Volume 2010, Article ID 154598,10pages doi:10.1155/2010/154598
Research Article
Weak
ψ
-Sharp Minima in
Vector Optimization Problems
S. Xu and S. J. Li
College of Mathematics and Statistics, Chongqing University, Chongqing 400030, China
Correspondence should be addressed to S. Xu,[email protected]
Received 23 April 2010; Revised 15 July 2010; Accepted 13 August 2010
Academic Editor: N. J. Huang
Copyrightq2010 S. Xu and S. J. Li. 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 present a sufficient and necessary condition for weakψ-sharp minima in infinite-dimensional spaces. Moreover, we develop the characterization of weak ψ-sharp minima by virtue of a nonlinear scalarization function.
1. Introduction
The notion of a weak sharp minimum in general mathematical program problems was first introduced by Ferris in1. It is an extension of sharp minimum in2. Weak sharp minima play important roles in the sensitivity analysis 3,4 and convergence analysis of a wide range of optimization algorithms5. Recently, the study of weak sharp solution set covers real-valued optimization problems5–8 and piecewise linear multiobjective optimization problems9–11.
Most recently, Bednarczuk12 defined weak sharp minima of order m for vector-valued mappings under an assumption that the order cone is closed, convex, and pointed and used the concept to prove upper H ¨olderness and H ¨older calmness of the solution set-valued mappings for a parametric vector optimization problem. In 13, Bednarczuk discussed the weak sharp solution set to vector optimization problems and presented some properties in terms of well-posedness of vector optimization problems. In14, Studniarski gave the definition of weakψ-sharp local Pareto minimum in vector optimization problems under the assumption that the order cone is convex and presented necessary and sufficient conditions under a variety of conditions. Though the notions in 12, 14 are different for vector optimization problems, they are equivalent for scalar optimization problems. They are a generalization of the weak sharp local minimum of orderm.
infinite-dimensional spaces. In addition, we develop the characterization of weakψ-sharp minima in terms of a nonlinear scalarization function.
This paper is organized as follows. In Section 2, we recall the definitions of the local Pareto minimizer and weak ψ-sharp local minimizer for vector-valued optimization problems. InSection 3, we present a sufficient and necessary condition for weakψ-sharp local minimizer of vector-valued optimization problems. We also give an example to illustrate the optimality condition.
2. Preliminary Results
Throughout the paper,XandY are normed spaces.Bx, δdenotes the open ball with center
x∈ Xand radiusδ > 0.Nxis the family of all neighborhoods ofx, and distx, Wis the distance from a pointxto a setW ⊂ X. The symbolsSc, intSandbdsdenote, respectively,
the complement, interior and boundary ofS.
LetD ⊂Y be a convex conecontaining 0. The cone defines an order structure onY, that is, a relation “≤” inY ×Y is defined by y1 ≤ y2 ⇔ y2−y1 ∈ D.Dis a proper cone if
{0}/D /Y.
LetΩbe an open subset of X,S ⊂ Ω. Given a vector-valued map f : Ω → Y, the following abstract optimization is considered:
Minfx:x∈S. 2.1
In the sequel, we always assume thatDis a proper closed and convex cone.
Definition 2.1. One says that x0 is a local Pareto minimizer for 2.1, denoted by x0 ∈ LMinf, S, if there existsU∈ Nxfor which there is nox∈S∩Usuch that
fx−fx0∈−D\D. 2.2
If one can chooseU X, one will say thatx0 is a Pareto minimizer for 2.1, denoted by x0∈Minf, S.
Note that2.2may be replaced by the simple conditionfx−fx0∈−D\ {0}if
we assume that the coneDis pointed.
Definition 2.2see14. Letψ : 0,∞ → 0,∞be a nondecreasing function with the propertyψt 0⇔t0such a family of functions is denoted byΨ. Letx0 ∈S. One says
thatx0 is a weakψ-sharp local Pareto minimizer for2.1, denoted byx0 ∈ WSLψ, f, S, if
there exist a constantα >0 andU∈ Nx0such that
fx D∩Bfx0, αψdistx, W
∅, ∀x∈S∩U\W, 2.3
where
W :x∈S:fx fx0
If one can chooseU X, one says x0 is a weakψ-sharp minimizer for2.1, denoted by x0∈WSψ, f, S. In particular, letψmt:tmform1,2, . . . .Then, one says thatx0is a weak ψ-sharp local Pareto minimizer of ordermfor2.1ifx0 ∈WSLψm, f, S, and one says that x0is a weak sharp Pareto minimizer of ordermfor2.1ifx0∈WSψm, f, S.
Remark 2.3. IfWis a closed set, condition2.3can be expressed as the following equivalent forms:
fx∈fx0 B
0, αψdistx, W−Dc, ∀x∈S∩U\W, 2.5
dfx−fx0,−D
≥αψdistx, W, ∀x∈S∩U\W. 2.6
Remark 2.4. In theDefinition 2.2, ifY R,D 0,∞, andψ ψm, then the relation2.6
becomes the following form:
fx−fx0≥αdistx, Wm, ∀x∈S∩U, 2.7
which is the well-known definition of a weak sharp minimizer of ordermfor2.1; see16.
3. Main Results
In this section, we first generalize the result of Theorem 1 in Studniarski 14 to infinite-dimensional spaces. Finally, we develop the characterization of weakψ-sharp minimizer by means of a nonlinear scalarization function.
LetD ⊂Y be a proper closed convex cone with intD /∅. The topological dual space ofY is denoted byY∗. The polar cone toDisD∗ {λ ∈Y∗ :λ, y ≥0, ∀y ∈D}. It is well known that the coneD∗contains aw∗-compact convex setΛwith 0/∈Λsuch that
D∗coneΛ {rλ:r≥0, λ∈Λ}. 3.1
The setΛis called a base for the dual coneD∗. Recall that a pointλis an extremal point of a setΛif there exist no different pointsλ1, λ2∈Λandt∈0,1such thatλtλ1 1−tλ2.
Theorem 3.1. Suppose thatf :X → Yis a vector-valued map. LetD⊂Y be a proper closed convex cone withintD /∅,x0 ∈S, andψ ∈Ψ.
iLetΛbe aw∗-compact convex base ofD∗ andQthe set of extremal points ofΛ. Suppose thatWdefined by2.4is a closed set. Then,x0 ∈WSLψ, f, Sif and only if there exist U∈ Nx, a constantα >0, a covering{Sλ:λ∈Q}ofS∩U, and
λ, fx>λ, fx0
αψdistx, W, ∀x∈Sλ∩U\W, ∀λ∈Q. 3.2
iiLetQ⊂D∗\ {0}and assume thatD∗cl cone coQ. Thenx0 ∈LMinf, Sif and only if there exists a covering{Sλ:λ∈Q}ofS∩Usuch that
λ, fx>λ, fx0
Proof. iPart “only if”: by assumption, there existβ >0 andU∈ Nx0such that
fx−fx0 D
∩B0, βψdistx, W∅, ∀x∈S∩U\W. 3.4
Lete ∈ intD be a fixed point. Setβ0 infλ∈Λλ, e. SinceΛisw∗-compact, the infimum is
attained at a point ofQ. Namely,β0 minλ∈Qλ, e. Clearly,λ, e>0 for anyλ∈Λ. Hence, β0>0.
For eachλ∈Q, we define
Sλ
x∈S∩U:λ, fx≥λ, fx0
β
2eψdistx, Wβ0 . 3.5
We will show that
S∩U⊂ λ∈Q
Sλ. 3.6
Letx ∈ S∩U. Ifx ∈ W, thenfx fx0by2.4, hence,x ∈Sλfor allλ ∈ Q. Ifx /∈W,
suppose thatx /∈Sλfor anyλ∈Q, then
λ, fx<λ, fx0
β
2eψdistx, Wβ0, ∀λ∈Q. 3.7
This relation, together with statementλ, e ≥β0yields
λ, fx0−fx β
2eψdistx, We
>0, ∀λ∈Q. 3.8
Obviously, for anyλ∈D∗, the above relation becomes the following form:
λ, fx0−fx β
2eψdistx, We
≥0. 3.9
Consequently, by the bipolar theorem, one has
d:fx0−fx β
2eψdistx, We∈D. 3.10
Therefore,
fx−fx0 d β
2eψdistx, We, 3.11
andfx−fx0 d∈B0, βψdistx, W, which is a contradiction to3.4. We have thus
Now, letx∈Sλ∩U\Wandλ ∈Q. From the procedure of the above proof, we see
thatS∩U\W⊂ ∪λ∈QSλ. Hence, by3.5, setαββ0/4e, inequality3.2is true.
Part “if”: we defineβ1supλ∈Λλ, e. The supremum is attained at an extremal point
because of thew∗-compactness ofΛ. Soβ1maxλ∈Qλ, e>0 andβ−11λ, e ≤1 for anyλ∈Q.
Hence, by assumption, we have
λ, fx>λ, fx0
αψdistx, W≥λ, fx0
β1−1αψdistx, Wλ, e, 3.12
forx∈Sλ∩U\Wandλ∈Q.
Now, suppose that for allβ > 0,3.4is false, then there existx ∈ S∩U\W and
d∈Dsuch that
fx−fx0 d∈B
0, βψdistx, W. 3.13
Lete∈intDbe a fixed point, and sinceDis a cone, there isk >0 such thatB0,1⊂ke−D. Consequently,
B0, βψdistx, W⊂kβψdistx, We−D. 3.14
Therefore,
fx−fx0 d∈kβψdistx, We−D. 3.15
There isd∈Dfrom3.15such that
fx−fx0 kβψdistx, We−
dd. 3.16
Sincex ∈S∩U\W ⊂ λ∈QSλ\W, there isλ ∈Qsuch thatx ∈Sλ. Moreover,Λ ⊂D∗
anddd∈D. Hence,
λ, fx−λ, fx0
kβψdistx, Wλ, e−λ, dd≤kβψdistx, Wλ, e. 3.17
By choosingββ−11αk−1, we obtain a contradiction to3.12.
iiPart “only if”: for eachλ∈Q, we define,
Sλ
x∈S∩U:λ, fx≥λ, fx0
. 3.18
Now, we will check that3.6holds true. Pick anyx ∈ S∩U. Suppose thatx /∈Sλ for any λ∈Q, then
λ, fx−fx0
Hence, for anyλ∈cl cone coQD∗,λ, fx−fx0 ≤0. By applying the bipolar theorem,
we have
fx−fx0∈ −D, 3.20
Combing it with the assumption, we have
fx−fx0∈−D∩D, 3.21
which is a contradiction to3.19. So3.6holds and3.3is satisfied by the definition ofSλ.
Part “if”: suppose thatx0/∈LMinf, S, then there existsx∈S∩Usuch that
fx−fx0∈ −D\D. 3.22
Indeed,x∈S∩Ucan be replace byx∈S∩U\W, becausex∈W,fx−fx0 0, which
is contradiction to3.22. Hence, forx∈S∩U\W, we haveλ, fx−fx0 ≤0, ∀λ∈D∗.
In particular,
λ, fx−fx0 ≤0, ∀λ∈Q. 3.23
It follows from the assumption that
∪λ∈QSλ∩U
\W⊃S∩U\W. 3.24
Therefore, by3.3, we obtain
λ, fx−fx0
>0, ∀λ∈Q, ∀x∈Sλ∩U\W, 3.25
which contradicts relation3.23.
Remark 3.2. By takingUXin parti resp.,iiofTheorem 3.1, we obtain a necessary and sufficient condition forx0 to be in WSψ, f, S resp., Minf, S. In particular, if we choose Y RpandDRp
andQ{λ1, λ2, . . . , λp}, then, we obtain Theorem 1 in14.
Finally, we apply the nonlinear scalarization function to discuss the weak ψ-sharp minimizer in vector optimization problems.
LetD ⊂ Y be a closed and convex cone with nonempty interior intD. Given a fixed pointe∈intDandy∈Y, the nonlinear scalarization functionξ:Y → Ris defined by
ξyinft:te∈yD. 3.26
In what follows, we present several properties about the nonlinear scalarization function.
Lemma 3.3see17. For any fixede∈intD,y∈Y, andr∈R. One has
iξy< r ⇔re∈yintD,
iiξy> r ⇔re/∈yD.
iiiξy r ⇔re∈ybdD.
Given a vector-valued mapf :X → Y, definef:X → Y by
fx fx−fx0. 3.27
Next, we consider weakψ-sharp local minimizer for a vector-valued mapfthrough a weak sharp local minimizer of a scalar functionξ◦f:X → R.
Theorem 3.4. Letx0∈S⊂X. Suppose thatWdefined by2.4is a closed set. Then,
x0∈WSL
ψ, f, S⇐⇒x0∈WSL
ψ, ξ◦f, S . 3.28
Proof. Part “only if”: let us assume thatx0 ∈ WSLψ, f, S. Thus, there existα >0 andU ∈
Nx0such that
fx−fx0 D
∩B0, αψdistx, W∅, ∀x∈S∩U\W. 3.29
Note that, whenWis a closed set,
α
4eψdistx, We∈B
0, αψdistx, W ∀x∈S∩U\W. 3.30
Therefore,
α
4eψdistx, We /∈fx−fx0 D ∀x∈S∩U\W. 3.31
By usingLemma 3.3ii, one has
ξfx−fx0
> α
4eψdistx, W ∀x∈S∩U\W. 3.32
According toLemma 3.3iii, one has
ξfx0−fx0
This relation, together with3.32yields
ξfx−fx0
> ξfx0−fx0
α
4eψdistx, W, ∀x∈S∩U\W. 3.34
Namely,
ξ◦fx>ξ◦fx0 α
4eψdistx, W, ∀x∈S∩U\W, 3.35
that is,x0∈WSLψ, ξ◦f, S .
Part “if”: by assumption, there existβ >0 andU∈ Nx0such that
ξfx> ξfx 0
βψdistx, W, ∀x∈S∩U\W. 3.36
In terms ofLemma 3.3iii, we have
ξfx 0
ξfx0−fx0
0. 3.37
Hence,
ξfx−fx0
> βψdistx, W, ∀x∈S∩U\W. 3.38
Once more usingLemma 3.3ii, one has
βψdistx, We /∈fx−fx0 D, ∀x∈S∩U\W, 3.39
which implies that
βψdistx, We−D∩fx−fx0 D
∅, ∀x∈S∩U\W. 3.40
Sincee∈intD, there exists some number >0 such thatB0, ⊂e−D. Moreover,
B0, λ⊂λe−D, ∀λ >0. 3.41
Hence, it follows from the relation that
B0, βψdistx, W⊂βψdistx, We−D, ∀x∈S∩U\W. 3.42
Combing it with relation3.40, we deduce that
B0, βψdistx, W∩fx−fx0 D
Letαβ, by the definition of weakψ-sharp local minimizer, we havex0∈WSLψ, f, S.
It is possible to illustrateTheorem 3.4by means of adapting a simple example given in
14.
Example 3.5. Letnp2,S Ω R2, andDR2
and letf f1, f2:R2 → R2be defined
by
f1
x1, x2:max0,minx1, x2 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
x1, ifx2≥x1>0,
x2, ifx1> x2>0,
0, ifx1≤0 orx2≤0,
f2
x1, x2:max0,min−x1, x2 ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
−x1, ifx2≥ −x1 >0,
x2, if −x1> x2>0,
0, ifx1≥0 orx2≤0,
3.44
We chooseUR2. UsingDefinition 2.2, we derive thatx
0 0,0∈WSψ1, f, S.
Lete 1,1. From Corollary 1.46 in17, we haveξ◦fx max1≤i≤2fix. Observe
that
W x:fx 0,0x:x2≤0∪x:x10. 3.45
It is easy to verify thatfix distx, Wfor allx∈S\W. Using relation2.7, we show that x0 0,0∈WSψ1, ξ◦f, S . Hence, condition3.28withψψ1holds forα∈0,1.
Acknowledgments
This paper was partially supported by the National Natural Science Foundation of China
Grant no. 10871216and Chongqing University Postgraduates Science and Innovation Fund
Project no. 201005B1A0010338. The authors would like to thank the anonymous referees for their valuable comments and suggestions, which helped to improve the paper, and are grateful to Professor M. Studniarski for providing the paper14.
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