Volume 2009, Article ID 928968,8pages doi:10.1155/2009/928968
Research Article
A Note on Essential Components and Essential
Weakly Efficient Solutions for Multiobjective
Optimization Problems
Qun Luo
Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China
Correspondence should be addressed to Qun Luo,[email protected]
Received 19 June 2009; Revised 21 September 2009; Accepted 24 November 2009
Recommended by Paolo Ricci
The concept of essential component of weakly efficient solution set is introduced first. Then we obtain some sufficient conditions for the existence of an essential component or an essential weakly efficient solution in the weakly efficient solution sets for multiobjective optimization problems.
Copyrightq2009 Qun Luo. 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.
1. Introduction
In 1–7 and the references therein, the authors have studied the existence and the stability ofvector-valued, set-valued, semi-infinite vectoroptimization and multiobjective optimization problems, while the author in 2 shows that most of the weakly efficient solution sets of multiobjective optimization problems in the sense of Baire category are stable. By the fact that there are still quite a few weakly efficient solution sets of multiobjective optimization problems which are not stable, in this paper, we discusses the stability of solution set for multiobjective optimization problems from the perspective of essential components.
2. Definitions and Lemmas
LetXbe a nonempty compact subset of a Banach spaceE,R −∞,∞
f:X → Rn, whereff1, f2, . . . , fn
Consider the following multiobjective optimization problem:
minfx, s.t. x∈X. VMP
Definition 2.1. 1A pointx∗∈Xis called a weakly efficient solution toVMP, if there is no
x∈Xsuch that
fix< fix∗, ∀i1,2, . . . , n. 2.2
2A pointx∗∈Xis called an efficient solution toVMP, if there is nox∈Xx /x∗ such that
fix≤fix∗, ∀i1,2, . . . , n. 2.3
WEf, Xor WEfis used to denote the weakly efficient solution set ofVMP, and
Ef, XorEfis used to denote the efficient solution set ofVMP.
Clearly,Ef, X⊂WEf, X, but the reverse containment may not hold.
Example 2.2see8. LetX 0,2, define
fix x, i∈ {1,2, . . . , n−1},
fnx
⎧ ⎨ ⎩
1, 0≤x <1,
2−x, 1≤x≤2.
2.4
ThenEf, X {0} ∪1,2, WEf, X 0,2.
It is easy to see thatn 1, andVMPis just a scalar optimization problem, in this case, we still denote by WEf, Xor WEfthe set of optimal solution.
Remark 2.3. Letf f1, f2, . . . , fn : X → Rn, for alli ∈ {1,2, . . . , n}, one has WEfi, X ⊂ WEf, X.
Denote
Y f |f:X → Rn continuous, Xbe a nonempty compact subset of a Banach space.
2.5
For anyf f1, f2, . . . , fn,g g1, g2, . . . , gn∈Y, define
ρf, g n
i1 sup
x∈X
fix−gix. 2.6
Clearly,Y, ρis a complete metric space.
Lemma 2.4. The mapping WE:Y → 2Xis upper semicontinuous with nonempty compact values. For anyf∈Y, the component of a pointx∈WEf, Xis the union of all connected subsets of WEf, Xwhich contain the pointx. From [9, page 356], one knows that components are connected
closed subsets of WEf, X and thus they are also compact as WEf, Xis compact. It is easy to
see that the components of two distinct points of WEf, Xeither coincide or are disjoint, so that all components constitute a decomposition of WEf, Xinto connected pairwise disjoint compact subsets, that is,
WEf, X
α∈Λ
Cαf, 2.7
where Λ is an index set, for anyα ∈ Λ,Cαfis a nonempty connected compact and for any α,
β∈Λ α /β, Cαf∩Cβf ∅.
Definition 2.5. Forf∈Y,Cαfis called an essential component of WEf, Xif, for any open
setOcontainingCαf, there existsδ >0 such that for allg ∈Y withρf, g< δ, WEg, X∩ O /∅.
The followingDefinition 2.6is from2.
Definition 2.6. For f ∈ Y,x ∈ WEf, Xis said to be an essential weakly efficient solution
toVMPif, for any open neighborhoodNxofxinX, there exists an open neighborhood
OfatfinY such that WEg, X∩Nx/∅for allg ∈Of.
Remark 2.7. Forf∈Y, ifx∈WEf, Xis an essential weakly efficient solution toVMP, then
the component which contains the pointxis an essential component.
Remark 2.8. Forf ∈Y, maybe, there is no essential component in WEf, X, and no essential
weakly efficient solution in WEf, X.
Example 2.9. LetX 0,6, define
f1x f2x · · ·fnx
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1−x, x∈0,1,
0, x∈1,2,
x−2, x∈2,3,
4−x, x∈3,4,
0, x∈4,5,
x−5, x∈5,6.
2.8
Then, WEf, X 1,2∪4,5, and for alli∈ {1,2, . . . , n}, WEf, X WEfi, X. By
Example 2.10. LetX 0,2, define
f1x f2x · · ·fnx
⎧ ⎨ ⎩
x, x∈0,1,
2−x, x∈1,2. 2.9
Then, WEf, X {0} ∪ {2}, and for alli∈ {1,2, . . . , n}, WEf, X WEfi, X. By3, Theorem 3.2, WEf, X WEfi, Xcontains no essential weakly efficient solution.
Definition 2.11. LetX be a nonempty convex subset of a Banach space, andfi :X → R, the
functionfiis said to be strongly quasiconvex onX, if
fitx1 1−tx2<max
fix1, fix2
2.10
for allx1, x2∈X,x1/x2,t∈0,1.
3. Essential Component and Essential Weakly Efficient Solution
Theorem 3.1. Forf ∈Y,x∗ ∈WEf, X, if there isi∈ {1,2, . . . , n}such that WEfi, X {x∗},
thenx∗ is an essential weakly efficient solution of VMP. Hence, the component that contains the
pointx∗is an essential component.
Proof. Suppose thatf ∈Y,x∗∈WEf, and there existsi∈ {1,2, . . . , n}such that WEfi, X
{x∗}. For any open neighborhoodNx∗ofx∗inX, there is an open neighborhoodMx∗of
x∗inXsuch thatMx∗⊂Nx∗, whereMx∗denotes the closure ofMx∗.
Sincefix∗ minx∈Xfix, and WEfi, X {x∗}, then
inf
x∈X\Mx∗fix−fix ∗>0.
3.1
Takeδ >0 such that
inf
x∈X\Mx∗fix−δ > fix ∗ δ.
3.2
Then for anyg g1, g2, . . . , gn∈Ywithρf, g< δ, one has
fix−gix< δ, ∀x∈X. 3.3
Then
by3.2and3.4, one has
min
x∈Mx∗gix≤x∈minMx∗fix δfix ∗ δ,
fix∗ δ < inf
x∈X\Mx∗fix−δ≤x∈Xinf\Mx∗gix,
3.5
therefore
min
x∈Mx∗gix<x∈Xinf\Mx∗gix. 3.6
Then, WEgi, X∩Mx∗/∅, and hence WEgi, X∩Nx∗/∅, which implies WEg, X∩ Nx∗/∅. ByDefinition 2.6,x∗is an essential weakly efficient solution toVMP. Hence, the
component that contains the pointx∗is an essential component.
Corollary 3.2see2. Whenn1, forf∈Y, if WEf {x}is a singleton, thenxis an essential optimum solution.
Lemma 3.3. LetXbe a nonempty compact convex subset of a Banach space, the functionfi:X → R
continuous and strongly quasiconvex, then WEfi, Xis a singleton.
Proof. Supposex1, x2∈WEfi, X, andx1/x2. ByDefinition 2.11,
fitx1 1−tx2<max
fix1, fix2
min
x∈Xfix, t∈0,1, 3.7
which is a contradiction, then WEfi, Xis a singleton.
ByTheorem 3.1andLemma 3.3, we have the followingTheorem 3.4.
Theorem 3.4. Let X be a nonempty compact convex subset of a Banach space, for f f1,
f2, . . . , fn ∈Y, if there existsi∈ {1,2, . . . , n}such thatfiis strongly quasiconvex, then WEf, X
has an essential weakly efficient solution, consequently, WEf, Xhas an essential component.
Theorem 3.5. Forf ∈ Y, if WEf, Xhas only one componentCf, then Cfis an essential component.
Proof. ByLemma 2.4, the mapping WE :Y → 2Xis upper semicontinuous atf ∈ Y, hence,
for any open set Owith O ⊃ WEf, X Cf, there existsδ > 0 such that for allg ∈ Y with ρf, g < δ, one has WEg, X ⊂ O, and hence, WEg, X∩O /∅. By Definition 2.5, WEf, X Cfis an essential component.
Remark 3.6. When n 1, by3, the optimum solution set WEf, X WEf1, X where
Example 3.7. LetX 0,3, define
f1x f2x · · ·fn−1x x, 0≤x≤3,
fnx
⎧ ⎨ ⎩
1x, x∈0,1,
3−x, x∈1,3.
3.8
WEf, X {0} ∪2,3is disconnected; however,x∗ 0 ∈ WEf, Xis an essential weakly efficient solution, C1f {0}is an essential component, andC2f 2,3is an essential component.
Example 3.8. LetX 0,1, define
f1x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
x, x∈
0,1 2
,
1−x, x∈
1 2,1
,
f2x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1−x, x∈
0,1 2
,
x, x∈
1 2,1
.
3.9
Then WEf, X 0,1, WEf1, X {0,1}, WEf2, X {1/2}. ByTheorem 3.1,x1 1/2 is an essential weakly efficient solution. But,x2 0 andx3 1 are not essential weakly efficient solutions. In fact, forx2 0, takeN0,1/4 0,1/4,N0,1/4 0,1/4, for all
ε: 0< ε <1, takeg g1, g2as the following:
g1x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
xε
2
1 4−x
, x∈
0,1 4
,
f1x, x∈
1 4,1
,
g2x
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
f2x, x∈
0,3 4
,
x−ε
2
x−3
4
, x∈
3 4,1
.
3.10
Then WEg, X/∅, and
ρf, gsup
x∈X
f1x−g1xsup
x∈X
But WEg, X∩N0,1/4 ∅. In fact, for allt ∈N0,1/4, ift 0,g10 ε/8, g20 1, takex∗1∈X, we have
g1x∗ 0< g10, g2x∗ 1−
ε
8 < g20. 3.12
Ift∈0,1/4, g1t t ε/21/4−t, g2t 1−t, takex∗1−t∈X, we have
g1x∗ 1−x∗t < g1t,
g2x∗ x∗−
ε
2
x∗−3
4
1−t−ε 2
1 4−t
< g2t.
3.13
Thus WEg, X∩N0,1/4 ∅; therefore,x20 is not an essential weakly efficient solution. Similarly,x31 is not an essential weakly efficient solution.
4. Conclusions
Whenn 1, by3, forf ∈ Y, WEfhas an essential component if and only if WEfis connected, and WEfhas an essential optimum solution if and only if WEfis a singleton. When n ≥ 2, we obtain some sufficient conditions for the existence of an essential component or an essential weakly efficient solution in WEf, X.Example 3.7 shows that WEf, X disconnected, but WEf, X has an essential component and WEf, X is not a singleton, but WEf, Xhas an essential weakly efficient solution. Example 3.8shows that, if WEfi, Xis not a singleton, for somei, then for anyx ∈ WEfi, X,xis not an essential weakly efficient solution.
Acknowledgments
The author expresses her sincerely thanks to anonymous referees for their comments and suggestions leading to the present version of this paper. This research was supported by the Natural Science Foundation of Guangdong Province9251064101000015, China.
References
1 P. L. Yu,Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, vol. 30 ofMathematical Concepts and Methods in Science and Engineering, Plenum Press, New York, NY, USA, 1985.
2 J. Yu, “Essential weak efficient solution in multiobjective optimization problems,” Journal of Mathematical Analysis and Applications, vol. 166, no. 1, pp. 230–235, 1992.
3 Q. Luo, “Essential component and essential optimum solution of optimization problems,”Journal of Optimization Theory and Applications, vol. 102, no. 2, pp. 433–438, 1999.
4 E. Miglierina and E. Molho, “Scalarization and stability in vector optimization,”Journal of Optimization Theory and Applications, vol. 114, no. 3, pp. 657–670, 2002.
5 X. X. Huang, “Stability in vector-valued and set-valued optimization,” Mathematical Methods of Operations Research, vol. 52, no. 2, pp. 185–193, 2000.
6 S. W. Xiang and W. S. Yin, “Stability results for efficient solutions of vector optimization problems,” Journal of Optimization Theory and Applications, vol. 134, no. 3, pp. 385–398, 2007.
8 A. R. Warburton, “Quasiconcave vector maximization: connectedness of the sets of Pareto-optimal and weak Pareto-optimal alternatives,”Journal of Optimization Theory and Applications, vol. 40, no. 4, pp. 537–557, 1983.
