# On the Generic Uniqueness of Pareto-Efficient Solutions of Vector Optimization Problems

Authors: Dejin Zhang; Shuwen Xiang; Yanlong Yang; Xicai Deng
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, the generic uniqueness of Pareto weakly efficient solutions, especially Pareto-efficient solutions, of vector optimization problems is studied by using the nonlinear and linear scalarization methods, and some further results on the generic uniqueness are proved. These results present that, for most of the vector optimization problems in the sense of the Baire category, the Pareto weakly efficient solution, especially the Pareto-efficient solution, is unique. Furthermore, based on these results, the generic Tykhonov well-posedness of vector optimization problems is given.


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## Body

## 1. Introduction

In optimization theory, the uniqueness of the solution is critical to the stability and calculation of the solution, but it is difficult to guarantee the uniqueness of solutions, even for the optimal solution of scalar optimization problems. For this reason, some researchers focus on the generic uniqueness problems, that is, when most of the optimization problems (in the sense of Baire category) have a unique solution (see, e.g., [1] and references therein). Kenderov [1] proved the generic uniqueness of the solution for scalar optimization problems. Beer [2] extended the generic uniqueness to constrained optimization problems in the Čech complete space. The generic uniqueness of optimal solutions was proved for some classes of infinite dimensional linear programming problems (see, e.g., [3, 4]), and some generic uniqueness results were proposed in linear optimization problems (see, e.g., [5, 6]). Ioffe and Lucchetti [7] presented several types of concepts of generic uniqueness and Tykhonov well-posedness for various classes of optimization problems. Based on the method of genericity, the generic uniqueness of solution was generalized to zero-sum games, variational inequalities, equilibrium problems, and so on (see, e.g., [8–15]).Tykhonov well-posedness plays a very important role in solving optimization problems by the iterative method (see, e.g., [16]). In vector optimization theory, there are a lot of approaches which give rise to various well-posedness notions (see, e.g., [17–25]). The Pareto-efficient solutions of vector optimization problems are based on partial order, so they are not generally unique and even not generically unique (see Example 1). In view of the multiplicity of Pareto-efficient solutions, the generalization of Tykhonov well-posedness for the vector case is less developed. There are two interesting problems that deserve attention. (1) Can some results about the generic uniqueness of the solutions in vector optimization problems be given under certain conditions? (2) Based on the generic uniqueness, what can be given about the generic results of Tykhonov well-posedness under given conditions?This paper studies the two problems above by the scalarization method, mainly by means of the generic uniqueness and pointwise method of vector optimization problems, and presents the generic uniqueness of Pareto weakly efficient solutions, especially Pareto-efficient solutions. Based on the generic uniqueness results above, the generic Tykhonov well-posedness of vector optimization problems is given.The outline of the paper is as follows. In Section2, the authors introduce some preliminaries, especially review the nonlinear and linear scalarization method. In Section 3, using nonlinear and linear scalarization methods, we present that, in the sense of the Baire category, for most of the vector optimization problems, the Pareto weakly efficient solution, especially the Pareto-efficient solution, is unique. It follows that the vector optimization problem is generic Tykhonov well-posed at the corresponding value. In Section 4, some conclusions are given.


## 2. Preliminaries

Throughout this paper, unless stated otherwise, we always assume that the feasible domain of optimization problem is a metric space, whileBrx denotes the open ball with center x and radius r, and ℝn denotes n-dimensional Euclidean space. For convenience, we first recall some definitions and conclusions.Definition 1.
LetX,Y be Hausdorff topological spaces and F:X⟶2Y be a set-valued mapping.(i)
F is said to be upper semicontinuous at x∈X, iff for each open set G in Y with G⊃Fx, there exists an open neighborhood Ox of x such that G⊃Fx′ for all x′∈Ox.(ii)
F is said to be lower semicontinuous at x∈X, iff for each open set G in Y with G∩Fx≠∅, there exists an open neighborhood Ox of x such that G∩Fx′≠∅ for all x′∈Ox.(iii)
F is said to be continuous at x∈X, iff F is both upper semicontinuous and lower semicontinuous at x.(iv)
F is said to be an usco mapping, iff for each x∈X, Fx is compact and F is upper semicontinuous at x.First of all, consider the following scalar optimization problem:(1)X,f:minx∈Xfx,where f:X⟶ℝ.The definitions of Tykhonov well-posedness and generalized Tykhonov well-posedness of scalar optimization problem are introduced (for more details and results, see [17, 26]).Definition 2.
LetX be a Hausdorff topological space and f:X⟶ℝ be a function.(i)
The optimization problemX,f is called Tykhonov well-posed, iff f has a unique optimal solution x¯ on X, and for any sequence xn⊂X with fxn⟶infx∈Xfx, it holds that xn⟶x¯.(ii)
The optimization problemX,f is called generalized Tykhonov well-posed, iff the set of solutions of X,f is nonempty, and for every sequence xn⊂X with fxn⟶infx∈Xfx, there exists a subsequence xnk of xn converging to an optimal solution of X,f.In what follows, we will investigate the vector optimization problem as below:(2)VX,f:minx∈Xfx,where f:X⟶ℝn, fx=f1x,…,fnx, and fix:X⟶ℝ, i∈1,…,n.Definition 3.
LetX be a Hausdorff topological space, f:X⟶ℝn, and fX=y:y=fx,x∈X; then,(i)
y¯=y¯1,…,y¯n∈fX is said to be a Pareto weakly efficient point of fX, iff there exists no y=y1,…,yn∈fX, such that yi<y¯i for all i∈1,…,n. Denote the set of all Pareto weakly efficient points of fX by MinwfX. x¯∈X is called a Pareto weakly efficient solution of VX,f, iff fx¯∈MinwfX; denote the set of all Pareto weakly efficient solutions of VX,f by EwX,f.(ii)
y¯=y¯1,…,y¯n∈fX is said to be a Pareto-efficient point of fX, iff there is no y=y1,…,yn∈fX, such that yi≤y¯i for all i∈1,…,n, and at least one strict inequality holds. Denote the set of all Pareto-efficient points of fX by MinfX. x¯∈X is called a Pareto-efficient solution of VX,f, iff fx¯∈MinfX; denote the set of all Pareto-efficient solutions of VX,f by EX,f.Some results of existence of Pareto-efficient solutions for vector optimization problems are restated as follows (see, e.g., [27, 28]).Lemma 1.
IfY⊂ℝn is nonempty and compact, then MinY≠∅, and thus MinwY≠∅.Lemma 2.
IfX is nonempty and compact, f:X⟶ℝn is continuous; then, EX,f≠∅, and thus EwX,f≠∅.The scalarization method is a powerful tool for studying vector optimization problems. Two cases, nonlinear scalarization and linear scalarization, are considered in this paper. The nonlinear scalarization function plays an important role in vector optimization problems, and it was first proposed by Gerstewitz [29] and further investigated by Luc [30] and Chen et al. [28].Definition 4.
LetY be a Hausdorff topological linear space, C be a closed convex cone of Y, and intC≠∅. For given e∈intC and a∈Y, the nonlinear scalarization function is defined by(3)ξeay=mint∈ℝ:y∈te+a−C,for any y∈Y.Clearly, for anyy∈Y, there exists a unique t∈ℝ, such that ξeay=t. In particular, when Y=ℝn, C=ℝ+n, e=e1,…,en∈intℝ+n and a=a1,…,an∈ℝn; for the sake of argument, an equivalent definition of the nonlinear scalarization function is given as follows:(4)ξeay=maxyi−aiei:i=1,…,n,∀y∈Y.Specifically, for0∈ℝn, denote ξe0:=ξe. It is easy to check that ξea is continuous and convex. Moreover, ξea is strictly monotone, i.e.,(5)y2>y1⇒ξeay2>ξeay1,∀y1,y2∈Y.The another typical scalarization method is linear scalarization, that is, the weight method. Let(6)W=ω:ω=ω1,…,ωn∈ℝ+n,∑i=1nωi=1,where ω is called a weight vector. For the vector optimization problem (2), consider the following weighted optimization problem:(7)minx∈X∑i=1nωifix.For anyx∈X, denote wf,ωx:=∑i=1nωifix. For x¯∈X, if wf,ωx¯=minx∈Xwf,ωx holds, then x¯ is called a weighted solution of vector optimization problem (2) with respect to ω. Denote(8)Fwf,ω≔x¯:wf,ωx¯=minx∈Xwf,ωx,the set of all weighted solution of the vector optimization problem (2) with respect to ω. In fact, for each f:X⟶ℝn and w∈W, Fwf,ω⊂EwX,f.


## 3. Main Results

In this section, letX be a nonempty compact set, C=ℝ+n, and e=e1,…,en∈intℝ+n. Let(9)CnX=f=f1,…,fn|fi:X⟶ℝis continuous,∀i=1,…,n.The norm onCnX is defined by(10)fx∞=maxx∈Xmaxf1x,…,fnx.It is easy to verify thatCnX,·∞ is a complete metric space.For any fixede∈intC, a∈ℝn, and f∈CnX, define the nonlinear scalarization function ξeaf:X↦ℝ:(11)ξeafx≔ξeafx,∀x∈X.From the continuity off and ξea, we get that ξeaf is continuous on X. Next, we will present some properties of nonlinear scalarization function ξeaf.Lemma 3.
Forf,a∈CnX×ℝn,(i)
ξeaf has a minimum value on X.(ii)
x¯∈Ewf,X⇔ξefx¯fx¯=minx∈Xξefx¯fx=0.Proof.
(i)
By the continuity ofξeaf and compactness of X, it is easy to verify that ξeaf has a minimum value on X.(ii)
“⇐” Suppose by contradiction that x¯∉Ewf,X; then, there exists x′∈X, such that fx′<fx¯, and by the strict monotony of ξea, we have(12)ξefx¯fx′=ξefx¯fx′<ξefx¯fx¯=ξefx¯fx¯,which is in contradiction with the minimal point x¯ of ξefx¯f on X, and thus x¯∈Ewf,X.
“⇒” For every x¯∈Ewf,X, firstly, by the definition of ξeaf and ξea, let a=fx¯, and we have ξefx¯fx¯=0. In what follows, we will show that ξefx¯fx¯=minx∈Xξefx¯fx. Suppose by contradiction that there exists x′∈X, such that ξefx¯fx′=ξefx¯fx′<0; by the definition of ξea, there exists t′<0 such that fx′∈fx¯+t′e−C; hence, fx′−fx¯∈−intC, which is in contradiction with x¯∈Ewf,X.For everyf∈CnX and a∈ℝn, denote Tξf,a:=ξeaf; then, the result is as follows.Lemma 4.
Tξ:CnX×ℝn↦CX is a continuous mapping.Proof.
Assume by contradiction that there exists a sequencefm,am⊂CnX×ℝn with fm,am⟶f,a such that Tξfm,am do not converge to Tξf,a; it means that there exist ε0>0 and a subsequence of fm,am, without loss of generality, still denoted by fm,am, and exist corresponding xm∈X, such that(13)ξeamfmxm−ξeafxm=ξeamfmxm−ξeafxm≥ε0.
Without loss of generality, letξeamfmxm−ξeafxm≥ε0; otherwise, consider ξeafxm−ξeamfmxm≥ε0.
Byξeamfmxm−ξeafxm≥ε0, ξeamfmxm−ε0≥ξeafxm. From ξeay=maxyi−ai/ei:i=1,…,n, we have(14)maxfimxm−aimei:i=1,…,n−ε0≥maxfixm−aiei:i=1,…,n.
Letfi0mxm−ai0m/ei0=maxfimxm−aim/ei:i=1,…,n; then,(15)fi0mxm−ai0mei0−ε0≥maxfixm−aiei:i=1,…,n.
Therefore,(16)fi0mxm−ai0mei0−ε0≥fi0xm−ai0ei0.
Hence,(17)fi0mxm−fi0xm≥ε0ei0+ai0m−ai0.
Note thatei0>0 and from am⟶a that ai0m−ai0>−1/2ε0ei0 as m is large enough, then(18)f−fm∞≥fi0mxm−fi0xm≥12ε0ei0,as m is large enough, which is in contradiction with fm⟶f.For eachf∈CnX, let Ff:=EwX,f. By Lemma 2, Ff≠∅; therefore, F:CnX⟶2X is a set-valued mapping with nonempty values. For each fixed e∈intC and every f,a∈CnX×ℝn, there exists a corresponding nonlinear scalarization function ξeaf. Denote all minimal points of ξeaf by Mf,a:=Fξeaf; by Lemma 3i, Mf,a≠∅, and thus we define(19)M:CnX×ℝn⟶2X.Moreover, ifa∈EwX,f, then(20)Mf,a⊂EwX,f=Ff.We introduce the Fort theorem and some results of continuity of Pareto-efficient solutions (for more details and results, see [31, 32]).Lemma 5.
(Fort theorem) LetX be a complete metric space and T:X⟶2Y be an usco mapping; then, there exists a dense residual subset Q of X, such that T is continuous at each x∈Q.It is easy to see that the following result holds (see [32]).Lemma 6.
F:CnX⟶2X is an usco mapping.Remark 1.
From Fort theorem and Lemma6, it is easy to observe that the usco mapping F is generic continuous on X; in other words, it is continuous at most of the points of X in the sense of Baire category.Lemma 7.
For each fixed pointe∈intC, M:CnX×ℝn⟶2X is an usco mapping.Proof.
By the definition ofM,F, and Tξ, for f,a∈CnX×ℝn, we have(21)Mf,a=Fξeaf=FTξf,a.
SinceF is usco on CX and Tξ is continuous on CnX×ℝn, we get that M is usco on CnX×ℝn.We introduce the following results (see Theorem 3.1 of [32]).Proposition 1.
Letx∗∈EX,f be an Pareto-efficient solution of f. If there is some w∈ℝ+n such that x∗ is the unique weighted solution of problem (7) with weight vector ω, then x∗ is essential, that is, for given ε>0, there exist some δ>0 such that EX,g∩Bεx∗≠∅ whenever g−f∞<δ.The following example shows that Pareto-efficient solutions of vector optimization problems do not have the generic uniqueness in general; that is, the setQ of functionals for which there is a unique Pareto-efficient solution to problem (2) is not always a dense residual subset of CnX.Example 1.
LetX=−1,1 and f:X↦ℝ2 be defined by(22)fx=0,−x,0≤x≤1;x,0,−1≤x<0.
Then,f is continuous and quasiconvex on compact convex set X (see Figure 1).
In this case, we have(23)MinfX=−1,0,0,−1EX,f=−1,1.
Then,x=−1 and x=1 are the unique weighted solutions of f subject to weight vectors 2/3,1/3 and 1/3,2/3, respectively, so that they are essential. The concept of essential solution implies that for given ε0=1/3, there exist some δ>0 such that EX,f′∩−4/3,−2/3≠∅ and EX,f′∩2/3,4/3≠∅ whenever g−f∞<δ. Let x1∈EX,f′∩−4/3,−2/3 and x2∈EX,f′∩2/3,4/3. Then, x1≠x2 and x1,x2∈EX,f′ for all f′∈Bδf, which suggests that the set of functions for which there is a unique Pareto-efficient solution is not dense in C2X.
Based on Lemmas5 and 7, the generic uniqueness result of Pareto weakly efficient solution of vector optimization problems can be obtained as follows.Figure 1
The image of f.Theorem 1.
There exists a dense residual subsetQ⊂CnX×ℝn, such that for every f,a∈Q, if a∈fEwX,f, then the Pareto weakly efficient solution of vector optimization problem VX,f corresponding to a is unique, that is, f−1a is a singleton.Proof.
By Lemmas5 and 7, for each fixed e∈intC, there exists a dense residual subset Q⊂CnX×ℝn, such that M is continuous at each f,a∈Q. In what follows, we will prove that for each f,a∈Q and a∈fEwX,f, f−1a is a singleton.
Assume by contradiction that there arex¯1,x¯2∈EwX,f=Ff with x¯1≠x¯2, satisfying fx¯1=fx¯2=a.
It follows fromx¯1≠x¯2 that there exists δ>0, such that Bδx¯1¯∩Bδx¯2¯=∅. Since x¯1,x¯2∈EwX,f and by Lemma 3, we have x¯1,x¯2∈Mf,a.
Define the functionα:X↦ℝ by(24)αx=dx,Bδx2¯¯dx,Bδx1¯¯+dx,Bδx2¯¯,x∈X.
Letδm be a sequence of positive numbers satisfying δm⟶0; define the vector-valued functions fm:X⟶ℝn by(25)fmx=fx−αxδme,x∈X.
Then,fm is a sequence of vector-valued functions; obviously, fm⊂CnX. Since(26)fm−f∞=maxx∈Xmaxf1mx−f1x,…,fnmx−fnx,=maxx∈Xmaxαxδme1,…,αxδmen=δmmaxe1,…,en⟶0,it means that fm⟶f. For every x2∈Bδx¯2, we have(27)fmx2=fx2−αx2δme=fx2,fmx¯1=fx¯1−αx¯1δme=fx¯1−δme.
In what follows, we only need to show that(28)x2∉Mfm,a.
If it is not true, assume thatx2∈Mfm,a=Fξeafm; then,(29)ξeafmx2=minx∈Xξeafmx≤ξeafmx¯1.
Therefore,(30)ξeafx2≤ξeafx¯1−αx¯1δme=ξeafx¯1−δme.
By the definition ofξeafx¯1, we have(31)fx¯1∈a+ξeafx¯1e−C.
Hence,(32)fx¯1−δme∈a+ξeafx¯1e−δme−C.
That is,(33)fx¯1−δme∈a+ξeafx¯1−δme−C.
Therefore,(34)ξeafx¯1−δme≤ξeafx¯1−δm.
From the foregoing, we have(35)ξeafx2≤ξeafx¯1−δm.
Hence,ξeafx2<ξeafx¯1, that is, ξeafx2<ξeafx¯1, which is in contradiction with(36)ξeafx¯1=minx∈Xξeafx.
Therefore, for everyx2∈Bδx¯2, we have x2∉Mfm,a, and thus Bδx¯2∩Mfm,a=∅, this is a contradiction with that M is continuous at f,a.The following corollary is an immediate consequence of Theorem1.Corollary 1.
There exists a dense residual subsetQ⊂CnX×ℝn, such that for every f,a∈Q, if a∈fEX,f, then the Pareto-efficient solution of vector optimization problem VX,f corresponding to a is unique, that is, f−1a is a singleton.Next, we consider weighted optimization problem (7). For every f∈CnX and each ω=ω1,…,ωn∈W, it is clear that Fwf,ω≠∅, and then Fw:CnX×W↦2X is a set-valued mapping with nonempty values. We recall a result on the continuity of weighted solutions (see, e.g., [32]).Lemma 8.
Fw:CnX×W⟶2X is an usco mapping.It follows from Lemmas5 and 8 that the generic uniqueness result of Pareto weakly efficient solutions of vector optimization problems can be given as follows.Theorem 2.
There exists a dense residual subsetQ⊂CnX×W, such that for every f,ω∈Q, if a∈fFwf,w, then the vector a corresponds to the unique Pareto weakly efficient solution of vector optimization problem VX,f, that is, f−1a is a singleton.Proof.
By Lemmas5 and 8, there exists a dense residual Q⊂CnX×W, such that Fω is continuous at each f,ω∈Q. Next, we will prove that for every f,ω∈Q, if a∈fFwf,ω, then the vector a corresponds to a unique Pareto weakly efficient solution of vector optimization problem (7).
Suppose by contradiction that there existx¯1,x¯2∈EwX,f with x¯1≠x¯2, satisfying fx¯1=fx¯2=a. On the other hand, since a∈fFwf,ω, there exists x¯∈Fwf,ω, satisfying fx¯=a=a1,…,an. Since a∈fFwf,w and x¯1,x¯2∈EwX,f, we have(37)∑i=1nωifix¯1=∑i=1nωifix¯2=∑i=1nωiai.
It means that(38)x¯1,x¯2∈Fwf,ω=x¯∈X:wf,ωx¯=minx∈Xwf,ωx.
Moreover, it follows fromx¯1≠x¯2 that there exists δ>0, such that(39)Bδx¯1¯∩Bδx¯2¯=∅.
Define the mapping by(40)αx=dx,Bδx2¯¯dx,Bδx1¯¯+dx,Bδx2¯¯,x∈X.
Letδm be a sequence of positive numbers satisfying δm⟶0. For every m∈1,2,…, define the vector-valued function fm:X⟶ℝn by(41)fimx=fix−αxδm,i=1,…,n,fmx=f1mx,…,fnmx,x∈X.
Then, we obtain the vector-valued function sequencefm; it is easy to check fm⊂CnX.
Since(42)fm−f∞=maxx∈Xmaxf1mx−f1x,…,fnmx−fnx,=maxx∈Xmaxαxδm,…,αxδm=δmmaxx∈Xαx≤δm⟶0,then fm⟶f.
Takeωm⟶ω; as Fω is continuous at f,ω and f,ωm⟶f,ω, there exists xm∈Fwf,ωm, such that xm⟶x¯1. From xm∈Fwf,ωm, we have(43)wf,ωmxm=∑i=1nωimfixm=minx∈X∑i=1nωimfix.
Sincexm⟶x¯1, there exists N>0, such that xm∈Bδx¯1 for all m>N. Next, we prove that when m>N, Fwfm,ωm∩Bδx2¯=∅. If it is not true, then for some m>N, there exists x2∈Bδx¯2 satisfying x2∈Fwfm,ωm, that is,(44)∑i=1nωimfimx2=minx∈X∑i=1nωimfimx≤∑i=1nωimfimxm.
Note that(45)fimx2=fix2+αx2δm=fix2,i=1,…,n.
Then,(46)∑i=1Nωimfix2=∑i=1nωimfimx2≤∑i=1nωimfimxm=∑i=1nωimfixm−δm,=∑i=1nωimfixm−δm.
Hence,(47)∑i=1nωimfix2<∑i=1nωimfixm.
It is in contradiction with∑i=1nωimfixm=minx∈X∑i=1nωimfix.
Hence, whenm>N,Fwfm,ωm∩Bδx2¯=∅, it is in contradiction with the continuity of Fω at f,ω. Therefore, a corresponds to the unique Pareto weakly efficient solution.Based on Theorem1, we obtain a result about pointwise Tykhonov well-posedness in vector optimization problems as below.Definition 5.
The vector optimization problem (2) is called Tykhonov well-posed at a∈fEwX,f, if there exists a unique x¯ in EwX,f, such that fx¯=a; then, for any xm⊂X satisfying fxm⟶a, we have that xm⟶x¯.Remark 2.
If the vector optimization problem (2) is a scalar optimization, then Definition 5 is Tykhonov well-posed for the scalar case.Theorem 3.
There exists a dense residual subsetQ⊂CnX×ℝn, such that for every f,a∈Q, if a∈fEwX,f, the vector optimization problem (2) is Tykhonov well-posedness at a.Proof.
By Theorem1, there exists a dense residual subset Q⊂CnX×ℝn, such that for every f,a∈Q, if a∈fEwX,f, then the vector a corresponds to a unique Pareto weakly efficient solution of Vf,a. As a∈fEwX,f, there exist some x¯∈EwX,f, such that fx¯=a. Assume that xm⊂X with fxm⟶a; since X is compact, xm has a convergent subsequence xmk, and let xmk⟶x^. Since f−1a=x¯ and fxm⟶a, it is easy to see that for all subsequences of xm converging to x^, we have that xm⟶x¯. Therefore, the vector optimization problem (2) is Tykhonov well-posedness at a.


## 4. Conclusion

In this paper, the authors solve the problems mentioned in the introduction. By use of nonlinear scalarization method, in the sense of Baire category, for most pointsf,a constituting a dense residual subset Q of the product space CnX×ℝn, if the vector a is a Pareto weakly efficient point of the function f, then f−1a is the unique Pareto weakly efficient solution of vector optimization problem VX,f corresponding to a. Obviously, if the vector a is a Pareto-efficient point of the function f, the result still holds. As applications of the uniqueness, in the sense of Baire category, for most f,a, if the vector a is a Pareto weakly efficient point, the authors show that vector optimization problem (2) is Tykhonov well-posed at the vector a. By means of the weighting method which is a linear scalarization method, similar uniqueness result is obtained.



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