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,
1, 2I. Ahmad,
1, 3Z. Husain,
4and A. Jayswal
51Department 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
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 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 belocally Lipschitz at a pointx ∈ Rn if there exist scalarsζ >0 and >0 such that
fx1−fx2ζx1−x2, ∀x1, x2∈x B, 2.1
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
fy tv−fy
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{ξTv:ξ∈∂cfx}.
We consider the following nonlinear multiobjective programming problem:
Minimize fx f1x, f2x, . . . , fkx,
subject to x∈Sx∈X:gx0, MP
where X ⊆ Rn is an open set and the functions f f1, f2, . . . , fk : X → Rk and g
g1, g2, . . . , gm:X → Rmare 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 solutionofMPif there exists nox∈S such thatfx≤fx.
Definition 2.2. A pointx∈Sis said to be a weakly efficient solutionofMPif 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:gjx 0}andgJdenotes 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
fix−fiuαix, uξiηx, u,
−gjuβjx, uζjηx, u.
2.4
Definition 2.5. The pairf, g is said to be V-pseudo-quasi-type I invex atu∈ Xif for each
x ∈ S and for anyξi ∈ ∂cfiu, ζj ∈ ∂cgju, there exist vectorsαi and βj, where αi,βj :
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
ξiηx, u>0⇒ k
i1
αifix> k
i1
αifiu,
−m
j1
βjgju<0⇒ m
j1
ζjηx, 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 functionsfi : Rn → Ri 1,2,· · ·, k be locally
Lipschitzian at a pointx∗∈Rn. Then, for weightswi∈R, one has
∂c
k
i1 wifi
x∗⊂k
i1
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,
fλdenotes the vectorλ
1f1, λ2f2, . . . , λkfkandgJμdenotes the vector whose components are
μjgj, j ∈Jx.
Theorem 3.1. Suppose that there exist a feasible solutionxforMPand scalarsλi>0, i∈K, μj 0, j ∈Jxsuch that
i0∈ki1λi∂cfix j∈Jxμj∂cgjx,
ii f, gJis V-type I invex atx.
Then,xis an efficient solution forMP.
Proof. Hypothesisiimplies that there existξi ∈ ∂cfix, i ∈ K and ζj ∈ ∂cgjx, j ∈ Jx satisfying
0 k
i1 λiξi
j∈Jx
μjζj. 3.1
Sincef, gJis V-type I invex atx, we have for allx∈S
fix−fixαix, xξiηx, x, for anyξi∈∂cfix, i∈K,
0−gjxβjx, xζjηx, x, for anyζj∈∂cgjx, j ∈Jx.
3.2
By usingαix, x>0, i∈Kandβjx, x>0, j ∈Jx, we get
1
αix, xfix− 1
αix, xfixξiηx, x, for anyξi∈∂ cf
ix, i∈K,
0ζjηx, x for any ζj∈∂cgjx, j ∈Jx.
3.3
Asλi>0, i∈Kandμj0, j∈Jx, using3.3, we obtain
k
i1 λi
αix, xfix− k
i1 λi
αix, xfix
⎛ ⎝k
i1 λiξi
j∈Jx
μjζj
⎞
which on using3.1yields
k
i1 λi
αix, xfix k
i1 λi
αix, xfix. 3.5
Suppose thatxis not an efficient solution forMP. Then, there exist a feasible solutionxfor
MPand an indexrsuch that
frx< frx,
fixfix, ∀i /r.
3.6
Becauseλi>0, andαix, x>0, i∈K,we have
k
i1 λi
αix, xfix< k
i1 λi
αix, xfix. 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, μj 0, j ∈Jxsuch that
i0∈ki1λi∂cfix j∈Jxμj∂cgjx,
ii fλ, gJμis 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
frx< frx,
fixfix, ∀i /r.
3.8
Sinceλi>0 andαix, x>0,i∈K,above inequalities give
k
i1
λiαix, xfix< k
i1
λiαix, xfix. 3.9
Alsogjx 0,j∈Jxyields
j∈Jx
The hypothesisiiand inequalities3.9and3.10imply
k
i1 ξ
iηx, x<0, for anyξi∈∂c
λifi
x,
j∈Jx
ζ
jηx, x0, for anyζj∈∂c
μjgj
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 someξi ∈ ∂cλifixandζj ∈ ∂cμjgjx, there existξi ∈ ∂cfix andζj∈∂cgjxsuch that
ξ
iλiξi, i∈K andζj μjζj, j ∈Jx. 3.13
Hence, the above inequality becomes
⎛ ⎝k
i1 λiξi
j∈Jx
μjζj
⎞
⎠ηx, x<0. 3.14
This contradictsi, as forξi∈∂cfix, ζj∈∂cgjx,ki1λiξi
j∈Jxμjζj0.Hence,xis an efficient solution forMP.
Remark 3.3. If we takeλi0,i∈K,ki1λi1,then the above theorem still holds under the
assumption thatfλ, gJμis V-strictly pseudo-quasi-type I invex atx.
Theorem 3.4. Suppose that there exist a feasible solutionxforMPand scalarsλi 0,i ∈ K,
k
i1λi1, μj0, j ∈Jxsuch that
i0∈ki1λi∂cfix j∈Jxμj∂cgjx,
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
αix, xfix k
i1 λi
Suppose thatxis not a weakly efficient solution forMP. Then, there exists a feasible solution
xx /xforMPsuch that
fix< fix, i∈K. 3.16
Becauseλi0,i∈K,ki1λi 1, andαix, x>0,i∈K,we have
k
i1 λi
αix, xfix< k
i1 λi
αix, xfix. 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 λi 0, i ∈
K,ki1λi1andμj0, j ∈Jxsuch that
i0∈ki1λi∂cfix j∈Jxμj∂cgjx,
ii fλ, gJμis 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,ki1λi1, 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 λi 0, i ∈
K,ki1λi1andμj0, j ∈Jxsuch that
i0∈ki1λi∂cfix j∈Jxμj∂cgjx,
ii fλ, gJμis 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
Sinceλi0, i∈K,ki1λi1, andαix, x>0, i∈K,the above inequality gives
k
i1
λiαix, xfix< k
i1
λiαix, xfix. 3.21
Alsogjx 0, j ∈Jxyields
j∈Jx
βjx, xμjgjx 0. 3.22
If hypothesisiiholds, we have
k
i1 ξ
iηx, x>0⇒ k
i1
λiαix, xfix> k
i1
λiαix, xfix, for any ξi∈∂c
λifi
x,
3.23
−
j∈Jx
βjx, xμjgjx0⇒
j∈Jx
ζ
jηx, x<0, for anyζj∈∂c
μjgj
x. 3.24
In view of3.22,3.24implies
j∈Jx
ζ
jηx, x<0, for anyζj∈∂c
μjgj
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 ξi∈∂c
λifi
x. 3.27
In view of3.27,3.23yields
k
i1
λiαix, xfix> k
i1
λiαix, xfix, 3.28
4. Mond-Weir Type Duality
We now consider the following Mond-Weir type dual forMP:
MWDMaximize fy
subject to
4.1
0∈ k
i1
λi∂cfiy m
j1
μj∂cgjy, 4.2
μjgjy0, j∈M, 4.3
λi0, 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 thatfλ, gμis V-pseudo-quasi-type
I invex aty. Then the following cannot hold
fx< fy. 4.7
Proof. Suppose the contrary to the result that4.7holds, that is,fx< fy.Usingαix, y> 0,λi0,i∈Kandki1λi1, we get
k
i1
αix, yλifix< k
i1
αix, yλifiy. 4.8
Also, asβjx, y>0, j∈M, inequality4.3yields
−m
j1
βjx, yμjgjy0. 4.9
By V-pseudo-quasi-type I invexity offλ, gμaty,4.8and4.9give
k
i1 ξ
iη
x, y<0, for any ξi∈∂cλifiy,
m
j1 ζ
jη
x, y0, for any ζj∈∂cμjgjy.
Adding the above inequalities, we get
⎛ ⎝k
i1 ξ
i m
j1 ζ
j
⎞
⎠ηx, y<0. 4.11
However, byTheorem 2.7, for some ξi ∈ ∂cλifiy and ζj ∈ ∂cμjgjy, there exist ξi ∈
∂cf
iyandζj∈∂cgjysuch that
ξ
iλiξi, i∈K andζj μjζj, j∈M. 4.12
Hence, the above inequality changes to
⎛ ⎝k
i1 λiξi
m
j1 μjζj
⎞
⎠ηx, y<0, 4.13
which contradicts the dual constraint 4.2, because ξi ∈ ∂cfiy and ζj ∈ ∂cgjy imply
k
i1λiξi
m
j1μjζj0.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 eithergju<0 for allj ∈Mor 0∈conv{∂cgju:gju 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,ki1λi1andμj0, j ∈Msuch that
0∈ k
i1
λi∂cfix m
j1 μj∂cg
jx,
μjgjx 0, j ∈M.
4.14
Theorem 4.4 strong duality. Let x be a weakly efficient solution for MP at which Cottle’s
Proof. Sincexis a weakly efficient solution ofMPand the Cottle’s constraint qualification is satisfied atx, fromTheorem 4.3, there existλi0,i∈K,ik1λi1, andμj 0,j ∈Msuch that
0∈ k
i1
λi∂cfix m
j1 μj∂cg
jx,
μjgjx 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< fy, 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
λifiy. 4.17
If
i fλ, gμis V-strictly pseudo-quasi-type I invex aty, iiα1ix, y 1,i∈K,
thenxy.
Proof. We assume thatx /yand exhibit a contradiction. Sincey, λ, μ ∈T, from4.2, there existξi∈∂cfiy,i∈Kandζj∈∂cgjy,j ∈Msuch that
k
i1 λiξi
m
j1
μjζj 0. 4.18
The hypothesisialong with4.3andβjx, y>0, j ∈Myields
m
j1 ζ
jη
x, y0, for any ζj ∈∂cμjgjy. 4.19
Also by Theorem 2.7, for some ξi ∈ ∂cfiy, i ∈ K and ζj ∈ ∂cgjy, j ∈ M, there exist
ξ
Hence,4.18gives
⎛ ⎝k
i1 ξ
i m
j1 ζ
j
⎞
⎠ηx, y0, 4.20
which with4.19gives
k
i1 ξ
iη
x, y0, for anyξi∈∂cλifiy. 4.21
Therefore the hypothesisiagain yields
k
i1
αix, yλifix> k
i1
αix, yλifiy. 4.22
Sinceαix, y 1, i∈K,we have
k
i1
λifix> k
i1
λifiy, 4.23
which contradicts4.17. This completes the proof.
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