Abstract

We introduce several types of the Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints. Criteria and characterizations of these types of the Levitin-Polyak well-posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. The results in this paper unify, generalize, and extend some known results in the literature.

1. Introduction

The vector variational inequality in a finite-dimensional Euclidean space has been introduced in [1] and applications have been given. Chen and Cheng [2] studied the vector variational inequality in infinite-dimensional space and applied it to vector optimization problem. Since then, many authors [3–11] have intensively studied the vector variational inequality on different assumptions in infinite-dimensional spaces. Lee et al. [12, 13], Lin et al. [14], Konnov and Yao [15], Daniilidis and Hadjisavvas [16], Yang and Yao [17], and Oettli and SchlΓ€ger [18] studied the generalized vector variational inequality and obtained some existence results. Chen and Li [19] and Lee et al. [20] introduced and studied the generalized vector quasi-variational inequality and established some existence theorems.

On the other hand, it is well known that the well-posedness is very important for both optimization theory and numerical methods of optimization problems, which guarantees that, for approximating solution sequences, there is a subsequence which converges to a solution. The study of well-posedness originates from Tykhonov [21] in dealing with unconstrained optimization problems. Its extension to the constrained case was developed by Levitin and Polyak [22]. The study of generalized Levitin-Polyak well-posedness for convex scalar optimization problems with functional constraints originates from Konsulova and Revalski [23]. Recently, this research was extended to nonconvex optimization problems with abstract set constraints and functional constraints (see [24]), nonconvex vector optimization problem with abstract set constraints and functional constraints (see [25]), variational inequality problems with abstract set constraints and functional constraints (see [26]), generalized inequality problems with abstract set constraints and functional constraints [27], generalized quasi-inequality problems with abstract set constraints and functional constraints [28], generalized vector inequality problems with abstract set constraints and functional constraints [29], and vector quasivariational inequality problems with abstract set constraints and functional constraints [30]. For more details on well-posedness on optimizations and related problems, please also see [31–37] and the references therein. It is worthy noting that there is no study on the Levitin-Polyak well-posedness for a generalized vector quasi-variational inequality problem.

In this paper, we will introduce four types of Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with an abstract set constraint and a functional constraint. In Section 2, by virtue of a nonlinear scalarization function and a gap function for generalized vector quasi-varitional inequality problems, we show equivalent relations between the Levitin-Polyak well-posedness of the optimization problem and the Levitin-Polyak well-posedness of generalized vector quasi-varitional inequality problems. In Section 3, we derive some various criteria and characterizations for the (generalized) Levitin-Polyak well-posedness of the generalized vector quasi-variational inequality problems. The results in this paper unify, generalize, and extend some known results in [26–30].

2. Preliminaries

Throughout this paper, unless otherwise specified, we use the following notations and assumptions.

Let (𝑋,β€–β‹…β€–) be a normed space equipped with norm topology, and let (𝑍,𝑑1) be a metric space. Let 𝑋1βŠ‚π‘‹, πΎβŠ‚π‘ be nonempty and closed sets. Let π‘Œ be a locally convex space ordered by a nontrivial closed and convex cone 𝐢 with nonempty interior int𝐢, that is, 𝑦1≀𝑦2 if and only if 𝑦2βˆ’π‘¦1∈𝐢 for any 𝑦1,𝑦2βˆˆπ‘Œ. Let 𝐿(𝑋,π‘Œ) be the space of all the linear continuous operators from 𝑋 to π‘Œ. Let π‘‡βˆΆπ‘‹1β†’2𝐿(𝑋,π‘Œ) and π‘†βˆΆπ‘‹1β†’2𝑋1 be strict set-valued mappings (i.e., 𝑇(π‘₯)β‰ βˆ… and 𝑆(π‘₯)β‰ βˆ…, forallπ‘₯βˆˆπ‘‹1), and let π‘”βˆΆπ‘‹1→𝑍 be a continuous vector-valued mapping. We denote by βŸ¨π‘§, π‘₯⟩ the value 𝑧(π‘₯), where π‘§βˆˆπΏ(𝑋,π‘Œ), π‘₯βˆˆπ‘‹1. Let 𝑋0={π‘₯βˆˆπ‘‹1βˆΆπ‘”(π‘₯)∈𝐾} be nonempty. We consider the following generalized vector quasi-variational inequality problem with functional constraints and abstract set constraints.

Find π‘₯βˆˆπ‘‹0 such that π‘₯βˆˆπ‘†(π‘₯) and there exists π‘§βˆˆπ‘‡(π‘₯) satisfying𝑧,π‘₯βˆ’π‘₯ξ¬ξ€·βˆ‰βˆ’int𝐢,βˆ€π‘₯βˆˆπ‘†π‘₯ξ€Έ.(GVQVI) Denote by 𝑋 the solution set of (GVQVI).

Let 𝑍1,𝑍2 be two normed spaces. A set-valued map 𝐹 from 𝑍1 to 2𝑍2 is

(i) closed, on 𝑍3βŠ†π‘1, if for any sequence {π‘₯𝑛}βŠ†π‘3 with π‘₯𝑛→π‘₯ and π‘¦π‘›βˆˆπΉ(π‘₯𝑛) with 𝑦𝑛→𝑦, one has π‘¦βˆˆπΉ(π‘₯);

(ii) lower semicontinuous (l.s.c. in short) at π‘₯βˆˆπ‘1, if {π‘₯𝑛}βŠ†π‘1,π‘₯𝑛→π‘₯, and π‘¦βˆˆπΉ(π‘₯) imply that there exists a sequence {𝑦𝑛}βŠ†π‘2 satisfying 𝑦𝑛→𝑦 such that π‘¦π‘›βˆˆπΉ(π‘₯𝑛) for 𝑛 sufficiently large. If 𝐹 is l.s.c. at each point of 𝑍1, we say that 𝐹 is l.s.c. on 𝑍1;

(iii) upper semicontinuous (u.s.c. in short) at π‘₯βˆˆπ‘1, if for any neighborhood 𝑉 of 𝐹(π‘₯), there exists a neighborhood π‘ˆ of π‘₯ such that 𝐹(π‘₯β€²)βŠ†π‘‰, forallπ‘₯β€²βˆˆπ‘ˆ. If 𝐹 is u.s.c. at each point of 𝑍1, we say that 𝐹 is u.s.c. on 𝑍1.

It is obvious that any u.s.c. nonempty closed-valued map 𝐹 is closed.

Let (𝑃,𝑑) be a metric space, 𝑃1βŠ‚π‘ƒ, and π‘₯βˆˆπ‘ƒ. We denote by 𝑑𝑃1(π‘₯)=inf{𝑑(π‘₯,π‘ξ…ž)βˆΆπ‘ξ…žβˆˆπ‘ƒ1} the distance from the point π‘₯ to the set 𝑃1. For a topological vector space 𝑉, we denote by π‘‰βˆ— its dual space. For any set Ξ¦βŠ‚π‘‰, we denote the positive polar cone of Ξ¦ byΞ¦βˆ—=ξ€½πœ†βˆˆπ‘‰βˆ—ξ€ΎβˆΆπœ†(π‘₯)β‰₯0,βˆ€π‘₯∈Φ.(2.1)

Let π‘’βˆˆint𝐢 be fixed. DenoteπΆβˆ—0=ξ€½πœ†βˆˆπΆβˆ—ξ€Ύ.βˆΆπœ†(𝑒)=1(2.2)

Definition 2.1. (i) A sequence {π‘₯𝑛}βŠ†π‘‹1 is called a type I Levitin-Polyak (LP in short) approximating solution sequence if there exist {πœ–π‘›}βŠ†π‘1+={π‘Ÿβ‰₯0|π‘Ÿisarealnumber} with πœ–π‘›β†’0 and π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) such that 𝑑𝑋0ξ€·π‘₯π‘›ξ€Έβ‰€πœ–π‘›,π‘₯(2.3)𝑛π‘₯βˆˆπ‘†π‘›ξ€Έ,(2.4)βŸ¨π‘§π‘›,π‘₯βˆ’π‘₯π‘›βŸ©+πœ–π‘›ξ€·π‘₯π‘’βˆ‰βˆ’int𝐢,βˆ€π‘₯βˆˆπ‘†π‘›ξ€Έ.(2.5)
(ii) {π‘₯𝑛}βŠ†π‘‹1 is called a type II LP approximating solution sequence if there exist {πœ–π‘›}βŠ†π‘1+ with πœ–π‘›β†’0 and π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) such that (2.3)–(2.5) hold, and, for any π‘§βˆˆπ‘‡(π‘₯𝑛), there exists 𝑀(𝑛,𝑧)βˆˆπ‘†(π‘₯𝑛) satisfying βŸ¨π‘§,𝑀(𝑛,𝑧)βˆ’π‘₯π‘›βŸ©βˆ’πœ–π‘›π‘’βˆˆβˆ’πΆ.(2.6)
(iii) {π‘₯𝑛}βŠ†π‘‹1 is called a generalized type I LP approximating solution sequence if there exist {πœ–π‘›}βŠ†π‘1+ with πœ–π‘›β†’0 and π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) such that 𝑑𝐾𝑔π‘₯π‘›ξ€Έξ€Έβ‰€πœ–π‘›(2.7) and (2.4), (2.5) hold.
(iv) {π‘₯𝑛}βŠ†π‘‹1 is called a generalized type II LP approximating solution sequence if there exist {πœ–π‘›}βŠ†π‘1+ with πœ–π‘›β†’0, π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) such that (2.4), (2.5), and (2.7) hold, and, for any π‘§βˆˆπ‘‡(π‘₯𝑛), there exists 𝑀(𝑛,𝑧)βˆˆπ‘†(π‘₯𝑛) such that (2.6) holds.

Definition 2.2. (GVQVI) is said to be type I (resp., type II, generalized type I, generalized type II) LP well-posed if the solution set 𝑋 of (GVQVI) is nonempty, and, for any type I (resp., type II, generalized type I, generalized type II) LP approximating solution sequence {π‘₯𝑛}, there exists a subsequence {π‘₯𝑛𝑗} of {π‘₯𝑛} and π‘₯βˆˆπ‘‹ such that π‘₯𝑛𝑗→π‘₯.

Remark 2.3. (i) It is clear that any (generalized) type II LP approximating solution sequence is a (generalized) type I LP approximating solution sequence. Thus, (generalized) type I LP well-posedness implies (generalized) type II LP well-posedness.
(ii) Each type of LP well-posedness of (GVQVI) implies that the solution set 𝑋 is compact.
(iii) Suppose that 𝑔 is uniformly continuous functions on a set 𝑋1𝛿0ξ€Έ=ξ€½π‘₯βˆˆπ‘‹1βˆΆπ‘‘π‘‹0(π‘₯)≀𝛿0ξ€Ύ,(2.8) for some 𝛿0>0. Then generalized type I (resp., generalized type II) LP well-posedness of (GVQVI) implies its type I (resp., type II) LP well-posedness.
(iv) If π‘Œ=𝐑1, 𝐢=𝐑1+, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the generalized quasi-variational inequality problem defined by Jiang et al. [28]. If π‘Œ=𝐑1, 𝐢=𝐑1+, 𝑆(π‘₯)=𝑋0 for all π‘₯βˆˆπ‘‹1, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the generalized variational inequality problem defined by Huang, and Yang [27] which contains as special cases for the type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the variational inequality problem in [26].
(v) If 𝑆(π‘₯)=𝑋0 for all π‘₯βˆˆπ‘‹1, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the generalized vector variational inequality problem defined by Xu et al. [29].
(vi) If the set-valued map 𝑇 is replaced by a single-valued map 𝐹, then type I (resp., type II, generalized type I, generalized type II) LP well-posedness of (GVQVI) reduces to type I (resp., type II, generalized type I, generalized type II) LP well-posedness of the vector quasivariational inequality problems defined by Zhang et al. [30].
Consider the following statement: π‘₯π‘‹β‰ βˆ…andforanytypeIresp.,typeII,generalizedtypeI,generalizedtypeIILPapproximatingsolutionsequence𝑛,wehave𝑑𝑋π‘₯𝑛.⟢0(2.9)

Proposition 2.4. If (GVQVI) is type I (resp., type II, generalized type I, generalized type II) LP well-posed, then (2.9) holds. Conversely if (2.9) holds and 𝑋 is compact, then (1) is type I (resp., type II, generalized type I, generalized type II) LP well-posed.

The proof of Proposition 2.4 is elementary and thus omitted.

To see the various LP well-posednesses of (1) are adaptations of the corresponding LP well-posednesses in minimizing problems by using the Auslender gap function, we consider the following general constrained optimization problem:min𝑓(π‘₯)s.t.π‘₯βˆˆπ‘‹β€²1𝑔(π‘₯)∈𝐾,(P)

where π‘‹ξ…ž1βŠ†π‘‹1 is nonempty and fβˆΆπ‘‹1→𝐑1βˆͺ{+∞} is proper. The feasible set of (P) is 𝑋′0, where π‘‹ξ…ž0={π‘₯βˆˆπ‘‹ξ…ž1βˆΆπ‘”(π‘₯)∈𝐾}. The optimal set and optimal value of (P) are denoted by π‘‹ξ…ž and 𝑣, respectively. Note that if Dom(𝑓)βˆ©π‘‹ξ…ž0β‰ βˆ…, whereξ€½Dom(𝑓)=π‘₯βˆˆπ‘‹1ξ€ΎβˆΆπ‘“(π‘₯)<+∞,(2.10) then 𝑣<+∞. In this paper, we always assume that 𝑣>βˆ’βˆž.

Definition 2.5. (i) A sequence {π‘₯𝑛}βŠ†π‘‹ξ…ž1 is called a type I LP minimizing sequence for (P) if limsupπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛≀𝑑𝑣,(2.11)𝑋′0ξ€·π‘₯π‘›ξ€ΈβŸΆ0.(2.12)
(ii) {π‘₯𝑛}βŠ†π‘‹ξ…ž1 is called a type II LP minimizing sequence for (P) if limπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛=𝑣(2.13) and (2.12) hold.
(iii) {π‘₯𝑛}βŠ†π‘‹ξ…ž1 is called a generalized type I LP minimizing sequence for (P) if 𝑑𝐾𝑔π‘₯π‘›ξ€Έξ€ΈβŸΆ0(2.14) and (2.11) hold.
(iv) {π‘₯𝑛}βŠ†π‘‹ξ…ž1 is called a generalized type II LP minimizing sequence for (P) if (2.13) and (2.14) hold.

Definition 2.6. (P) is said to be type I (resp., type II, generalized type I, generalized type II) LP well-posed if the solution set π‘‹ξ…ž of (P) is nonempty, and for any type I (resp., type II, generalized type I, generalized type II) LP minimizing sequence {π‘₯𝑛}, there exists a subsequence {π‘₯𝑛𝑗} of {π‘₯𝑛} and π‘₯βˆˆπ‘‹ξ…ž such that π‘₯𝑛𝑗→π‘₯.
The Auslender gap function for (GVQVI) is defined as follows: 𝑓(π‘₯)=infπ‘§βˆˆπ‘‡(π‘₯)supπ‘₯β€²βˆˆπ‘†(π‘₯)infπœ†βˆˆπΆβˆ—0πœ†ξ€·ξ«π‘§,π‘₯βˆ’π‘₯ξ…žξ¬ξ€Έ,βˆ€π‘₯βˆˆπ‘‹1.(2.15)
Let 𝑋2βŠ†π‘‹ be defined by 𝑋2={π‘₯βˆˆπ‘‹βˆ£π‘₯βˆˆπ‘†(π‘₯)}.(2.16)
In the rest of this paper, we set π‘‹ξ…ž1 in (P) equal to 𝑋1βˆ©π‘‹2. Note that if 𝑆 is closed on 𝑋1, then π‘‹ξ…ž1 is closed.
Recall the following widely used function (see, e.g., [38]) πœ‰βˆΆπ‘ŒβŸΆπ‘1ξ€½βˆΆminπ‘‘βˆˆπ‘1ξ€Ύ.βˆΆπ‘¦βˆ’π‘‘π‘’βˆˆβˆ’πΆ(2.17)
It is known that πœ‰ is a continuous, (strictly) monotone (i.e., for any 𝑦1,𝑦2βˆˆπ‘Œ, 𝑦1βˆ’π‘¦2∈𝐢 implies that πœ‰(𝑦1)β‰₯πœ‰(𝑦2) and (𝑦1βˆ’π‘¦2∈int𝐢 implies that πœ‰(𝑦1)>πœ‰(𝑦2)), subadditive and convex function. Moreover, it holds that πœ‰(𝑑𝑒)=𝑑,forallπ‘‘βˆˆπ‘1 and πœ‰(𝑦)=supπœ†βˆˆπΆβˆ—0πœ†(𝑦),forallπ‘¦βˆˆπ‘Œ.
Now we given some properties for the function 𝑓 defined by (2.15).

Lemma 2.7. Let the function 𝑓 be defined by (2.15), and let the set-valued map 𝑇 be compact-valued on 𝑋1. Then(i)𝑓(π‘₯)β‰₯0,forallπ‘₯βˆˆπ‘‹β€²1;(ii)for any π‘₯βˆˆπ‘‹ξ…ž0, 𝑓(π‘₯)=0 if and only if π‘₯βˆˆπ‘‹.

Proof. (i) Let π‘₯βˆˆπ‘‹ξ…ž1. Suppose to the contrary that 𝑓(π‘₯)<0. Then, there exists a 𝛿>0 such that 𝑓(π‘₯)<βˆ’π›Ώ. By definition, for 𝛿/2>0, there exists a π‘§βˆˆπ‘‡(π‘₯), such that supπ‘₯β€²βˆˆπ‘†(π‘₯)infπœ†βˆˆπΆβˆ—0πœ†ξ€·ξ«π‘§,π‘₯βˆ’π‘₯ξ…žπ›Ώξ¬ξ€Έβ‰€π‘“(π‘₯)+2𝛿<βˆ’2<0.(2.18) Thus, we have infπœ†βˆˆπΆβˆ—0πœ†ξ€·ξ«π‘§,π‘₯βˆ’π‘₯ξ…žξ¬ξ€Έ<0,βˆ€π‘₯ξ…žβˆˆπ‘†(π‘₯),(2.19) which is impossible when π‘₯ξ…ž=π‘₯. This proves (i).
(ii) Suppose that π‘₯βˆˆπ‘‹ξ…ž0 such that 𝑓(π‘₯)=0.
Then, it follows from the definition of π‘‹ξ…ž0 that π‘₯βˆˆπ‘†(π‘₯). And from the definition of 𝑓(π‘₯) we know that there exist π‘§π‘›βˆˆπ‘‡(π‘₯) and 0<πœ–π‘›β†’0 such that infπœ†βˆˆπΆβˆ—0πœ†π‘§ξ€·ξ«π‘›,π‘₯βˆ’π‘₯ξ…žξ¬ξ€Έβ‰€π‘“(π‘₯)+πœ–π‘›=πœ–π‘›,βˆ€π‘₯ξ…žβˆˆπ‘†(π‘₯),(2.20) that is, πœ‰π‘§ξ€·ξ«π‘›,π‘₯ξ…žβˆ’π‘₯β‰₯βˆ’πœ–π‘›,βˆ€π‘₯ξ…žβˆˆπ‘†(π‘₯).(2.21) By the compactness of 𝑇(π‘₯), there exists a sequence {𝑧𝑛𝑗} of {𝑧𝑛} and some π‘§βˆˆπ‘‡(π‘₯) such that π‘§π‘›π‘—βŸΆπ‘§.(2.22) This fact, together with the continuity of πœ‰ and (2.21), implies that πœ‰ξ€·ξ«π‘§,π‘₯ξ…žβˆ’π‘₯β‰₯0,βˆ€π‘₯ξ…žβˆˆπ‘†(π‘₯).(2.23) It follows that π‘₯βˆˆπ‘‹.
Conversely, assume that π‘₯βˆˆπ‘‹. It follows from the definition of 𝑋 that π‘₯βˆˆπ‘†(π‘₯). Suppose to the contrary that 𝑓(π‘₯)>0. Then, for any π‘§βˆˆπ‘‡(π‘₯), supπ‘₯β€²βˆˆπ‘†(π‘₯)infπœ†βˆˆπΆβˆ—0πœ†ξ€·ξ«π‘§,π‘₯βˆ’π‘₯ξ…žξ¬ξ€Έ>0.(2.24)
Thus, there exist 𝛿>0 and π‘₯0βˆˆπ‘†(π‘₯) such that infπœ†βˆˆπΆβˆ—0πœ†ξ€·βŸ¨π‘§,π‘₯βˆ’π‘₯0βŸ©ξ€Έβ‰₯𝛿.(2.25) It follows that πœ‰ξ€·βŸ¨π‘§,π‘₯0ξ€Έβˆ’π‘₯βŸ©β‰€βˆ’π›Ώ<0.(2.26) As a result, we have βŸ¨π‘§,π‘₯0βˆ’π‘₯βŸ©βˆˆβˆ’int𝐢.(2.27) This contradicts the fact that π‘₯βˆˆπ‘‹. So, 𝑓(π‘₯)=0. This completes the proof.

Lemma 2.8. Let 𝑓 be defined by (2.15). Assume that the set-valued map 𝑇 is compact-valued and u.s.c. on 𝑋1 and the set-valued map 𝑆 is l.s.c. on 𝑋1. Then 𝑓 is l.s.c. function from 𝑋1 to 𝐑1βˆͺ{+∞}. Further assume that the solution set 𝑋 of (GVQVI) is nonempty, then Dom(𝑓)β‰ βˆ….

Proof. First we show that 𝑓(π‘₯)>βˆ’βˆž, forallπ‘₯βˆˆπ‘‹1. Suppose to the contrary that there exists π‘₯0βˆˆπ‘‹1 such that 𝑓(π‘₯0)=βˆ’βˆž. Then, there exist π‘§π‘›βˆˆπ‘‡(π‘₯0) and {𝑀𝑛}βŠ‚π‘…1+ with 𝑀𝑛→+∞ such that supπ‘₯β€²ξ€·π‘₯βˆˆπ‘†0ξ€Έinfπœ†βˆˆπΆβˆ—0πœ†π‘§ξ€·ξ«π‘›,π‘₯0βˆ’π‘₯ξ…žξ¬ξ€Έβ‰€βˆ’π‘€π‘›.(2.28) Thus, πœ‰π‘§ξ€·ξ«π‘›,π‘₯ξ…žβˆ’π‘₯0β‰₯𝑀𝑛,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†0ξ€Έ.(2.29) By the compactness of 𝑇(π‘₯0), there exist a sequence {𝑧𝑛𝑗}βŠ‚{𝑧𝑛} and some 𝑧0βˆˆπ‘‡(π‘₯0) such that 𝑧𝑛𝑗→𝑧0. This fact, together with (2.29) and the continuity of πœ‰ on π‘Œ, implies that πœ‰π‘§ξ€·ξ«0,π‘₯ξ…žβˆ’π‘₯0β‰₯+∞,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†0ξ€Έ(2.30) which is impossible, since πœ‰ is a finite function on π‘Œ.
Second, we show that 𝑓 is l.s.c. on 𝑋1. Let π‘Žβˆˆπ‘1. Suppose that {π‘₯𝑛}βŠ‚π‘‹1 satisfies 𝑓(π‘₯𝑛)β‰€π‘Ž,forall𝑛, and π‘₯𝑛→π‘₯0βˆˆπ‘‹1. It follows that, for each 𝑛, there exist π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) and 0<𝛿𝑛→0 such that ξ€·βˆ’πœ‰βŸ¨π‘§π‘›,π‘¦βˆ’π‘₯π‘›βŸ©ξ€Έβ‰€π‘Ž+𝛿𝑛π‘₯,βˆ€π‘¦βˆˆπ‘†π‘›ξ€Έ.(2.31)
For any π‘₯β€²βˆˆπ‘†(π‘₯0), by the l.s.c. of 𝑆, we have a sequence {𝑦𝑛} with {𝑦𝑛}βˆˆπ‘†(π‘₯𝑛) converging to π‘₯ξ…ž such that ξ€·βˆ’πœ‰βŸ¨π‘§π‘›,π‘¦π‘›βˆ’π‘₯π‘›βŸ©ξ€Έβ‰€π‘Ž+𝛿𝑛.(2.32)
By the u.s.c. of 𝑇 at π‘₯0 and the compactness of 𝑇(π‘₯0), we obtain a subsequence {𝑧𝑛𝑗} of {𝑧𝑛} and some 𝑧0βˆˆπ‘‡(π‘₯0) such that 𝑧𝑛𝑗→𝑧0. Taking the limit in (2.32) (with 𝑛 replaced by 𝑛𝑗), by the continuity of πœ‰, we have π‘§βˆ’πœ‰ξ€·ξ«0,π‘₯ξ…žβˆ’π‘₯0ξ¬ξ€Έβ‰€π‘Ž,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†0ξ€Έ.(2.33)
It follows that 𝑓(π‘₯0)=infπ‘§βˆˆπ‘‡(π‘₯0)supπ‘₯β€²βˆˆπ‘†(π‘₯0)βˆ’πœ‰(βŸ¨π‘§,π‘₯ξ…žβˆ’π‘₯0⟩)β‰€π‘Ž. Hence, 𝑓 is l.s.c. on 𝑋1. Furthermore, if π‘‹β‰ βˆ…, by Lemma 2.7, we see that Dom(𝑓)β‰ βˆ….

Lemma 2.9. Let the function 𝑓 be defined by (2.15), and let the set-valued map 𝑇 be compact-valued on 𝑋1. Then,(i){π‘₯𝑛}βŠ†π‘‹1 is a sequence such that there exist {πœ–π‘›}βŠ†π‘1+ with πœ–π‘›β†’0 and π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) satisfying (2.4) and (2.5) if and only if {π‘₯𝑛}βŠ†π‘‹ξ…ž1 and (2.11) hold with 𝑣=0,(ii){π‘₯𝑛}βŠ†X1 is a sequence such that there exist {πœ–π‘›}βŠ†π‘1+ with πœ–π‘›β†’0 and π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) satisfying (2.4) and (2.5), and for any π‘§βˆˆπ‘‡(π‘₯𝑛), there exists 𝑀(𝑛,𝑧)βˆˆπ‘†(π‘₯𝑛) satisfying (2.6) if and only if {π‘₯𝑛}βŠ†π‘‹β€²1 and (2.13) hold with 𝑣=0.

Proof. (i) Let {π‘₯𝑛}βŠ†π‘‹1 be any sequence if there exist {πœ–π‘›}βŠ†π‘1+ with πœ–π‘›β†’0 and π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) satisfying (2.4) and (2.5), then we can easily verify that ξ€½π‘₯π‘›ξ€ΎβŠ†π‘‹ξ…ž1ξ€·π‘₯,π‘“π‘›ξ€Έβ‰€πœ–π‘›.(2.34) It follows that (2.11) holds with 𝑣=0.
For the converse, let {π‘₯𝑛}βŠ†π‘‹ξ…ž1 and (2.11) hold with 𝑣=0. We can see that {π‘₯𝑛}βŠ†π‘‹1 and (2.4) hold. Furthermore, by (2.11), we have that there exists {πœ–π‘›}βŠ†π‘1+withπœ–π‘›β†’0 such that 𝑓(π‘₯𝑛)β‰€πœ–π‘›. By the compactness of 𝑇(π‘₯𝑛), we see that for every 𝑛 there exists π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) such that πœ‰π‘§ξ‚€ξ‚¬π‘›,π‘₯β€²βˆ’π‘₯𝑛β‰₯βˆ’πœ–π‘›,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†π‘›ξ€Έ.(2.35)
It follows that for every 𝑛 there exists π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) such that (2.5) holds.
(ii) Let {π‘₯𝑛}βŠ†π‘‹1 be any sequence we can verify that liminfπ‘›β†’βˆžπ‘“ξ€·π‘₯𝑛β‰₯0(2.36)
holds if and only if there exists {𝛼𝑛}βŠ†π‘1+ with 𝛼𝑛→0 and, for any π‘§βˆˆπ‘‡(π‘₯𝑛), there exists 𝑀(𝑛,𝑧)βˆˆπ‘†(π‘₯𝑛) such that βŸ¨π‘§,𝑀(𝑛,𝑧)βˆ’π‘₯π‘›βŸ©βˆ’π›Όπ‘›π‘’βˆˆβˆ’πΆ.(2.37)
From the proof of (i), we know that limsupπ‘›β†’βˆžπ‘“(π‘₯𝑛)≀0 and {π‘₯𝑛}βŠ†π‘‹ξ…ž1 hold if and only if {π‘₯𝑛}βŠ†π‘‹1 such that there exist {𝛽𝑛}βŠ†π‘1+ with 𝛽𝑛→0π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) satisfying (2.4) and (2.5) (with πœ–π‘› replaced by 𝛽𝑛). Finally, we let πœ–π‘›=max{𝛼𝑛,𝛽𝑛} and the conclusion follows.

Proposition 2.10. Assume that π‘‹β‰ βˆ… and 𝑇 is compact-valued on 𝑋1. Then(i)(GVQVI) is generalized type I (resp., generalized type II) LP well-posed if and only if (P) is generalized type I (resp., generalized type II) LP well-posed with 𝑓(π‘₯) defined by (2.15).(ii)If (GVQVI) is type I (resp., type II) LP well-posed, then (P) is type I (resp., type II) LP well-posed with 𝑓(π‘₯) defined by (2.15).

Proof. Let 𝑓(π‘₯) be defined by (2.15). Since π‘‹β‰ βˆ…, it follows from Lemma 2.7 that π‘₯βˆˆπ‘‹ is a solution of (GVQVI) if and only if π‘₯ is an optimal solution of (5) with 𝑣=𝑓(π‘₯)=0.(i)Similar to the proof of Lemma 2.9, it is also routine to check that a sequence {π‘₯𝑛} is a generalized type I (resp., generalized type II) LP approximating solution sequence if and only if it is a generalized type I (resp., generalized type II) LP minimizing sequence of (P). So (GVQVI) is generalized type I (resp., generalized type II) LP well-posed if and only if (P) is generalized type I (resp., generalized type II) LP well-posed with 𝑓(π‘₯) defined by (2.15).(ii)Since π‘‹ξ…ž0βŠ†π‘‹0,𝑑𝑋0(π‘₯)≀𝑑𝑋′0(π‘₯) for any π‘₯. This fact together with Lemma 2.9 implies that a type I (resp., type II) LP minimizing sequence of (P) is a type I (resp., type II) LP approximating solution sequence. So the type I (resp., type II) LP well-posedness of (GVQVI) implies the type I (resp., type II) LP well-posedness of (P) with 𝑓(π‘₯) defined by (2.15).

3. Criteria and Characterizations for Generalized LP Well-Posedness of (GVQVI)

In this section, we shall present some necessary and/or sufficient conditions for the various types of (generalized) LP well-posedness of (GVQVI) defined in Section 2.

Now consider a real-valued function 𝑐=𝑐(𝑑,𝑠,π‘Ÿ) defined for 𝑑,𝑠,π‘Ÿβ‰₯0 sufficiently small, such that𝑠𝑐(𝑑,𝑠,π‘Ÿ)β‰₯0,βˆ€π‘‘,𝑠,π‘Ÿ,𝑐(0,0,0)=0,π‘›βŸΆ0,𝑑𝑛β‰₯0,π‘Ÿπ‘›ξ€·π‘‘=0,𝑐𝑛,𝑠𝑛,π‘Ÿπ‘›ξ€ΈβŸΆ0implythatπ‘‘π‘›βŸΆ0,(3.1)

Theorem 3.1. Let the set-valued map 𝑇 be compact-valued on 𝑋1. If (GVQVI) is type II LP well-posed, the set-valued map 𝑆 is closed-valued, then there exist a function 𝑐 satisfying (3.1) such that ||𝑓||𝑑(π‘₯)β‰₯𝑐𝑋(π‘₯),𝑑𝑋0(π‘₯),𝑑𝑆(π‘₯)ξ€Έ(π‘₯),βˆ€π‘₯βˆˆπ‘‹1,(3.2) where 𝑓(π‘₯) is defined by (2.15). Conversely, suppose that 𝑋 is nonempty and compact and (3.2) holds for some 𝑐 satisfying (3.1). Then (GVQVI) is type II LP well-posed.

Proof. Define 𝑐||𝑓||(𝑑,𝑠,π‘Ÿ)=inf(π‘₯)∢π‘₯βˆˆπ‘‹1,𝑑𝑋(π‘₯)=𝑑,𝑑𝑋0(π‘₯)=𝑠,𝑑𝑆(π‘₯)ξ€Ύ.(π‘₯)=π‘Ÿ(3.3) Since π‘‹β‰ βˆ…, it is obvious that 𝑐(0,0,0)=0. Moreover, if 𝑠𝑛→0,𝑑𝑛β‰₯0,π‘Ÿπ‘›=0, and 𝑐(𝑑𝑛,𝑠𝑛,π‘Ÿπ‘›)β†’0, then there exists a sequence {π‘₯𝑛}βŠ†π‘‹1 with 𝑑𝑋(π‘₯𝑛)=𝑑𝑛, 𝑑𝑆(π‘₯𝑛)(π‘₯𝑛)=π‘Ÿπ‘›=0, 𝑑𝑋0ξ€·π‘₯𝑛=π‘ π‘›βŸΆ0,(3.4) such that ||𝑓π‘₯𝑛||⟢0.(3.5)
Since 𝑆 is closed-valued, π‘₯π‘›βˆˆπ‘†(π‘₯𝑛) for any 𝑛. This fact, combined with (3.4) and (3.5) and Lemma 2.9 (ii) implies that {π‘₯𝑛} is a type II LP approximating solution sequence of (GVQVI). By Proposition 2.4, we have that 𝑑𝑛→0.
Conversely, let {π‘₯𝑛} be a type II LP approximating solution sequence of (GVQVI). Then, by (3.2), we have ||𝑓π‘₯𝑛||𝑑β‰₯𝑐𝑋π‘₯𝑛,𝑑𝑋0ξ€·π‘₯𝑛,𝑑𝑆(π‘₯𝑛)ξ€·π‘₯𝑛.ξ€Έξ€Έ(3.6)
Let 𝑑𝑛=𝑑𝑋π‘₯𝑛,𝑠𝑛=𝑑𝑋0ξ€·π‘₯𝑛,π‘Ÿπ‘›=𝑑𝑆(π‘₯𝑛)ξ€·π‘₯𝑛.(3.7)
Then 𝑠𝑛→0 and π‘Ÿπ‘›=0,forallπ‘›βˆˆπ‘. Moreover, by Lemma 2.9, we have that |𝑓(π‘₯)|β†’0. Then, 𝑐(𝑑𝑛,𝑠𝑛,π‘Ÿπ‘›)β†’0. These facts together with the properties of the function 𝑐 imply that 𝑑𝑛→0. By Proposition 2.4, we see that (GVQVI) is type II LP well-posed.

Theorem 3.2. Let the set-valued map 𝑇 be compact-valued on 𝑋1. If (GVQVI) is generalized type II LP well-posed, the set-valued map 𝑆 is closed, then there exist a function 𝑐 satisfying (3.1) such that ||𝑓||𝑑(π‘₯)β‰₯𝑐𝑋(π‘₯),𝑑𝐾(𝑔(π‘₯)),𝑑𝑆(π‘₯)ξ€Έ(π‘₯),βˆ€π‘₯βˆˆπ‘‹1,(3.8) where 𝑓(π‘₯) is defined by (2.15). Conversely, suppose that 𝑋 is nonempty and compact and (3.8) holds for some 𝑐 satisfying (3.4) and (3.5). Then, (GVQVI) is generalized type II LP well-posed.

Proof. The proof is almost the same as that of Theorem 3.1. The only difference lies in the proof of the first part of Theorem 3.1. Here we define 𝑐||𝑓||(𝑑,𝑠,π‘Ÿ)=inf(π‘₯)∢π‘₯βˆˆπ‘‹1,𝑑𝑋(π‘₯)=𝑑,𝑑𝐾(𝑔(π‘₯))=𝑠,𝑑𝑆(π‘₯)ξ€Ύ.(π‘₯)=π‘Ÿ(3.9)
Next we give the Furi-Vignoli-type characterizations [39] for the (generalized) type I LP well-posedness of (GVQVI).
Let (𝑋,β€–β‹…β€–) be a Banach space. Recall that the Kuratowski measure of noncompactness for a subset 𝐻 of 𝑋 is defined as ξ‚†ξšπ»πœ‡(𝐻)=infπœ–>0βˆΆπ»βŠ†π‘–ξ€·π»,diam𝑖<πœ–,𝑖=1,…,𝑛,(3.10)
where diam(𝐻𝑖) is the diameter of 𝐻𝑖 defined by 𝐻diam𝑖=supβ€–π‘₯1βˆ’π‘₯2β€–βˆΆπ‘₯1,π‘₯2βˆˆπ»π‘–ξ€Ύ.(3.11)
Given two nonempty subsets A and B of a Banach space (𝑋,β€–β‹…β€–), the Hausdorff distance between A and B is defined by β„Žξ€½ξ€½π‘‘(𝐴,𝐡)=maxsup𝐡𝑑(π‘Ž)βˆΆπ‘Žβˆˆπ΄,sup𝐴(𝑏)βˆΆπ‘βˆˆπ΅ξ€Ύξ€Ύ.(3.12)
For any πœ–β‰₯0, two types of approximating solution sets for (GVQVI) are defined, respectively, by Ξ©1(πœ–)=π‘₯βˆˆπ‘‹1∢π‘₯βˆˆπ‘†(π‘₯),𝑑𝑋0(π‘₯)β‰€πœ–,βˆƒπ‘§βˆˆπ‘‡(π‘₯),s.t.𝑧,π‘₯β€²ξ‚­βˆ’π‘₯+πœ–π‘’βˆ‰βˆ’int𝐢,βˆ€π‘₯ξ…žξ‚‡,Ξ©βˆˆπ‘†(π‘₯)2(πœ–)=π‘₯βˆˆπ‘‹1∢π‘₯βˆˆπ‘†(π‘₯),𝑑𝐾(𝑔(π‘₯))β‰€πœ–,βˆƒπ‘§βˆˆπ‘‡(π‘₯),s.t.𝑧,π‘₯β€²ξ‚­βˆ’π‘₯+πœ–π‘’βˆ‰βˆ’int𝐢,βˆ€π‘₯ξ…žξ‚‡.βˆˆπ‘†(π‘₯)(3.13)

Theorem 3.3. Assume that 𝑇 is u.s.c. and compact-valued on 𝑋1and 𝑆 is l.s.c. and closed on 𝑋1. Then
(a) (GVQVI) is type I LP well-posed if and only if limπœ–β†’0πœ‡ξ€·Ξ©1ξ€Έ(πœ–)=0,(3.14)
(b) (GVQVI) is generalized type I LP well-posed if and only if limπœ–β†’0πœ‡ξ€·Ξ©2ξ€Έ(πœ–)=0.(3.15)

Proof. (a) First we show that, for every πœ–>0, Ξ©1(πœ–) is closed. In fact, let π‘₯π‘›βˆˆΞ©1(πœ–) and π‘₯𝑛→π‘₯0. Then (2.4) and the following formula hold: 𝑑𝑋0ξ€·π‘₯π‘›ξ€Έβ‰€πœ–,βˆƒπ‘§π‘›ξ€·π‘₯βˆˆπ‘‡π‘›ξ€Έξ‚¬π‘§,s.t.𝑛,π‘₯β€²βˆ’π‘₯𝑛+πœ–π‘’βˆ‰βˆ’int𝐢,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†π‘›ξ€Έ.(3.16) Since π‘₯𝑛→π‘₯0, by the closedness of 𝑆 and (2.4), we have π‘₯0βˆˆπ‘†(π‘₯0). From (3.16), we get 𝑑𝑋0ξ€·π‘₯0ξ€Έβ‰€πœ–,(3.17)βˆƒπ‘§π‘›ξ€·π‘₯βˆˆπ‘‡π‘›ξ€Έπ‘§,s.t.πœ‰ξ‚€ξ‚¬π‘›,π‘₯β€²βˆ’π‘₯𝑛β‰₯βˆ’πœ–,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†π‘›ξ€Έ.(3.18) For any π‘£βˆˆπ‘†(π‘₯0), by the lower semi-continuity of 𝑆 and (3.18), we can find π‘£π‘›βˆˆπ‘†(π‘₯𝑛) with 𝑣𝑛→𝑣 such that πœ‰ξ€·βŸ¨π‘§π‘›,π‘£π‘›βˆ’π‘₯π‘›βŸ©ξ€Έβ‰₯βˆ’πœ–.(3.19)
By the u.s.c. of 𝑇 at π‘₯0 and the compactness of 𝑇(π‘₯0), there exist a subsequence {𝑧𝑛𝑗}βŠ‚{𝑧𝑛} and some 𝑧0βˆˆπ‘‡(π‘₯0) such that π‘§π‘›π‘—βŸΆπ‘§0.(3.20)
This fact, together with the continuity of πœ‰ and (3.19), implies that πœ‰ξ€·βŸ¨π‘§0,π‘£βˆ’π‘₯0βŸ©ξ€Έξ€·π‘₯β‰₯βˆ’πœ–βˆ€π‘£βˆˆπ‘†0ξ€Έ.(3.21)
It follows that βŸ¨π‘§0,π‘£βˆ’π‘₯0ξ€·π‘₯⟩+πœ–π‘’βˆ‰βˆ’intπΆβˆ€π‘£βˆˆπ‘†0ξ€Έ.(3.22) Hence, π‘₯0∈Ω1(πœ–).
Second, we show that ⋂𝑋=πœ–>0Ξ©1(πœ–). It is obvious that β‹‚π‘‹βŠ†πœ–>0Ξ©1(πœ–). Now suppose that πœ–π‘›>0 with πœ–π‘›β†’0 and π‘₯βˆ—βˆˆβ‹‚πœ–>0Ξ©1(πœ–π‘›). Then 𝑑𝑋0ξ€·π‘₯βˆ—ξ€Έβ‰€πœ–π‘›π‘₯,βˆ€π‘›,(3.23)βˆ—ξ€·π‘₯βˆˆπ‘†βˆ—ξ€Έ,ξ€·π‘₯(3.24)βˆƒπ‘§βˆˆπ‘‡βˆ—ξ€Έξ‚¬,s.t.𝑧,π‘₯β€²βˆ’π‘₯βˆ—ξ‚­+πœ–π‘›π‘’βˆ‰βˆ’int𝐢,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†βˆ—ξ€Έ.(3.25) From (3.23), we have π‘₯βˆ—βˆˆπ‘‹0.(3.26) From (3.25), we have 𝑧,π‘₯ξ…žβˆ’π‘₯βˆ—ξ¬βˆ‰βˆ’int𝐢,βˆ€π‘₯ξ…žξ€·π‘₯βˆˆπ‘†βˆ—ξ€Έ,(3.27)
that is π‘₯βˆ—βˆˆπ‘‹. Hence, ⋂𝑋=πœ–>0Ξ©1(πœ–).
Now we assume that (GVQVI) is type I LP well-posed. By Remark 2.3, we know that the solution 𝑋 is nonempty and compact. For every positive real number πœ–, since π‘‹βˆˆΞ©1(πœ–), one gets Ξ©1ξ‚€Ξ©(πœ–)β‰ βˆ…,β„Ž1(πœ–),𝑋=maxsupπ‘’βˆˆΞ©1(πœ–)𝑑𝑋(𝑒),supπ‘£βˆˆπ‘‹π‘‘Ξ©1(πœ–)ξƒ°(𝑣)=supπ‘’βˆˆΞ©1(πœ–)𝑑𝑋(𝑒).(3.28)
For every π‘›βˆˆπ‘, the following relations hold: πœ‡ξ€·Ξ©1ξ€Έξ‚€Ξ©(πœ–)≀2β„Ž1(πœ–),𝑋+πœ‡π‘‹ξ‚ξ‚€Ξ©=2β„Ž1(πœ–),𝑋,(3.29) where πœ‡(𝑋)=0 since 𝑋 is compact. Hence, in order to prove that limπœ–β†’0πœ‡(Ξ©1(πœ–))=0, we only need to prove that limπœ–β†’0β„Žξ‚€Ξ©1(πœ–),𝑋=limπœ–β†’0supπ‘’βˆˆΞ©1(πœ–)𝑑𝑋(𝑒)=0.(3.30)
Suppose that this is not true, then there exist 𝛽>0, πœ–π‘›β†’0, and sequence {𝑒𝑛}, π‘’π‘›βˆˆΞ©1(πœ–π‘›), such that 𝑑𝑋𝑒𝑛>𝛽,(3.31) for 𝑛 sufficiently large.
Since {𝑒𝑛} is type I LP approximating sequence for (GVQVI), it contains a subsequence {π‘’π‘›π‘˜} conversing to a point of 𝑋, which contradicts (3.31).
For the converse, we know that, for every πœ–>0, the set Ξ©1(πœ–) is closed, ⋂𝑋=πœ–>0Ξ©1(πœ–), and limπœ–β†’0πœ‡(Ξ©1(πœ–))=0. The theorem on Page. 412 in [40, 41] can be applied, and one concludes that the set 𝑋 is nonempty, compact, and limπœ–β†’0β„Žξ‚€Ξ©1(πœ–),𝑋=0.(3.32)
If {π‘₯𝑛} is type I LP approximating sequence for (GVQVI), then there exists a sequence {πœ–π‘›} of positive real numbers decreasing to 0 such that π‘₯π‘›βˆˆΞ©1(πœ–π‘›), for every π‘›βˆˆπ‘. Since 𝑋 is compact and lim𝑛→+βˆžπ‘‘π‘‹ξ€·π‘₯𝑛≀lim𝑛→+βˆžβ„Žξ‚€Ξ©1ξ€·πœ–π‘›ξ€Έ,𝑋=0,(3.33) by Proposition 2.4, (GVQVI) is type I LP well-posed.
(b) The proof is Similar to that of (a), and it is omitted here. This completes the proof.

Definition 3.4. (i) Let 𝑍 be a topological space, and let 𝑍1βŠ†π‘ be nonempty. Suppose that β„ŽβˆΆπ‘β†’π‘…1βˆͺ{+∞} is an extended real-valued function. β„Ž is said to be level-compact on 𝑍1 if, for any π‘ βˆˆπ‘1, the subset {π‘§βˆˆπ‘1βˆΆβ„Ž(𝑧)≀𝑠} is compact.
(ii) Let 𝑋 be a finite-dimensional normed space, and let 𝑍1βŠ‚π‘ be nonempty. A function β„ŽβˆΆπ‘β†’π‘1βˆͺ{+∞} is said to be level-bounded on 𝑍1 if 𝑍1 is bounded or limπ‘§βˆˆπ‘1,‖𝑧‖→+βˆžβ„Ž(𝑧)=+∞.(3.34)
Now we establish some sufficient conditions for type I (resp., generalized I type) LP well-posedness of (GVQVI).

Proposition 3.5. Suppose that the solution set 𝑋 of (GVQVI) is nonempty and set-valued map 𝑆 is l.s.c. and closed on 𝑋1, the set-valued map T is u.s.c. and compact-valued on 𝑋1. Suppose that one of the following conditions holds:
(i) there exists 0<𝛿1≀𝛿0 such that 𝑋1(𝛿1) is compact, where 𝑋1𝛿1ξ€Έ=ξ€½π‘₯βˆˆπ‘‹1βˆ©π‘‹2βˆΆπ‘‘π‘‹0(π‘₯)≀𝛿1ξ€Ύ;(3.35)
(ii) the function 𝑓 defined by (2.15) is level-compact on 𝑋1βˆ©π‘‹2;
(iii) 𝑋 is finite-dimensional and limπ‘₯βˆˆπ‘‹1βˆ©π‘‹2,β€–π‘₯β€–β†’+βˆžξ€½π‘“max(π‘₯),𝑑𝑋0ξ€Ύ(π‘₯)=+∞,(3.36)
where 𝑓 is defined by (2.15);
(iv) there exists 0<𝛿1≀𝛿0 such that 𝑓 is level-compact on 𝑋1(𝛿1) defined by (3.35). Then (GVQVI) is type I LP well-posed.

Proof. First, we show that each of (i), (ii), and (iii) implies (iv). Clearly, either of (i) and (ii) implies (iv). Now we show that (iii) implies (iv). Indeed, we need only to show that, for any π‘‘βˆˆπ‘1, the set 𝐴=π‘₯βˆˆπ‘‹1𝛿1ξ€Έξ€ΎβˆΆπ‘“(π‘₯)≀𝑑(3.37) is bounded since 𝑋 is finite-dimensional space and the function 𝑓 defined by (2.15) is l.s.c. on 𝑋1 and thus 𝐴 is closed. Suppose to the contrary that there exists π‘‘βˆˆπ‘1 and {π‘₯ξ…žπ‘›}βŠ†π‘‹1(𝛿1) such that β€–π‘₯ξ…žπ‘›β€–β†’+∞ and 𝑓(π‘₯ξ…žπ‘›)≀𝑑. From {π‘₯ξ…žπ‘›}βŠ†π‘‹1(𝛿1), we have 𝑑𝑋0(π‘₯ξ…žπ‘›)≀𝛿1.
Thus, 𝑓π‘₯maxξ…žπ‘›ξ€Έ,𝑑𝑋0ξ€·π‘₯ξ…žπ‘›ξ€½ξ€Έξ€Ύβ‰€max𝑑,𝛿1ξ€Ύ,(3.38) which contradicts (3.36).
Therefore, we only need to we show that if (iv) holds, then (GVQVI) is type I LP well-posed. Let {π‘₯𝑛} be a type I LP approximating solution sequence for (GVQVI). Then, there exist {πœ–π‘›}βŠ†π‘1+ with πœ–π‘›β†’0 and π‘§π‘›βˆˆπ‘‡(π‘₯𝑛) such that (2.3), (2.4), and (2.5) hold. From (2.3) and (2.4), we can assume without loss of generality that {π‘₯𝑛}βŠ†π‘‹1(𝛿1). By Lemma 2.9, we can assume without loss of generality that {π‘₯𝑛}βŠ†{π‘₯βˆˆπ‘‹1(𝛿1)βˆΆπ‘“(π‘₯)≀1}. By the level-compactness of 𝑓 on 𝑋1(𝛿1), we can find a subsequence {π‘₯𝑛𝑗} of {π‘₯𝑛} and π‘₯βˆˆπ‘‹1(𝛿1) such that π‘₯𝑛𝑗→π‘₯. Taking the limit in (2.3) (with π‘₯𝑛 replaced by π‘₯𝑛𝑗), we have π‘₯βˆˆπ‘‹0. Since 𝑆 is closed and (2.4) holds, we also have π‘₯βˆˆπ‘†(π‘₯).
Furthermore, from the u.s.c. of 𝑇 at π‘₯ and the compactness of 𝑇(π‘₯), we deduce that there exist a subsequence {𝑧𝑛𝑗} of {𝑧𝑛} and some π‘§βˆˆπ‘‡(π‘₯) such that 𝑧𝑛𝑗→𝑧. From this fact, together with (2.5), we have 𝑧,π‘₯β€²βˆ’π‘₯ξ‚βˆ‰βˆ’int𝐢,βˆ€π‘₯ξ…žξ€·βˆˆπ‘†π‘₯ξ€Έ.(3.39) Thus, π‘₯βˆˆπ‘‹.
The next proposition can be proved similarly.

Proposition 3.6. Suppose that the solution set 𝑋 of (GVQVI) is nonempty and set-valued map 𝑆 is l.s.c. and closed on 𝑋1, the set-valued map T is u.s.c. and compact-valued on 𝑋1. Suppose that one of the following conditions holds:
(i) there exists 0<𝛿1≀𝛿0 such that 𝑋2(𝛿1) is compact, where 𝑋2𝛿1ξ€Έ=ξ€½π‘₯βˆˆπ‘‹1βˆ©π‘‹2βˆΆπ‘‘πΎ(𝑔(π‘₯))≀𝛿1ξ€Ύ;(3.40)
(ii) the function 𝑓 defined by (2.15) is level-compact on 𝑋1βˆ©π‘‹2;
(iii) 𝑋 is finite-dimension and limπ‘₯βˆˆπ‘‹1∩X2,β€–π‘₯β€–β†’+βˆžξ€½π‘“max(π‘₯),𝑑𝐾𝑔(π‘₯)=+∞,(3.41)
where 𝑓 is defined by (2.15),
(iv) there exists 0<𝛿1≀𝛿0 such that 𝑓 is level-compact on 𝑋2(𝛿1) defined by (3.40). Then (GVQVI) is generalized type II LP well-posed.

Remark 3.7. If 𝑋 is finite-dimensional, then the β€œlevel-compactness” condition in Propositions 3.1 and 3.6 can be replaced by β€œlevel boundedness” condition.

Remark 3.8. It is easy to see that the results in this paper unify, generalize and extend the main results in [26–30] and the references therein.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant no. 11171363 and Grant no. 10831009), the Natural Science Foundation of Chongqing (Grant No. CSTC, 2009BB8240)  and the special fund of Chongqing Key Laboratory (CSTC 2011KLORSE01).