i ∈ K,_k

i1λi1 andμ_j0, j ∈Jxsuch that i0∈_k

i1λ_i∂^cf_ix

j∈Jxμ_j∂^cg_jx,

ii f^λ, g_J^μis V-quasistrictly pseudo-type I invex atx.

Then,xis a weakly efficient solution forMP.

Proof. Suppose thatxis not a weakly efficient solution forMP. Then, there exists a feasible solutionxx /xforMPsuch that

fix< fix, i∈K. 3.20

Sinceλi0, i∈K,_k

i1λi1, andαix, x>0, i∈K,the above inequality gives k

i1

λ_iα_ix, xf_ix<^k

i1

λ_iα_ix, xf_ix. 3.21

Alsogjx 0, j ∈Jxyields

j∈Jx

β_jx, xμ_jg_jx 0. 3.22

If hypothesisiiholds, we have k

i1

ξ_iηx, x>0⇒^k

i1

λ_iα_ix, xfix>^k

i1

λ_iα_ix, xfix, for any ξ_i∈∂^c λ_if_i

x, 3.23

−

j∈Jx

β_jx, xμ_jg_jx0⇒

j∈Jx

ζ_jηx, x<0, for anyζ_j∈∂^c μ_jg_j

x. 3.24

In view of3.22,3.24implies

j∈Jx

ζ_jηx, x<0, for anyζ_j∈∂^c μ_jgj

x. 3.25

Also, by assumptioniandTheorem 2.7, we have k

i1

ξ_i

j∈Jx

ζ_j0. 3.26

Therefore,3.25becomes k

i1

ξ_iηx, x>0, for any ξ_i∈∂^c λ_if_i

x. 3.27

In view of3.27,3.23yields k

i1

λ_iα_ix, xfix>^k

i1

λ_iα_ix, xfix, 3.28

which contradicts3.21. Hence,xis a weakly efficient solution forMP.

4. Mond-Weir Type Duality

We now consider the following Mond-Weir type dual forMP:

MWDMaximize f y

subject to 4.1

0∈^k

i1

λi∂^cfi

y ^m

j1

μj∂^cgj y

, 4.2

μjgj y

0, j∈M, 4.3

λ_i0, i∈K, 4.4

μj0, j ∈M, 4.5

k i1

λi1. 4.6

LetTbe the set of all feasible solutions ofMWD.

Theorem 4.1weak duality. Letx∈Sandy, λ, μ∈Tsuch thatf^λ, g^μis V-pseudo-quasi-type I invex aty. Then the following cannot hold

fx< f y

. 4.7

Proof. Suppose the contrary to the result that4.7holds, that is,fx< fy.Usingα_ix, y>

0,λi0,i∈Kand_k

i1λi1, we get k

i1

αi

x, y

λifix<^k

i1

αi

x, y λifi

y

. 4.8

Also, asβ_jx, y>0, j∈M, inequality4.3yields

−^m

j1

βj x, y

μjgj y

0. 4.9

By V-pseudo-quasi-type I invexity off^λ, g^μaty,4.8and4.9give k

i1

ξ_iη x, y

<0, for any ξ_i∈∂^c λifi

y ,

m j1

ζ_jη x, y

0, for any ζ_j∈∂^c μjgj

y .

4.10

Adding the above inequalities, we get

⎛

⎝^k

i1

ξ_i ^m

j1

ζ_j

⎞

⎠η x, y

<0. 4.11

However, byTheorem 2.7, for some ξ_i ∈ ∂^cλif_iy and ζ_j ∈ ∂^cμjg_jy, there exist ξ_i ∈

∂^cf_iyandζ_j∈∂^cg_jysuch that

ξ_iλ_iξ_i, i∈K andζ_jμ_jζ_j, j∈M. 4.12

Hence, the above inequality changes to

⎛

⎝^k

i1

λ_iξ_i ^m

j1

μ_jζ_j

⎞

⎠η x, y

<0, 4.13

which contradicts the dual constraint 4.2, because ξ_i ∈ ∂^cf_iy and ζ_j ∈ ∂^cg_jy imply _k

i1λ_iξ_{i m}

j1μ_jζ_j0.Hence,4.7cannot hold.

Definition 4.2 Cottle’s constraint qualification21, page 48. Letfi, i ∈ K and gj, j ∈ M be locally Lipschitz functions at a pointu ∈X. The problemMPis said to satisfy Cottle’s constraint qualification atuif eitherg_ju<0 for allj ∈Mor 0∈conv{∂^cg_ju:g_ju 0}, where convZdenotes the convex hull of the set Z.

Theorem 4.3Karush-Kuhn-Tucker type necessary conditions21, page 50. Assume thatxis a weakly efficient solution forMPat which Cottle’s constraint qualification is satisfied. Then, there exist scalarsλi0, i∈K,_k

i1λi1 andμ_j0, j ∈Msuch that

0∈^k

i1

λi∂^cfix ^m

j1

μ_j∂^cgjx, μ_jgjx 0, j ∈M.

4.14

Theorem 4.4 strong duality. Let x be a weakly efficient solution for MP at which Cottle’s constraint qualification is satisfied. Then, there existλ∈R^k,μ∈R^msuch thatx, λ, μis feasible for (MWD) and the objective values of MPand (MWD) are equal. Further, if the hypotheses of weak duality (Theorem 4.1) hold for all feasible solutionsy, λ, μfor (MWD), thenx, λ, μis a weakly efficient solution of (MWD).

Proof. Sincexis a weakly efficient solution ofMPand the Cottle’s constraint qualification is satisfied atx, fromTheorem 4.3, there existλ_i0,i∈K,_k

i1λ_i1, andμ_j 0,j ∈Msuch that

0∈^k

i1

λi∂^cfix ^m

j1

μ_j∂^cgjx, μ_jgjx 0, j ∈M,

4.15

which yields thatx, λ, μis feasible forMWDand the corresponding objective values are equal. Ifx, λ, μis not a weakly efficient solution forMWD, then there exists a feasible solutiony, λ, μforMWDsuch that

fx< f y

, 4.16

which contradicts the weak duality Theorem 4.1. Hence, x, λ, μ is a weakly efficient solution forMWD.

Theorem 4.5strict converse duality. Letx∈Sandy, λ, μ∈Tsuch that

k i1

λifix^k

i1

λifi y

. 4.17

If

i f^λ, g^μis V-strictly pseudo-quasi-type I invex aty, iiα¹_ix, y 1,i∈K,

thenxy.

Proof. We assume thatx /yand exhibit a contradiction. Sincey, λ, μ ∈T, from4.2, there existξ_i∈∂^cfiy,i∈Kandζ_j∈∂^cgjy,j ∈Msuch that

k i1

λiξ_i ^m

j1

μ_jζ_j 0. 4.18

The hypothesisialong with4.3andβ_jx, y>0, j ∈Myields m

j1

ζ_jη x, y

0, for any ζ_j ∈∂^c μ_jg_j

y

. 4.19

Also by Theorem 2.7, for some ξ_i ∈ ∂^cfiy, i ∈ K and ζ_j ∈ ∂^cgjy, j ∈ M, there exist ξ_i∈∂^cλif_iyandζ_j∈∂^cμ_jg_jy such thatξ_i λ_iξ_iandζ_jμ_jζ_j.

Hence,4.18gives

⎛

⎝^k

i1

ξ_i ^m

j1

ζ_j

⎞

⎠η x, y

0, 4.20

which with4.19gives k

i1

ξ_iη x, y

0, for anyξ_i∈∂^c λifi

y

. 4.21

Therefore the hypothesisiagain yields k

i1

αi

x, y

λifix>^k

i1

αi

x, y λifi

y

. 4.22

Sinceα_ix, y 1, i∈K,we have k

i1

λifix>^k

i1

λifi y

, 4.23

which contradicts4.17. This completes the proof.

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