The augmented weak sharpness of solution sets in equilibrium problems

This study delves into equilibrium problems, focusing on the identification of finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the solution set may not exhibit weak sharpness, we propose an augmented mapping approach to mitigate this limitation. The core of our research is the formulation of augmented weak sharpness for the solution set, a comprehensive concept that encapsulates both weak sharpness and strong non-degeneracy within feasible solution sequences. Crucially, we identify a necessary and sufficient condition for the finite termination of these sequences under the premise of augmented weak sharpness for the solution set in equilibrium problems. This condition significantly broadens the scope of existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in the context of mathematical programming and variational inequality problems. Our findings not only shed light on the termination conditions in equilibrium problems but also introduce a less stringent sufficient condition for the finite termination of various optimization algorithms. This research, therefore, makes a substantial contribution to the field by enhancing our understanding of termination conditions in equilibrium problems and expanding the applicability of established theories to a wider range of optimization scenarios.


Introduction
In this paper, we explore the equilibrium problem denoted as EP (φ, S): Find x ∈ S such that φ(x, y) ≥ 0, ∀y ∈ S, where S ⊂ R n is a closed convex set, and φ : R n × R n → R is a function such that φ(x, x) = 0, ∀x ∈ S. ( Let S = {x ∈ S | φ(x, y) ≥ 0, ∀y ∈ S} = ∅ be the solution set of EP (φ, S) and S = {x ∈ S | ∃u ∈ ∂ y φ(x, x), such that u, y − x ≥ 0, ∀y ∈ S} is the stationary points set of EP (φ, S), where ∂ y φ(x, x) is generalized subdifferential at x of φ(x, •) ([ [22].Definition 8.3]), or sub-differential for short.The relation between S and S is discussed, where it is noted that while the general relation does not always hold, under certain conditions, such as lower semi-continuity of φ(x, •) and the satisfaction of (BCQ) constraint qualification in S, this inclusion is established.This condition is particularly valid when φ(x, •) is locally Lipschitzian or convex on R n .The model EP (φ, S) serves as a unified model encompassing various optimization problems, including mathematical programming, variational inequality, Nash equilibrium, and saddle point problems, for example, [3,4,9,10].The study extends to vector optimization problems, demonstrating the versatility of EP (φ, S).Previous research efforts have expanded the model to include generalized quasi-variational inequality problems.The concept of equilibrium problem plays a central role in various applied sciences, such as physics, economics, engineering, transportation, sociology, chemistry, biology, and other fields [13,15,19].The theory of gap functions, developed in the variational inequalities, is extended to a general equilibrium problem in [18].Van et al. [25] provide sufficient conditions and characterizations for linearly conditioned bifunction associated with an equilibrium problem.
Regarding the finite termination of algorithms for EP (φ, S), particularly in mathematical programming and variational inequality problems, existing research has predominantly concentrated on concepts such as weak sharp minimum and strong non-degeneracy of the solution set.Pioneering studies by scholars like Rockafellar [23], Polyak [21], Ferris [12], and others have laid down conditions for finite termination utilizing specific algorithms.Nonetheless, the reliance on algorithmic frameworks has highlighted the necessity for more expansive research into conditions that ensure finite termination, irrespective of the algorithms employed.Early significant contributions in this area were made by Burke and Moré [7], who established necessary and sufficient conditions for the finite termination of feasible solution sequences in smooth programming problems that converge to strongly non-degenerate points.Subsequently, Burke and Ferris [5] broaden these findings to encompass differentiable convex programming.In a further extension, Marcotte and Zhu [17] generalize these principles to continuous variational inequality problems characterized by pseudo-monotonicity + .Al-Homidan et al. [1] consider weak sharp solutions for the generalized variational inequality problem, in which the underlying mapping is set-valued.Huang et al. [14] give several characterizations of the weak sharpness in terms of the primal gap function associated with the mixed variational inequality.Nguyen [20] presents the concept of weak sharpness in variational inequality problems, specifically within the context of Hadamard spaces.Subsequently, numerous researchers have explored the concept of weak sharpness of the solution set, particularly focusing on its implications for the finite convergence of diverse algorithms applied to variational inequality problems.This topic has been extensively investigated in various studies, as detailed in references [2,16,29,30], among others.
This paper introduces and elaborates on the concepts of weak sharpness and strong non-degeneracy of the solution set within the framework of EP (φ, S).
To tackle scenarios where these characteristics are not present, we propose an augmented mapping on the solution set.This leads to the definition of augmented weak sharpness of the solution set for feasible solution sequences.This novel concept not only generalizes weak sharpness and non-degeneracy but is also employed in establishing necessary and sufficient conditions for the finite termination of feasible solution sequences under the premise of augmented weak sharpness.
The remainder of the paper is organized as follows.Section 2 provides preliminaries and discusses several special cases of EP (φ, S).In Section 3, the notion of augmented weak sharpness is introduced for the solution set of EP (φ, S) under general conditions.Section 4 presents examples illustrating situations where weak sharpness or non-degeneracy is not satisfied, but aug-mented weak sharpness holds.Section 5 establishes the finite identification of feasible solution sequences under the condition of augmented weak sharpness and presents consequences, generalizing results from conditions of weak sharpness or strong non-degeneracy.We make a conclusion in Section 6.

Preliminary
In this section, we introduce fundamental concepts and specific cases relevant to EP (φ, S), laying the groundwork for subsequent discussions.
Consider an infinite sequence N ⊆ {1, 2, • • • } and sets C k ⊂ R n for k = 1, 2, • • • .Define the upper limit and lower limit of the sequence of sets as follows: Thus we have The tangent cone for C at x is defined as The regular normal cone for C at x is defined as In general meaning, the normal cone for C at x is defined as N C (x) = lim sup The polar cone of C is defined as ], Proposition 6.5], we have T C (x) • = NC (x).When C is convex, by [ [22], Theorem 6.9], we have The projection of a point x ∈ R n over a closed set C is defined as [22], Exercise 8.8], we have ∂ψ(x) = {∇ψ(x)}, which means the projected sub-differential is the projected gradient P TC (x) (−∇ψ(x)).
We call that a sequence {x k } ⊂ R n terminates finitely to C if there exists k 0 such that x k ∈ C for all k ≥ k 0 .In EP (φ, S), we call φ is monotonic on S × S, if φ(x, y) + φ(y, x) ≤ 0, for ∀(x, y) ∈ S × S. A function φ is said to be pseudo-monotone on S × S, if φ(x, y) ≥ 0 =⇒ φ(y, x) ≤ 0, for ∀(x, y) ∈ S × S.
where f : R n → R. Obviously, φ satisfies (1).Then EP (φ, S) is the following mathematical programming problem: where F : S → R n .Obviously, φ satisfies (1).Then EP (φ, S) is the following variational inequality problem: By (4), we know φ is monotonic on S × S ⇐⇒ F is monotonic on S, and φ is pseudo-monotonic on S × S ⇐⇒ F is pseudo-monotonic over S.
where ϕ : R n → R. Obviously, φ satisfies (1).Then EP (φ, S) is the following global saddle point problem: with the obvious modifications for the cases i = 1 and i = n.

Suppose
where f i : R n → R. Obviously, φ satisfies (1).Then EP (φ, S) is the following Nash equilibrium problem: (N EP ) Find x ∈ S, such that f i (x) ≤ f i (x i , y i ), ∀y i ∈ S i , for all i ∈ I.
3 The augmented weak sharpness in the equilibrium problem In this section, we present the notions of weak sharpness and strong nondegeneracy for the solution set S ⊂ S of EP (φ, S).Furthermore, in order to provide a more relaxed conditions on the finite identification of a feasible solution sequence, we introduce an augmented mapping over the solution set S, and establish the concept of augmented weak sharpness for the solution set S on feasible solution sequence.Under several different cases and very general assumptions, we prove that this new concept is a generalization of weak sharpness and strong non-degeneracy.First, we give the concept of weak sharp minimum in mathematical programming (see [6,11]) Definition 1 In mathematical programming (M P ), the solution set S ⊂ S is weak sharp minimal, if there exists a constant α > 0, such that for ∀x ∈ S, we have The constant α and the set S are called the modulus and domain of sharpness for f over S, respectively.Clearly, S is a set of global minima for f over S.
If (M P ) is a non-smooth convex programming, then S is a weak sharp minimal set with the module α, if and only if where B is a unit ball (see [[6], Theorem 2.6, c]).If (M P ) is a smooth convex programming, then S is a weak sharp minimal set with the module α, if and only if (see [[6], Corollary 2.7, c)]).Here, in the case that f is smooth, (9) and ( 8) are equivalent, as ∇f (•) is a constant vector on S (see [ [5], Corollary 6]).
To generalize the characteristics of the solution set into variational inequalities and the smooth non-convex programming problems, some literature ( [17,26,27,32,33]) utilize (9) to define the weak sharpness of the solution set in these two kinds of problems.Now we use (8) to define the weak sharpness of the solution set S of EP (φ, S).
Definition 2 In the EP (φ, S), for ∀x ∈ S, ∂ y φ(x, x) = ∅, the solution set S ⊂ S is said to be a weak sharp set with the module α, if there exists a constant α > 0, such that Remark 1 In (10), we have used N S (•) instead of N S (•), since in general, S is not necessarily convex.When S is non-convex, according to [ [22], Proposition 6.5], N S (•) is a closed convex cone, while N S (•) is only a closed cone, and it holds that When S is convex, according to [ [22], Theorem 6.9], the formula above holds inequality.So we can obtain Therefore, NS (•) makes the conditions of Definition 2 more relaxed.
Next, we introduce the notion of strong non-degeneracy in EP (φ, S).
and x is said to be a strongly non-degenerate point.
Now we give the main notion in this paper.
Definition 4 In the EP (φ, S), suppose S ⊂ S is a closed set.For any x ∈ S, there is ∂ y φ(x, x) = ∅, and {x k } ⊂ S. We call S is augmented weak sharp with respect to {x k }.For an infinite sequence K = {k | x k / ∈ S}, there exists an augmented mapping (set-valued mapping) H : S → 2 R n such that the following hold: (a) there exists a constant α > 0, such that Now we will discuss the inclusion relation between two concepts, augmented weak sharpness of the solution set S and the weak sharpness as well as the strong non-degeneracy in several cases.

The non-smooth case
Proposition 1 In the EP (φ, S), suppose S ⊂ S is a closed set, ∂ y φ(x, x) = ∅ for any x ∈ S, and ∂ y φ(x, x) is monotonic over S. If S is weakly sharp, then for every {x k } ⊂ S, S is augmented weakly sharp.
∈ S} be an infinite sequence.Set By (10) we know that (a) in Definition 4 holds, and (b) also holds by the monotonicity of ∂ y φ(x, x).
The following proposition provides a sufficient condition for the monotonicity of ∂ y φ(x, x).
Proposition 2 In the EP (φ, S), suppose that φ satisfies the following conditions: Proof By (i) we have Therefore, according to the pseudo-monotonicity of φ, it holds that

Remark 2
The following two examples show that the two assumptions in Proposition 2 are only sufficient and not necessary conditions for that ∂ y φ(x, x) is monotonic over S.
We note that [ [5], Theorem 5] gave the characteristic description for the solution set of a non-smooth convex programming.The result in [[5], Theorem 5] not only contributes to the understanding of the nature of the solution set of convex programming, but also plays an important role in the analysis of the weak sharp minimality of the solution set in convex programming.So next this result is generalized to the solution set of EP (φ, S), and further apply it to the analysis on the weak sharpness of the solution set of EP (φ, S).
Proposition 3 Under the assumptions of Theorem 1, and furthermore suppose that S ⊂ S is closed, ∂ y φ(x, x) is monotonic over S.
then S is augmented weakly sharp for all {x k } ⊂ S.
Proof First, by the assumption (i) in Theorem 1 and [ [24], Theorem 23.4], we get that ∂ y φ(x, x) is a nonempty compact set for ∀x ∈ R n .Furthermore, by Theorem 1, we obtain that ∂ y φ(x, x) (−N S (x)) is a nonempty compact constant set over S.So by (16), there exists a constant α > 0 such that for ∀x ∈ S, From the formula above, for ∀x ∈ S, we get that Therefore, S is a weak sharp set by Definition 2. According to the monotonicity of ∂ y φ(x, x) and Proposition 1, the proof is complete.

The smooth case
In this subsection, we assume that φ(x, •) is continuously differentiable on R n for ∀x ∈ S. At this time, we have ∂ y φ(x, •) = {∇ y φ(x, •)}.
Proposition 4 In the EP (φ, S), suppose S ⊂ S is a closed set, and {x k } ⊂ S satisfies If S is weak sharp, then S is augmented weakly sharp with respect to {x k }.
Proof Let K = {k | x k / ∈ S} be an infinite sequence, and set By (10), we know that (a) holds in Definition 4, and by ( 17), we immediately obtain that lim sup k∈K, k→∞ i.e., (b) holds in Definition 4.

Proposition 5
In EP (φ, S), suppose S ⊂ S is a closed set, {x k } ⊂ S, {∇ y φ(x k , x k )} is bounded and any one of its accumulations p satisfies −p ∈ intG.Then S is augmented weakly sharp with respect to {x k }.
Proof Let K = {k | x k / ∈ S} be an infinite sequence.According to the hypotheses, there must be an accumulation p of {∇ y φ(x k , x k )} k∈K satisfying −p ∈ intG, i.e., there exists a constant α > 0 such that Letting H(z) = p, ∀z ∈ S, by (18) we know that (a) holds in Definition 4. Furthermore let K 0 ⊆ K such that lim k∈K0,k→∞ By (19), we immediately get that lim sup k∈K, k→∞ i.e., (b) holds in Definition 4.
Finally, we give the relation of strong non-degeneracy and augmented weak sharpness.
Proposition 6 In the EP (φ, S), suppose ∇ y φ(x, x) is continuous over S, {x k } ⊂ S is bounded and any of its accumulations is strongly non-degenerate.Then S is augmented weakly sharp with respect to {x k }.
∈ S} be an infinite sequence.Then there must be an accumulation point x of {x k } k∈K .Suppose K 0 ⊆ K such that lim k∈K0,k→∞ By the assumptions of the strong non-degeneracy of x and the continuity of ∇ y φ(x, x), and [[28], Proposition 5.1], we know that x is an isolated point of S, Thus we have According to (11) and ( 21), we know that there exists a constant α > 0 such that Now we define the augmented mapping According to (22) and ( 23), we can immediately obtain that i.e., (a) holds in Definition 4. Since x is an isolated point of S, by (20), for all sufficiently large k ∈ K 0 it holds that P S (x k ) = x.Therefore, according to (20), (23), and the continuity of ∇ y φ(x, x), we obtain that lim sup k∈K, k→∞ i.e., (b) holds in Definition 4.

Some examples
In this section, we give some examples of EP (φ, S) to show that the solution set S of EP (φ, S) does not satisfy the weak sharpness but the augmented weak sharpness.
Example 7 Consider the following mathematical programming problems (M P ) as a special case of EP (φ, S): This is a non-smooth and non-convex programming problem, and the solution set S of it is non-convex.By Example 1, we know that Therefore, ∂ y φ(x, •) = ∂f (•).Now, let (t) + = max{0, t}.When x ∈ S \ S, we have When x ∈ S, we have Note that S ⊂ intS, for all x ∈ S, we have T S (x) = R 2 .So, for ∀x ∈ S, we get that By (25), the second and third formula of (26), we obtain that By (27) we know that the model α > 0 is not a constant in Definition 2, which is related to x ∈ S, and when x → 0, α → 0 + , i.e., the constant α > 0 does not exist.Therefore, S is not a weak sharp set.Now, take small enough ε > 0, and let Below we will prove that S is an augmented weak sharp set with respect to arbitrary sequences ∈ S} be an infinite sequence.We introduce the augmented mapping H : S ⇒ R 2 as follows: According to ( 26), (28), and the conditions (a) in Definition 4 holds, i.e., there exists a constant α > 0 such that for all x ∈ S, where u k ∈ ∂f (x k ), v k ∈ H(P S (x k )), and According to ( 24), (28), and (29), we can easily prove that {x k } k∈K ⊂ S\Ω ε satisfies the condition (b) in Definition 4. For simplicity, we only prove the case that {x k } k∈K lies in the third quadrant.At the same time, considering x k / ∈ Ω ε , we have (i) when x k 2 ≤ x k 1 < 0 and x k 2 ≤ −ε, by the first item of ( 24), (29), and the second item of (28), we have by the first item of ( 24), (29), and the third item of (28), we have By (i), (ii), we get that lim sup k∈K, k→∞ i.e., (b) in Definition 4 holds.
In addition, we note that Remark 1 is verified through Example 7. As in this example, the regular normal cone of S at (0, 0) is while the normal cone under the general meaning is Here, N S (0, 0) is a closed cone, but not a convex cone.Furthermore, according to T S (0, 0) = R 2 , it follows that The advantage of the regular cone has been shown here compared with the normal cone under the general meaning.
Example 8 Consider the following variational inequality problem V IP (F, S) as a special case of EP (φ, S): When x ∈ S, we have and This is a non-monotonic variational inequality problem.By ( 30) and ( 31), one can see that S is not a weak sharp set.
Next, we will prove that S is an augmented weak sharp set with respect to the sequence {x k } ⊂ S which satisfies the following conditions: ∈ S} be an infinite sequence.Take λ ∈ (0, 1  2 ), we introduce the augmented mapping H : S → R 2 as follows: By ( 31) and ( 32), we obtain that the condition (a) in Definition 4 holds.Now, for k ∈ K, let where By the condition (ii), the accumulations of the bounded sequence {x k } k∈K are only possibly x = (0, 0) or x = ( π 2 , 0).Without loss of generality, let x = ( π 2 , 0) be one of its accumulations.Then there exists an infinite subsequence According to ( 32), (33), and (34), when k ∈ K 0 is big enough, we obtain that Furthermore, by (34), we immediately get that lim sup k∈K, k→∞ i.e., (b) in Definition 4 holds.
By Example 3, one can see that this is a saddle point problem (SP P ), i.e., It can be easily seen that and When x ∈ S, we have By ( 36) and (37), one can see that S is not a weak sharp set and that S is not a strongly non-degenerate set.
Next, we will prove that S is an augmented weak sharp set with respect to the sequence {x k } ⊂ S which satisfies the following conditions: For this purpose, let K = {k | x k / ∈ S} be an infinite sequence.Take λ ∈ (0, 1), we introduce the augmented mapping H : S → R 2 as follows: By ( 37) and (38), we obtain that the condition (a) in Definition 4 holds.Now we will prove the condition (b) in Definition 4 also holds.By (ii) one can see that the accumulations of {x k } are x = (0, 1) or x = (0, −1).Without loss of generality, assume that there exists a sequence k 0 ⊆ K such that lim K∈K0,k→∞ For k ∈ K, let where According to (35), (38), (39), and (40), when k ∈ K 0 is big enough, we obtain that Thus we have lim sup k∈K, k→∞ ), , S = {(0, 0)}, x = (0, 0).By Example 4, one can see that this is a N ash equilibrium problem (N EP ), i.e., find x ∈ S such that where By ( 42) and (43), we obtain that S is not a weak sharp set, i.e., the point (0, 0) is not a strongly non-degenerate point.Now we will prove that S is an augmented weak sharp set for arbitrary sequences ∈ S} be an infinite sequence.Take λ ∈ (0, 1 2 ), we introduce the augmented mapping H : S → R 2 as follows: By ( 43) and (44), one can see that the condition (a) in Definition 4 holds.Furthermore, according to (41) and (44), for k ∈ K, we obtain that Thus the condition (b) in Definition 4 also holds.
The following are some examples similar to the above examples.1. 3.

Finite termination of feasible solution sequence
In this section, under the condition that the solution set S of EP (φ, S) is augmented weak sharp with respect to {x k } ⊂ S , we present the condition of finite identification of {x k } (see Theorem 2).Applying the result to four special cases of EP (φ, S) (see Examples 1-4), we derive a series of results for finite identification of feasible solution sequences in these cases.In the first two special cases, i.e., the mathematical programming and variational inequalities problems, these results are generalizations of the corresponding results in the literature under the condition that S is weakly sharp or strongly non-degenerate.In the last two special cases, i.e., the saddle point and N ash equilibrium problems, the finite identification problems of feasible solution sequences have not been studied by other authors in the literature.
On the other hand, from (50) one can see 0 ∈ lim inf k∈K, k→∞ Therefore, there exists ūk ∈ ∂ y φ(x k , x k ) such that lim k∈K, k→∞ Using ( 54), (57), and the properties of the projected gradient ([[8], Lemma 3.1]), we immediately get that According to (52) and (58), which leads to a contradiction.The proof is complete.
Applying Theorem 2 to the special cases (Examples 1-4) of EP (φ, S), we can obtain the following corollaries.
(3) In the (SP P ), suppose that S ⊂ S is a closed set, for ∀x ∈ S, ∂ϕ(x) = ∅, S is augmented weakly sharp with respect to {x k } ⊂ S.So the following conclusions are established.(i) Suppose that (2) holds.If {x k } terminates finitely to S, then we have Remark 4 By Remark 2, one can see that the monotonicity of ∂ y φ(x, x) does not imply the convexity of φ(x, •), and the reverse is also true.For example, φ(x, y) = x 2 y − x 3 , (x, y) ∈ R × R.
Notice that a finite convex function on R n is locally Lipschitzian, therefore by Corollary 3 and Proposition 2, we immediately get the following corollary.For the special cases (Examples 1-3) of EP (φ, S), we have the following corollaries.

Corollary 5
The following conclusions are established.
(1) In the convex programming (M P ), suppose S ⊂ S is a weak sharp minimal set.Then {x k } ⊂ S terminates finitely to S, if and only if (59) holds.
(2) In the V IP (F, S), suppose F (•) is monotonic over S, S ⊂ S is a closed and weak sharp set.Then {x k } ⊂ S terminates finitely to S, if and only if (60) holds.
(3) In the Proof According to the hypotheses in (1) and ( 2), one can see that they all meet the hypotheses conditions of Corollary 3.For (3), by its hypotheses and (5), one can see that the hypotheses conditions of Corollary 4 are satisfied.So we immediately get that these conclusions (1)-(3) hold.Since the convex programming (M P ), as a special case of EP (φ, S), satisfies the hypotheses in Corollary 6, we can obtain the following corollary.By Theorem 2 and a series of its corollaries, one can see that, under normal conditions, the weak sharpness or strong non-degeneracy of the solution set is a special case of the augmented weak sharpness with respect to the feasible solution sequence.On the other hand, for some algorithms in mathematical programming and variational inequalities, for example, the proximal point algorithm, the gradient projection algorithm and the SQP algorithm and so on (see [7,8,21,26,27,31,32]), the projected gradient of the point sequence generated by them all converge to zero, i.e. (50) holds.Therefore, the notion of augmented weak sharpness of the solution set presented by us provides weaker sufficient conditions than the weak sharpness or strong non-degeneracy for the finite termination of these algorithms.

Conclusion
In this paper, a novel concept concerning the solution set of equilibrium problems has been introduced, namely, the augmented weak sharpness of the solution set.This concept extends the traditional notions of weak sharpness and strong non-degeneracy in relation to feasible solution sequences.We have established that the necessary and sufficient conditions for the finite termination of feasible solution sequences are met when the lower limit of the projected sub-differential sequence, associated with the feasible solution sequence, encompasses the zero point.This finding has led to the derivation of several significant corollaries.Additionally, the augmented weak sharpness of the solution set presents a sufficient condition for the finite termination of certain equilibrium problem algorithms and their specific variants, such as mathematical programming problems, variational inequality problems, Nash equilibrium problems, and global saddle point problems.This condition is less stringent than the traditional criteria of weak sharpness and strong non-degeneracy.

Corollary 1
The following conclusions hold.(1)In the (M P ), suppose that S ⊂ S is a closed set, ∂f (x) = ∅ for ∀x ∈ S, and S is augmented weak sharp with respect to {x k } ⊂ S.So the following conclusions are established.(i)Suppose that (2) holds.If {x k } terminates finitely to S, then we have0 ∈ lim inf k→∞ P TS (x k ) (−∂f (x k )).(59)(ii) If (59) holds, then {x k } terminates finitely to S. (2) In the V IP (F, S), suppose that S ⊂ S is a closed set, S is augmented weak sharp with respect to {x k } ⊂ S. Then {x k } terminates finitely to S if and only if

Corollary 4
In the EP (φ, S), suppose S ⊂ S is a closed set, for ∀x ∈ S, φ(x, •) is convex function on R n , and φ(•, •) is monotonic over S × S. If S is a weak sharp set, then {x k } ⊂ S terminates finitely to S, if and only if (50) holds.

Corollary 7 Remark 6 Corollary 8 Corollary 9 Remark 7 Remark 8
Suppose that in the convex programming (M P ), it holds that for ∈ S, −∂f (x) ∩ (−N S (x)) ⊂ intG.Then {x k } ⊂ S terminates finitely to S, if and only if (59) holds.Corollary 7 is a generalization of [[6], Theorem 4.7] in the smooth convex programming, and in Corollary 7, the two hypotheses about {x k } and ∇f (•) in [[6], Theorem 4.7] are removed.Next, we consider the smooth situation, the case that φ(x, •) is locally Lipschitzian.Therefore, by Proposition 4 and Corollary 2, we obtain the following corollary.In the EP (φ, S), suppose S ⊂ S is a closed set, and {x k } ⊂ S satisfies(17).If S is weak sharp, then {x k } terminates finitely to S, if and only iflim k→∞ P TS (x k ) (−∇ y φ(x k , x k )) = 0. (63)By Proposition 5 and Corollary 2, we obtain the following corollary.In the EP (φ, S), suppose S ⊂ S is a closed set, and {x k } ⊂ S, {∇ y φ(x k , x k )} is bounded and any of its accumulation p satisfies −p ∈ intG.Then {x k } terminates finitely to S, if and only if (63) holds.In the special cases V IP (F, S) of EP (φ, S), Corollary 9 is a generalization and improvement of [[33], Theorem 3.3], i.e., [T S (x) ∩ N S (x)] • in [[33], Theorem 3.3] is replaced with [T S (x) ∩ N S (x)] • ,and the assumption of the continuity of F (•) is removed.It is worthwhile to note that [[33], Theorem 3.3] has ever improved [[32], Theorem 3.2].By Proposition 6 and Corollary 2, we have the following corollary.Corollary 10 In the EP (φ, S), suppose ∇ y φ(x, x) is continuous over S, {x k } ⊂ S is bounded and any of its accumulation is strongly non-degenerate.Then {x k } terminates finitely to S, if and only if the (63) holds.In the smooth programming problems (M P ), a special case of EP (φ, S), Corollary 10 is just [[27], Theorem 5.3], and the latter is an extension of [[7], Corollary 3.5].

)
(4)) If (61) holds, then {x k } terminates finitely to S.(4)In the (N EP ), suppose that S ⊂ S is a closed set, ∂ yi f i (x) = ∅ for ∀x ∈ S and i ∈ I = {1, 2, .., n}, S is augmented weak sharp with respect to {x k } ⊂ S.So the following conclusions are established.(i)Supposethat (2) holds.If {x k } terminates finitely to S, then we haveTS (x k ) (−(∂ yi f i (x k ), i ∈ I)).•)is locally Lipschitzian function on R n .For this function, by [[22], Theorem 9.13 and Theorem 8.15], we know (2) is established, and ∂ y φ(x, x) = ∅.Therefore by Theorem 2, we have the following corollary.In the EP (φ, S), suppose that S ⊂ S is a closed set, for ∀x ∈ S, φ(x, •) is locally Lipschitzian, S is augmented weak sharp with respect to {x k } ⊂ S, then {x k } ⊂ S terminates finitely to S, if and only if (50) holds.In the EP (φ, S), suppose S ⊂ S is a closed set, φ(x, •) is locally Lipschitzian function on R n for ∀x ∈ S, and ∂ y φ(x, x) is monotone over S. If S is a weak sharp set, then {x k } ⊂ S terminates finitely to S, if and only if (50) holds.
k→∞ P and S ⊂ S is a weak sharp set.Then {x k } ⊂ S terminates finitely to S, if and only if (61) holds.