Hausdorff upper semicontinuousHausdorff lower semicontinuous C-upper semicontinuous set-valued mapping C-lower semicontinuous set-valued mapping C-upper semicontinuous vector-valued mapp
The Set Optimization Problem Model
In this section, we introduce the model of a set optimization problems and the concepts of its solutions Let F : X = Y be a set-valued mapping and ẽ be a nonempty subset of X We consider the following set optimization problem
(SOP) Minimize F(x) subject to ôETL.
For any nonempty subset ẽ` of X, we denote
Using the notions of minimality defined in Sections 1.2, we define three kinds of minimality notions for (SOP) with respect to the set less order relation An element xo € T is called an efficient minimal (weakly minimal, ideally minimal, respectively) solution of (SOP) if F(a) is minimal (weakly minimal, ideally minimal, respectively) set of Fp At that time, the pair (xo, yo) with yo € F(x) is said to be an efficient minimizer (weak minimizer, ideal minimizer, respectively) of (SOP).
Properties of Elements in Image Spaces 0004 11 2 Stability Conditions for Weak and Ideal Solutions to Set Optimization
Now, we study some properties of elements in the image space with respect to a set-valued mapping #': X 3 Y.
Firstly, we discuss the following results playing important roles in our analysis.
Lemma 1.6.1 Let T andl, be nonempty subsets of X such that T, AY Then, for every A € Fp and any neighborhood Q of the origin Ủy in Y, the following assertions hold.
(i) If F is Hausdorff upper semicontinuous, then there exists a sequence {A,} of nonempty subsets of Y with A, € Fr, for alln such that A, C A+Q form to be sufficiently large.
(ii) If F is Hausdorff lower semicontinuous, then there exists a sequence {A,} of nonempty subsets of Y with A, € Fr, for alln such that A CC An +Q for n to be sufficiently large.
(iii) If F is lower semicontinuous and compact-valued, then there exists a sequence
{Aa} of nonempty subsets of Y with A, € Fp, for all n such that AC A, +2 for n to be sufficiently large.
(iv) If F is continuous and compact-valued, then there exists a sequence {A,} of nonempty subsets of Y with A, € Fr, for all n such that A C An +Q and
An C A+Q for n to be sufficiently large.
Proof For any given A € ¥p, there exists x € T such that F(a) = A Since T, x r, there exists a sequence {z„} with z, €T, such that #„ > x. i) For any neighborhood 2 of the origin Oy in Y, it follows from the Hausdorff upper semicontinuity of F that there exists no € N such that F(#,) C Ƒ'(+)+9 for all n > nạ. ii) Analogously, taking into account the Hausdorff lower semicontinuity of F at x, we also have F(x) C F(a,) + © for n to be large enough. iii) Applying Lemma 1.1.2 (v), we obtain F(x) C F(z„) +2 eventually, and the assertion is proved.
(iv) The upper semicontinuity of F ensures that # is also H-usc Combining this with (iii), we obtain the conclusion.
Next, the characterization for the convergence of a sequence of nonempty subsets to a compact subset is given.
Lemma 1.6.2 Let T andT, (n = 1,2, ) be nonempty subsets of X such that T, A
T and T is compact Then, for every sequence {x,} with tn € Tn, there exists a subsequence {#n,} converging to some point ứ inT.
Proof Assume, on the contrary, that for every x € I we can find the neighborhood V, of z and n, € N such that #„ £ V,,Vn > nz Noting that the family {V, : 2 € T} is an open cover of the compact set I, we can pick up a finite subcover V = Ul", Vz, containing I’ Thus, there exists an element no € N such that 2, £ V,Vn > no, which is in contradiction to T, AT This completes the proof.
Finally, the sufficient conditions for existence of convergent subsequence of any sequence in the image space are shown in the following theorem.
Theorem 1.6.1 Let T and T, (n = 1,2, ) be nonempty subsets of X such that
Ty AT andT is compact, then the following assertions hold.
(i) If F is Hausdorff upper semicontinuous then for any sequence {A,} with A, €
Fy,,, there exist a subsequence { A„,} and an element A in Fp such that An, FL A,
(ii) If F is Hausdorff lower semicontinuous then for any sequence {A,} with An €
Fry,,, there exist a subsequence {An, } and an element A in Fp such that An, FA,
Proof For every sequence {A,} with A, € ¥p,, there exists x, € I, satisfying F(a) = An, Vn According to Lemma 1.6.2, there is a subsequence {z„„} of {tn} such that z,, > x €T By setting A = F(x), we have A C Fp.
(i) Let Q be an arbitrary neighborhood of the origin Oy in Y It follows from the Haus- dorff upper semicontinuity of F at z that F(za„) C F(x) + ©, for k to be sufficiently large By Proposition 1.4.1, An, LA,
(ii) Similarly, let Q be an arbitrary neighborhood of the origin Oy in Y, from the Haus- dorff lower semicontinuity of f at x, we arrive at F(x) C F(z„,) + for k to be sufficiently large Proposition 1.4.1 yields that A, TA
Applying Proposition 1.4.2, Theorem 1.6.1, and taking into account Lemma 1.1.2
(v), we obtain the following result.
Theorem 1.6.2 Let T and T, (n = 1,2, ) be nonempty subsets of X such that
Tn AT andT is compact, then the following assertions hold.
(i) If F is upper semicontinuous and closed-valued, then for any sequence {A,} with
A, € Fp,, there exist a subsequence {Aa,} and an element A in Fp such that
(ii) If F is lower semicontinuous and compact-valued, then for any sequence {A,} with A, € Fy,, there exist a subsequence {An,} and an element A in Fp such that A, SA.
(iii) If F is continuous and compact-valued, then for any sequence {Ay} with Ay €
Fry,,, there exist a subsequence {An, } and an element A in Fp such that An, AA.
Stability Conditions for Weak and Ideal
Solutions to Set Optimization Problems
One of the significant topics in set optimization is to study the behavior of minimal sets under perturbations Stability conditions for solutions to set optimization problems with respect to the lower type set less order relation in the image space have been introduced by several authors In 2016, Gutiérrez et al [34] studied internal and external stability results in terms of convergence for a sequence of solutions of perturbed problems with convergent sequences of constraint sets to a solution of the given problem in the senses of Panilevé-Kuratowski and Hausdorff Or it might be more precise to say that for every set in the solution to the original problem, we can find an ” equivalent” one that can be expressed as a limit of a convergent sequence of solutions to perturbed problems and the limit of a convergent sequence of solutions to perturbed problems is a solution of the original problem Recently, Karuna and Lalitha [38] have employed the main result in [34] to study the convergence results for set optimization problems with respect to the lower type set less order relation by establishing the convergence of solution sets both in the image space and the given space Moreover, by using the notion of gamma convergence, Karuna and Lalitha [81] have investigated the external and internal stability in terms of Painlevé-Kuratowski convergence of a sequence o solution sets to perturbed set optimization problems in the image space as well as in the decision space Besides, Geoffroy [75] has proposed a topology on the collection of lower bounded subsets of partially ordered normed space, and introduced concepts of set convergence via this topology to study the stability conditions for minimal sets and minimal solutions to set optimization problems with respect to the lower type set less
14 order relation All of the above-mentioned papers have been devoted to the stability of solutions to set optimization problems with respect to the lower type set less order relation.
This chapter has contributed to stability results in terms of the convergence of solutions to set optimization problems with the set less order relation by perturbing the feasible set We discuss the convergence results (in the senses of Painlevé-Kuratowski and Hausdorff) of weak and ideal minimal elements in image spaces as introduced in Section 1.5 Under certain continuity, compactness assumptions, and properties of approximately minimal elements, the external and internal stability conditions for weak and ideal solutions to set optimization problems are established Specifically, we show that any sequence of solutions to perturbed problems has a subsequence converging to a solution of the original problem Conversely, each solution of the original problem can be expressed as a limit of a sequence of solutions to perturbed problems.
The remainder of this chapter is structured as follows Section 2.2 is devoted to properties concerning ideally minimal, weakly minimal, and e-minimal elements in the image spaces In Section 2.3 we discuss the external stability results in the sense of upper convergence of the sequences of solutions to a set optimization problem in image spaces Meanwhile, the internal stability results in the sense of lower convergence are presented in Section 2.4 Section 2.5 is devoted to the model of Welfare Economics as an application of the above results The last section, Section 2.6, contains some concluding remarks.
2.2 Relationships among the Kinds of Minimal Elements
In this chapter, we consider the set optimization problem (SOP), which is intro- duced in Section 1.5.
Analogously to the concept of strict lower set less order relation in [75], we propose the following notion: For each e € int C, the approximating quasi-order relation =Oy
Proof (i) For any e,ẽ € intC’ such that e m¡, which leads to
Similarly, inclusion (2.10) implies that there exists an nạ € N such that
Let nọ = max{n1, nạ}, then (2.11) and (2.12) yield that for all n > no,
Equivalently, B a Combining this with (2.21) and taking into account the openness of int C, we arrive at bạ, — an, € —intC eventually, which contradicts (2.20).
By using techniques given in the first part with suitable modifications, we also gain the fact that A, C B, + intC for n to be large enough Consequently, we obtain B, < A, for n to be sufficiently large According to Lemma 2.2.1 (ii), this is in contradiction with A, € WMin(.Zr,„),Vn This brings the proof to its end.
Remark 2.3.1 When X = Y and F is a single-valued identity mapping, the problem
(SOP) collapses to the problem studied by Miglierina and Molho [76] Moreover, our result, Theorem 2.3.1, has the same conclusion as in Theorem 4.2 of [76] but under weaker assumption I, — T K
2.1 Inroduelion ee 14 2.2 Relationships among the Kinds of Minimal Elements
External Stability Conditions for Solutions
In this section, we study some upper convergence results of weak solutions and ideal ones in the image spaces which are important factors in investigating external stability conditions for set optimization problems.
This subsection is devoted to external stability results by considering the upper convergence of a sequence of weakly minimal solutions in the image spaces Precisely, we prove that the limit of a convergent sequence of weakly minimal solution sets of perturbed problems is a weakly minimal solution set of the original problem.
Theorem 2.3.1 Let {T'„} be a sequence of nonempty subsets of X Assume that the following assumptions hold:
(ii) F is Hausdorff continuous and compact-valued.
If A, € WMin(.¥p,,) for alln and A, 4, A, where A € Fp, then AE WMIin(.2:).
Proof By contradiction, suppose that A £ WMin(.¥r) Because F is compact-valued, Remark 1.2.1 yields that wmax(A) # @ and wmin(A) # 0 According to Lemma 2.2.1(ii), there exists B € ¥Yp such that B < A or, equivalently,
Noting that #' is also continuous and compact-valued, by applying Lemma 1.6.1(iv) on
F, we get a sequence {„} with B, € Fp, such that B,, + B.
We begin by showing that B, C A, — int C for n to be large enough Otherwise, without loss of generality, we can assume that there exists a sequence {b„} with bạ € B,, such that b„ € An — int C (2.20)
Noting that B is a compact set, by Lemma 1.6.2, {b„} admits a subsequence {bp, } converging to some b in B Inclusion (2.19) ensures the existence of a € A such that b—a€ —intC (2.21)
Since A, 4 A and A is compact, Proposition 1.4.2 yields that 4z, 5 A, and so there exists a sequence {a„, } with a„, € An, such that a,, > a Combining this with (2.21) and taking into account the openness of int C, we arrive at bạ, — an, € —intC eventually, which contradicts (2.20).
By using techniques given in the first part with suitable modifications, we also gain the fact that A, C B, + intC for n to be large enough Consequently, we obtain B, < A, for n to be sufficiently large According to Lemma 2.2.1 (ii), this is in contradiction with A, € WMin(.Zr,„),Vn This brings the proof to its end.
Remark 2.3.1 When X = Y and F is a single-valued identity mapping, the problem
(SOP) collapses to the problem studied by Miglierina and Molho [76] Moreover, our result, Theorem 2.3.1, has the same conclusion as in Theorem 4.2 of [76] but under weaker assumption I, — T K
Remark 2.3.2 Theorem 2.3.1 corrects a miscalculation in the proof of Theorem 3.1 in [34] Specifically, Gutiérrez et al [34] investigated the upper convergence results for the weakly minimal solutions to set optimization problems via the I-type of less order relation and use the inequality inf, đ (hx, (Ca, + B(O,m,) + int K)) < m,, for hy € Y \ (Cy, + int K) in the proof of Theorem 3.1 in [34] Clearly, this inequality does not hold in the general The following example gives us a further description Let
K be an orthant cone in R?, C be defined as the Figure 2.1, and C,, = C for all nạ.
By using a new approach, we avoid the employing of the above inequality in the proof of Theorem 2.3.1, and hence fix Theorem 3.1 in [34].
Figure 2.1: An illustration of h € Y\(C+int K) but d(h,Y \ (C + B(0,n) + int K)) >
Example 2.3.1 Let X = Y = R’, C = R?,T = {(z,0) € R?,z € (0,1)}, Fạ = {(x, +) € R®,x € (0,1)} for all n > 1.
Then, all assumptions in Theorem 2.3.1 are satisfied It is easy to see that A, :F (a,+) in WMin(¥p,,) for all n and A, 3 A:= F(a,0) in WMin(2r), where a €
The following example illustrates the essentiality of the compact-valuedness of F’.
Ay = F _1\- (x 3) ER: 2, - 2-19 A={(i.z;) € R?: 2, = 0} n— n — 1,42 MAI n nn , — 1,02 NHƯ
All assumptions in Theorem 2.3.1 are satisfied except the compact-valuedness of
F It is not hard to see that A, € WMin(.Zr„),Vn and A, A AE Fy Noting that
The following corollary is obtained by applying Theorems 1.6.1 and 2.3.1.
Corollary 2.3.1 Let {1} be a sequence of nonempty subsets of X Assume that the following assumptions hold:
(iii) F is Hausdorff continuous and compact-valued.
Then, for any sequence {A,} with A, € WMIn(.2r„), there exists a subsequence {An, } of {An} such that An, 4 A for some A in WMin(Fp).
Remark 2.3.3 Corollary 2.3.1 improves Theorem 3.1 in [38] in the following meanings.
On the one hand, Corollary 2.3.1 does not employ any information on the perturbed problems, which is required in Theorem 3.1 of [38] Specifically, in Theorem 3.1 in
[38], the authors assumed that T is closed, cl(Uncnyl’,) was sequentially compact, and
Ty * YL It is worth noting that if [,, ST then Pc cl(Unenl’,) (indeed, if there
22 is € T but z € cl(OzenF„), then d(x, cl(Unenl,)) = 7 for some r > 0, and so d(z,T,) > r for all n Because x € T and T, * TP, there exists a sequence {z„} with
#„ ET, satisfying x, —> x which contradicts đ(z, „) > r for all n) This together with assumption (a) of Theorem 3.1 in [38] implies that T is compact.
On the other hand, in the proof of Theorem 3.1 in [38], the authors employed the proof of Theorem 3.1 in [34], which involved the miscalculation as mentioned in Remark 2.3.2, so Corollary 2.3.1 can be considered as a correction of Theorem 3.1 in [38].
The following example demonstrates that the assumptions of Corollary 2.3.1 are strictly weaker than that of Theorem 3.1 in [38].
Example 2.3.3 Let X = C'((0;1],R) be the space of all real continuous functions defined on the interval |0; 1] endowed with the sup norm lzll= sup |z0)|: te[0;1]
Let Y = R?,C = R‡,P = {0x}, and „ = (0x, +), Vn > 1, where B(a,r) is the closed ball centered at a with radius r Let F : X = Y, {An}n>2 and A be defined as
By, lzl)U {(t.te) €R2: +81} if Il) 0 and
Since A, € SMin(.2r„), An < By or, equivalently, B, C A, +C This and (2.23) yield that B C A„+ B(0,e„) +Œ It follows from A, LA that A, C A+ B(0,e,), and hence BCA+B(0,2e,)+C Since A is compact and C is closed, A+ B(0, 2é,) +C 5Š ALC which contradicts B £ A+C.
(a) In the assumption (ii), we can replace the compactness of F(x) by C-closedness and (—C)-closedness (that means F(x) + C and F(a) — C are closed).
(b) In special cases as mentioned in Remark 2.3.1, the authors only considered the upper convergence of the solutions in image space for weak minimality When we reformulate (SOP) in the special case as in Remark 2.3.1, the conclusion of Theorem 2.3.2 becomes as follows: Ls SMinT„ C SMinT, where SMin A := {a € A:ACa+C} Therefore, even for this special case, Theorem 2.3.2 is new.
Like Corollary 2.3.1, the upper convergence result for ideally minimal solutions to set optimization problems in the image space is shown in the following corollary.
Corollary 2.3.2 Let {T„} be a sequence of nonempty subsets of X Assume that the following assumptions hold:
(ii) Tạ ST andT, ŠT;
(ili) F is Hausdorff continuous and compact-valued.
Then, for any sequence {A,} with A, € SMin(¥p,), there exists a subsequence { A„,} of {An} such that An, AA for some A in SMin(2r).
2.4 Internal Stability Conditions for Solutions
We devote this section to some internal stability results on weakly and ideally minimal solutions to set optimization problems.
In this subsection, we show that for any weakly minimal solution to the original problem, we can express it as the limit of a sequence of approximately minimal solutions to perturbed problems.
Theorem 2.4.1 Let {I',} be a sequence of nonempty subsets of X Assume that the following assumptions hold:
(iii) F is Hausdorff continuous and compact-valued.
Then, for any A € WMin( Fp), there exist a sequence {e„} C int C with e„ > Oy and a sequence {A,} such that A, € Min.,(Fr,,) for every n and A, Aa
Proof Let A be an arbitrary element in WMin(.A;) Note that A € ¥p and F is continuous with compact values By applying Lemma 1.6.1(iv), we conclude that there exists a sequence {A,} such that A, € ¥p, for every n and A, 5A First, we show that
Ve € intC,dn, € ẹ,Vn > nz: An € Min.(Fp,,)- (2.24)
By contradiction, suppose that there exist € € int C and a subsequence {A,,} C {An} such that A,, ¢ Minz(Fp,,, ), Wk €N Since F is compact-valued, by Remark 1.2.1, wmin(A,,) and wmax(A,,) are nonempty According to Lemma 2.2.1(i), there exists a sequence {W,} with W, € Fp, and W;, a Combining this with (3.11) and noting that int C is open, we obtain bạ — a, € — int Ở for n to be large enough, which contradicts (3.10).
Next, we show that A, C B, + int C for n to be sufficiently large In fact, suppose in the contrary that there exists a sequence {c,} with c, € A, such that
Because A, € ¥p,,Vn, there is a sequence {z,} with x, € T, such that A, = F(z).
By Lemma 1.6.2, without the loss of generality, we can assume that z„ 3 ô € T.
Noting that F is upper semicontinuous and compact-valued, taking into account c, €
A, = F(aằ), we can assume that c, + c Since A, 5 A, we have c € A, which together with (3.7) yield that c—d€cintŒ, for some de B (3.13)
From (3.9), applying Proposition 1.4.2(ii), we arrive at B, x B, which ensures the existence of a sequence {d„}, with dạ € B, converging to d Combining this with (3.13), we obtain c, —d, € int C for n to be large enough, which contradicts (3.12) Therefore,
A, CC B,+int C, and so B, < A, Using Theorem 3.2.1, we get A, ¢ Min(.Zr„), which is a contradiction This brings the proof to its end.
The following example shows that the strict C-convexity assumption is essential for the conclusion of Theorem 3.3.1.
Example 3.3.1 Let X = Y = R’, C = R2, T = [2,3] x {0}, F„ = {(a,-n“'2) :2< x < 3} for alln > 1 Let `: X = Y be defined by
Then, the assumptions in Theorem 3.3.1 are satisfied, except for the strict C- convexity of F’ It is easy to check that and A, 4 A = {(3,0)} but A £ Min(.Zr) The reason is that F is not strictly
C-convex, although this mapping is C-convex.
Definition 3.3.1 (See [77,83]) A set-valued mapping #' from X into Y is said to have converse property at Xo with respect to #9 € X if and only if, either F (#9) 4 F(a) or for any sequences {#„},{Ê„} of X with #a > #o, Zn —> Zo, there exists nọ € N such that F(Ê„¿) < F(2nq)-
By employing the converse property, we obtain the external stability result for efficient minimal solutions to set optimization problems in the image space without the conditions to force efficient solutions to coincide with weak or ideal ones.
First, we illustrate the difference between the converse property and the strict C- convexity of set-valued mappings via the following example.
Example 3.3.2 Let X = Y=R, C = R,, I = [1,0], and F,G: RR be defined by
It is not hard to see that #! has converse property at every x € ẽ' with respect to each y € Ty # 2, but is not strictly C-convex on [ Meanwhile, G is strictly C-convex on I, but has not converse property at every x € I with respect to = 0.
Next, we present the external stability of efficient minimal solutions to a sequence of perturbed set optimization problems.
Theorem 3.3.2 Let {[,} be a sequence of nonempty subsets of X For the problem (SOP), assume that the following assumptions hold:
(iii) F is continuous and compact-valued;
(iv) F has converse property at every x €T with respect to eachy €T,y#x.
Then, every sequence {A,} in Min(.¥p,,) admits a subsequence {An,} such that An, 5
Proof Let {A,} be an arbitrary sequence with A, € Min(.¥p,,),Vn By Theorem 1.6.2, there exists a subsequence of {A,,}, which is still denoted as {A,} for simplicity, such that
Thus, there is a sequence {a,} with a, € T, satisfying A, = F(a,) From (i), (ii), and taking into account Lemma 1.6.2, we can assume that {a„} converges to some a € T. Since F is continuous and compact-valued, for any neighborhood U of the origin Oy in
Combining this with (3.14) and taking into account Lemma 1.4.1, we obtain F(a) = A. For any B € ?r with B = A, we claim that A < B Indeed, Lemma 1.6.1(iv) ensures the existence of a sequence {B,,} with B, € ¥p, such that for any neighborhood 2) of the origin Oy,
B,C B+Q and BC B,+Q, (3.15) for n to be large enough It follows from B, € 7r, that there exists a sequence {b,} with b, € [T, such that B, = F(b,) Again, applying Lemma 1.6.2, we can assume b, —> b € T (taking a subsequence if necessary) Since #' is continuous and compact-valued, for any neighborhood V of the origin 0y in Y, F(b,) C F(b)+V and F(b) C F(bn) + V for n to be sufficiently large By Proposition 1.4.2,
Combining (3.15) with (3.16) and taking into account Lemma 1.4.1, we arrive at F'(b) B Ifa =b then A = B, that is A € Min(.¥p), and hence the conclusion holds Thus, we assume that a # b For any wu in F(a), since A, 5 F(a), there exists a sequence
{un} with uy € F(a,) such that u, —> u Taking into account (iv), there is ứ € N such that F(bn,) < F(an,) Noting that F(an,) € Min(Zr, ), we have F(an,) < F\(bn,), and hence, tun, € F(an,) C F(bn,) — C Then, there exist two sequences {v,,} and {cx} with vn, € F(bn,), ce € C such that c, = Un, — tn,,Vk Because Ƒ' is usc and compact-valued, we can assume that {v,,} converges to v € F(b) = B This together
40 with u,, — u and the closedness of Œ imply that — u € C, and so
Now, let an arbitrary z € B = F(b) The lower semicontinuity of F ensures the existence of a sequence {z„} with z, € F'(b,) such that z„ + z Assumption (iv) again yields that there is a sequence of index nj € N such that F(z„,) < F(an,) It follows from Ƒf(a„,) € Mim(Zr,,) that F(an,) < F(2n,), which implies that there exist two sequences {t,,} and {dj} with tn, € F(dn,), dj € C such that z„, = tn, + d;,Vj Since
F is usc and compact-valued, we can assume that {f„,} converges to t € F(a) = A.
Combining this with z, —> z, we obtain d; + d € C Consequently, z = t+d, and hence
BC A+C This together with (3.17) yields that A < B Therefore, A € Min(.2r) and the proof follows.
To demonstrate that Theorem 3.3.2 guarantees the external stability of efficient minimal solutions in the cases that the family of efficient minimal solution sets is different from that of ideally and weakly minimal solution sets, we consider the following examples.
Example 3.3.3 Let X =R, Y = R”, C=R?.,T = {0,1}, T, = {ằ 1,1+m 1} for all n> 1 Let F: X 3 Y be defined by
Clearly, all assumptions in Theorem 3.3.2 are satisfied It is easy to check that
A, := F (4) in Min(¥p,) for all n and A, 5Š A:= (0) in Min(Zr) Similarly, we also have B, := F'(1+ +) in Min(.¥p,) for all n and Đụ S Bex F(1) in Min(2r).
Note further that A,,B, £ SMin(.¥p,) and A,B £ SMin(2r), where SMin(Y) :{AEG: AX B VBE}.
Example 3.3.4 Let X = R, Y = R?, T,, = {cos [(1 — 2n) (4m)~!z], cos [(1 — n) (4n)~1z]} forn > 1,1 = {0,V27!} Let C= R2, F: XY be defined by
Obviously, all assumptions of Theorem 3.3.2 are satisfied Direct calculations give
An := F (cos[(1—2n)(4n)~!r]) in Min(.¥p,,) for all n and A, Acs F(0) in
Min(.¥r) Moreover, B, = F (cos[(1 — n) (4n)~'r]) € WMin(¥p,,) and B,, £ Min(.Zr,,).
Similarly, B = F(V2-!) € WMin(.Zr) and B ¢ Min(Fp).
To illustrate the essentiality of assumption (iv) in Theorem 3.3.2, we consider the following example.
Example 3.3.5 Let X = Y = R?, T = {(—2,0), (-1,0)}, T, = {(-2, 2n74), (-1,n+)} for alln > 1 Let C = R2, ”':X— Y be defined by
Fứn,#2) = {Ú, 2) € RẺ: (yr — 11)? + (yo — #2)? < 1}, Vi, 2) € X.
Then, the assumptions in Theorem 3.3.2 are satisfied, except for assumption (iv) It is not hard to see that A, := F (—1,+) in Min(.¥p,) for all n and A„ Ass F(-1,0) in Fp but A £ Min(.Zr) The reason is that the assumption (iv) is violated.
Relationships for Well-Posedness Properties between an Optimistic Coun-
Optimistic Counterpart of P() and Corresponding Scalar
This section is devoted to investigating the relationships between well-posedness for optimistic counterparts of uncertain optimization problems and that for the corre-
4.3.1 The Concepts of Well-Posedness for the Optimistic Coun- terpart of P(⁄) and the Scalar Optimization Problems
Now we introduce two concepts of well-posedness for the optimistic counterpart of P().
Definition 4.3.1 Let p € int C A sequence {#„} C K is called
(a) ap-minimizing sequence for P() corresponding to s € Sol if there exists a sequence
{én} C Ry, €n —> 0, such that Vn € U, để € U satisfying f (tn, §) Se f(s.) + env;
(b) a generalized p-minimizing sequence for P(W) if there exist sequences {e„} C
Ry, én 2 0, and {s„} C Sol such that Vn € U, để € U satisfying
Noting that Definition 4.3.1 (a) and (b) do not depend on the choice of the vector p € int C as in the following results.
Proposition 4.3.1 For a given p € int C, the following assertions hold true.
(i) {2,} ts a p-minimizing sequence for P(U) corresponding to s € Sol if and only if there exists a sequence {z,} C C converging to Oy such that Vn € U,AE € 1 satisfying f(@n,§) Se f(s) + za.
(ii) {z„} ts a generalized p-minimizing sequence for P(U) if and only if there exist sequences {z„} C int C converging to Oy, and {s„} C Sol such that Vn € ?4, 3€ EU satisfying
Proof Since the proof techniques are similar, we only give the proof for (b) If {z„} is a generalized p-minimizing sequence for P(U/), then the conclusion is trivial with
Zn = Enp Conversely, let {z,} C int C converging to 0y, {z„} C K and {s„} C Sol be such that Vn € U, d£ € U satisfying that means ƒ(s„,0)) + 2n € ƒ(n,€) + C (4.1)
Noting that (p — int Ở) is a neighborhood of Oy, there exists ỉ > 0 such that ỉ(0y, 1) is contained in (p— int C), where B(a,r) is the closed ball centered at a with radius r.
For any n € N, en € [2ul|Be,1) C Hy — inte) c Maly — œ,
For each n € N, if we set e„ = 07'||z,||, then the sequence {z„} converges to 0 and
Enp — Zn € C Combining this with (4.1) we obtain or equivalently,
Therefore, {%,} is the generalized p-minimizing sequence for P(U/).
Definition 4.3.2 Let p € int C An optimistic counterpart of uncertain problem P() is said to be
(a) p-well-posed at s € Sol if for every p-minimizing sequence {#„} corresponding to s it holds that 2, —> s;
(b) generalized p-well-posed if for every generalized p-minimizing sequence {#„}, it admits a subsequence {#„,„} converging to some Z € Sol.
Let w: K C X > R, we recall the Tykhonov well-posedness notions considered for a scalar optimization problem
Definition 4.3.3 [66, Page 1] The optimization problem OP(K, ) is said to be
(a) Tykhonov well-posed, if it has a unique solution € K and every minimizing sequence {#„}, in the sense that u(x,) —> inf yu, converges to Z;
(b) generalized Tykhonov well-posed, if its solution set is nonempty and every minimiz- ing sequence {#„} has a subsequence converging to a minimizing point.
4.3.2 Relationships for Well-Posedness Properties between an
Optimistic Counterpart of P() and Corresponding Scalar
Definition 4.3.4 [104] Let g : X > R be an extended real-valued function and # € X.
The function g is said to be
(b) upper semicontinuous at # if —g is lower semicontinuous at Z.
Definition 4.3.5 [104] A function g : X —> R is said to be
(a) lower pseudocontinuous at # if and only if for all € X such that g(x) < g(Z),
(b) upper pseudocontinuous at # if —g is lower pseudocontinuous at Z.
Definition 4.3.6 [105] Let h : X > Y be a vector-valued mapping and # € X The mapping h is said to be
(a) C-lower semicontinuous (C-l.s.c) at if for each neighborhood V of h(Z), there is a neighborhood N of z such that
(b) C-upper semicontinuous (C-u.s.c) at # if (—h) is C-Ls.c at #;
(c) C-bounded from below on a subset A of X if for each neighborhood V of the origin in Y, there exists ý > 0 such that h(A) C £V +C;
(d) C-bounded from above on a subset A of X if it is (—C)-bounded from below on A.
Definition 4.3.7 [26, Definitions 3.1.16] Let zo € X A set-valued mapping F’ from
X into Y is said to be
(a) C-upper semicontinuous (C-usc) at xo if for any open set V containing '{zo), there exists a neighborhood U of zo such that for all x € U, F(x) CV +C;
(b) C-lower semicontinuous (C-lsc) at a9 if for any open subset V of Y with F(zo)nV # Ú, there exists a neighborhood U of zo such that for all z € U, F(x) N(V — C) 49;
(c) C-continuous at xo if it is both C-usc and C-lsc at xp.
Lemma 4.3.1 [104] Let K be a nonempty compact subset of a Hausdorff topological space X and h : K + R be a lower pseudocontinuous function Then, OP(K, ) is generalized Tykhonov well-posed Moreover, if argmin(K,) is a singleton, then OP(K, p) ¡s Tykhonov well-posed.
We first study various important properties of the image mapping ƒ(-,⁄), which play a crucial role in our analysis.
Lemma 4.3.2 Suppose that U is compact Then, the following assertions hold true.
(i) If f is C-Ls.c on K x U, then f(-,U) is C-use and C-closed on K.
(ii) For each x € K, if ƒ(%,-) is C-Ls.c on U, then f(x,U) is a C-closed set.
(iii) For each € EU, if f(-,€) is C-u.s.c on K, then f(-,U) is C-lsc on K.
Proof (i) First, suppose to the contrary that there exists # € K such that f(-,/) is not C-usc at # Then, there are an open subset V containing f(%,U/), and a sequence {z„} converging to # such that, for each n, there exists yn, € f(@n,U) but yn V4 C. Because yn € f(%n,U), there is €, € U such that y, = ƒ(z„.Éa) By the compactness
57 of U, one can assume that {£„} converges to some £ € UW Then, ƒ(#,€) € f(z,U) C V Since f is C-Ls.c at (Z,€),
Next, let an arbitrary sequence {(#„, „)} be such that #„ € Ky, Yn € ƒ(za,i) + C and (#a, 0a) —> (Z, 0ỉ) We claim that ÿ € f(%,U)+C Indeed, for each n € ẹ, because
Yn € f(an,U) + C, there is & € U such that y, € ƒ(#a,Ê„) + C Since U is compact, there exists a subsequence {€,,, } of {£„} converging to € € U For each neighborhood B of the origin Oy in Y, there exists a balanced neighborhood B, of Oy, that is, —B, = By, such that Bị + Bị C B It follows from yp, > ÿ that ÿ € yn, + Bi, and hence for k to be large enough On the other hand, the C-lower semicontinuity of f yields that ƒ(#a,Đứ,) € ƒ(#,€) + Bi + CƠ, (4.3) for k to be sufficiently large Combining (4.2) with (4.3), we obtain ÿ€ƒf(,©@+B.+C+C~+BịC ƒ(z,9)+ B+C.
Since B is arbitrary and ƒ(#, £) + Œ is closed, we arrive that ÿ € f(Z,€)+C Therefore, y € f(Z,U) + and consequently ƒ(-,) is C-closed.
(ii) For each x € K, let yn, € f(x,U) +C such that y, > ÿ Then, for each n € N, there is €„ € U satisfying „ € f(x,&,) + C Because #⁄4 is compact, without loss of generality, we can assume that {£„} converges to some € € U It follows from the C-lower semicontinuity of f(a,-) at £ that for any neighborhood V of the origin in Y, f(t,€n) € f(v,€) +V +C for n to be large enough.
Un € f(x, €) +V +C for n to be large enough.
Since C’ is closed and V is arbitrary, we get ỹ€ ƒ(œ,€) + Œ.
(iii) For an arbitrary point # € K, let V be an open set such that f(z,U)NV #49. This implies that there is £ € YU satisfying f(Z,€) € V Because ƒ(-,€) is C-u.s.c at #, there exists a neighborhood N of # such that Vx € N, ƒ(z,©)eV~—C.
Thus, ƒ(z,)fn(V —C) #,Vz€ N Asa result, f(-,U) is Ở-lsc at #,
Lemma 4.3.3 If U is compact and f is C-ls.c on K x U, then for any AC Y, the set {z€ K: AC f(x,U)+C} ts closed In addition, if f(-,€) is C-u.s.c on K for each € EU, then for any q € Y the set {(z,y) € Kx K: f(y) C ƒ(z,4) +ạ+C} is closed.
Proof To begin with, we prove the first assertion of the lemma For any given AC Y, let {x,} be an arbitrary sequence in K converging to x such that
Taking into account the closedness of K, we get ứ € K It follows from (4.4) that for alla € A there is &, € U, a € ƒ(z„,Ê„) + C Because #⁄ is compact, without loss of generality, we can assume that {§„} converging to some € € U (taking a subsequence if necessary) Because f is C-l.s.c at (z,€), for any neighborhood V of the origin in Y, f (tn, €n) € ƒ(+,€) +V + C, for n to be large enough.
Since V is arbitrary and C is closed, a € ƒ(z,£) + Œ Therefore, A C ƒ(z,i4) + Œ.
Next, for a given q € Ÿ, take an arbitrary sequence {(#„,1„)} C K x K converging to (x,y) such that
For each neighborhood B of the origin Oy in Y, there exists a balanced neighborhood
Bị of Oy, such that B, + By CB For any b € f(y,U), there is € € U such that
59 b= ƒ(u,€) Since ƒ(-,€) is C-u.s.c, for n to be sufficiently large, or ƒ0.£) © Fun, €) + Bi + Ơ (4.6)
It follows from (4.5) that for each n, there exists €, € U satisfying f(Yns§) € f(tn, &n) ++ C (4.7)
The compactness of U/ yields that there is a subsequence {£,,} of {€,} converging to
€ €U This, together with (4.7) and taking into account of the C-upper semicontinuity of f(-,€), we arrive at for k to be sufficiently large.
Combining (4.6) with (4.8), we obtain b= ƒ(u,€)€ ƒ(œ,€)+ B.+q~+C+B.+CC ƒ(z,£)+q+ B+C.
Because B is arbitrary and f(z,€) + q+ Œ is a closed set, we conclude that b € f(az,é)+q4+C Cc f(x,U)+q+C Therefore, f(y,U) C f(x,U) +¢4+C The proof is complete.
Now, we recall the definition of an extended version of Gerstewitz scalarization function [79] which plays an important role in our analysis.
Definition 4.3.8 [79] For a given e € —intC, the generalized Gerstewitz function
G.(A, B) := supinf{t€ R:bete+A+C} for A,B Ee P(Y). beB
A nonempty subset A in P(Y) is said to be C-closed if A + C is a closed set, C- bounded if for each neighborhood U of the origin Ủy, A C tU + C for some positive real number
Lemma 4.3.4 [79] Let p € intC, e = —p, A, B,Ae P(Y), A, A be C-closed and B be C-bounded Then, the following statements hold true.
The following results present an important characterization of a solution to the optimistic counterpart of an uncertain vector optimization problem.
Lemma 4.3.5 Let e € —intC be fired Suppose that U is compact, f is C-bounded from below on K x U, and ƒ(œ,-) is C-Ls.c onU for each x € K For each s € Sol, the following assertions hold true.
(ii) Fora € K, G,(ƒ(+,ữ ), f(s,U)) = 0 if and only if f(@,U) C ƒ(s.)+(Œ and f(s,U) C ƒ(œ,M) + Œ.
Proof (i) For a given s € Sol, suppose on the contrary that there are x € K andr >0 such that
According to Lemma 4.3.2(ii), f(y,U) is C-closed for all y € K, and hence by Lemma 4.3.4(iv), we have
—re + f(a,U) +intC C ƒ(z,) + int C, ¢ EU, f(x,6) inf G (f(2,U), ƒ(u,)) yeD is lower pseudocontinuous on K.
Proof For any given t € R, we consider the level set
According to Lemma 4.3.3, L is closed for allt € R, which implies that Œ, (f(-,U), ƒ(-,#4)) is lower semicontinuous on K x D Now, let # and x be such that inf Ge (f(#,U), J(,M)) < inf Ge (ƒ(2,M), f(y,4)) yeD yeD and {#„} be a sequence converging to Z We consider the following two cases.
Case 1 There exist z € K satisfying inf Ge (f(#,U), f(y,U)) < inf Ge (f(z,U4), Fly,U)) < inf Ge (F(@,U), f4): eD , eD , yeD
Thus, we can pick up an element 4 € D such that
We claim that infyen Ge (f(@n,U), f(y,U)) > Ge(f(z,U), f(g,U)) Indeed, suppose to the contrary that infyep G.(f(tn,U), f(y,U)) < G,(ƒ(z,) ƒ(0,04)), then for a sequence {z„} strictly decreasing to zero, there exists y, € D satisfying
Ge (F(an,U); f(YnsU)) + (en) < Ge (F(2,4), ƒ(0,01)): (4.11)
Since D is a closed set in the compact set K, we can assume that the sequence {yn} converges to some ¥ € D Combining (4.11) with the lower semicontinuity of Œ, (ƒ(,0 ƒ(,)) at (2, 9), we obtain
Ti—>OO which contradiets (4.10) Consequently, inf Ge (ƒa,Ù0, ƒ(,)) > Ge (ƒ(z,), ƒ(0./)) = int Ge (F(z,4), fly,Y)) (4-12) ụcD ụcD
This together with (4.9) leads to lim inf inf Ge (f(@n,U), f(y,U)) = inf Ge (ƒ(z.14) ƒ(u.14)) noo yeED
Case 2 There does not exist z € K satisfying the inequality (4.9) Similar to the above arguments for (4.12), we obtain inf Ge (f(an,U), fly,U)) > inf G(ƒ(x,0), fly,4))- eD yeD