Weak-stability and saddle point theorems for a multiobjective optimization problem with an infinite number of constraints

In this paper, we focus on weak-stability and saddle point theorems of multiobjective optimization problems that have an infinite number of constraints. The obtained results are based on the notion of weak-subdifferentials for vector functions. Some properties of weak stability for the problems are introduced. Relationships between strong duality and saddle points of the augmented Lagrange vector functions associated to the problems are investigated. Connections between weak-stability and saddle point theorems of the problems are established. An example is given.


Introduction
It is well known that the notion of subdifferentials for convex functions is important in convex analysis, optimization theory, and variational analysis [13]. Optimality conditions for convex optimization problems are usually represented in terms of subdifferentials of convex functions. Note that this notion may not be suitable for the cases of nonconvex optimization problems. Hence, it is necessary to generalize this concept for nonconvex functions. Such generalized subdifferentials for nonconvex functions were proposed by several mathematicians, such as Clarke [3], Rockafellar [12,14], Mordukhovich [11], and Azimov and Gazimov [2].
One of the generalized subdifferentials mentioned above, which has attracted several researchers recently, is the notion of weak-subdifferentials introduced by Azimov and Gazimov. In [2], based on the notion of supporting cones, the authors proposed the notion of weak-subdifferentials for scalar nonconvex functions. Moreover, with a type of augmented Lagrange function, which was constructed by using supporting cones to the epigraph of a perturbation function, the relationships between stability and duality of nonconvex problems via the augmented Lagrangians are investigated. Several papers related to this topic were published, where the obtained results were based on the use of weak-subdifferentials [4][5][6][7][8][9][10]15]. Recently, this notion was developed for vector functions defined on a normed space [9]. Inspired by this work, very recently, it was extended to vector functions defined on a linear space and was applied to a multiobjective semiinfinite optimization problem [16]. Then dual problems via augmented Lagrange vector functions were studied. In that paper, a concept of weak-stability for vector functions was proposed and the relationships between weak-stability and strong duality for the problems were investigated.
In optimization theory, there exists a close connection between the notion of strong duality and saddle point theorems (see [17] for more details). Hence, it is worth studying such relationships for the case of multiobjective semiinfinite optimization. Note that the relationships between weak-stability, strong duality, and saddle point theorems for a scalar nonconvex optimization problem based on weak-subdifferentials for scalar functions were introduced in [2]. Furthermore, the results were extended for scalar semiinfinite optimization problems in [15]. Very recently, the results were developed for multiobjective problems. In [16], the relationships between weak-stability and strong duality for semiinfinite optimization problems are studied. Motivated by the observations above, in this research, based on the notion of weak-subdifferentials for vector functions, we study the relationships between weak-stability and saddle point theorems for multiobjective optimization problems that have an infinite number of constraints. In our research, besides studying the relationships between weakstability and saddle point theorems, connections between strong duality and the existence of saddle points of the problems are also investigated. Hence, the relations among weak stability, strong duality, and saddle points theorems for the problems are studied.
The results in this paper are divided into two parts. The first part is devoted to investigating some properties of weak-stability for the multiobjective optimization problem. The second part concerns saddle point theorems for augmented Lagrange vector functions. We also show that there exists a close connection between the pairs of efficient solutions of a multiobjective optimization problem and its dual problem with saddle points of the corresponding vector Lagrange function. Then saddle point theorems for the multiobjective optimization problems are given.
The paper is organized as follows. In the second section, we recall the results on weak-subgradients for vector functions and the theorem that connects the property of weak-stability with the property of L-lower Lipschitz. The main results are given in Sections 3 and 4, where results on weak-stability for a multiobjective optimization problem are given and saddle point theorems are introduced. An example is given in the last part.

Preliminaries
Let T be an arbitrary index set. We use the following notations: Let λ : T → R be a real function such that λ(t) = λ t ∈ R for each t ∈ T but only finitely many λ t differ from zero. We need the following linear space: For λ ∈ R (T ) , its supporting set, denoted by T (λ), is defined by It is obvious that, for a given λ ∈ R (T ) , the set T (λ) is a finite set. The subset of R (T ) where its elements + is a convex cone of R (T ) . For λ,λ ∈ R (T ) , u ∈ R T , and α ∈ R, we define αλ := (αλ t ) t∈T , λ ±λ := (λ t ±λ t ) t∈T , Let us define the function ρ : R T × R (T ) → R as follows: The notation ∥u∥ λ is used instead of the value of the function ρ at (u, λ).
Let us denote D := R m + . The notations D • , D * , and 0 D stand for the interior of D , D \ {0 D }, and (0, 0, . . . , 0), respectively. Then, for y, z ∈ R m , we understand that We need some more notations below: and For convenience, we recall some basic concepts and results.

Definition 2.1 [16, Definition 3.2]
Given Λ ∈ L and u 0 ∈ R T , the Λ-neighborhood of u 0 ∈ R T with the radius We note that the two following definitions are developed from the original ones for real valued functions, which were introduced in [2].
for all i ∈ M , and let u 0 ∈ domh. The weak-subdifferential of the function h at u 0 is denoted and defined by the set The vector function h is called Λ -lower Lipschitz if the inequality above holds for all u ∈ domh .
The following proposition and corollary are developed with inspiration from Theorem 1 and Corollary 2, respectively, in [2] (see also [1]).
The following statements are equivalent: We understand that a vector function h : R T → R m is said to be bounded below if each component function of h is bounded below.

Corollary 2.5
Let h : R T → R m be a Λ -lower Lipschitz function at u 0 ∈ domh . If the function h is bounded below, then it is weakly subdifferentiable at u 0 .
Proof If the function h is bounded below then there exists a vector m ∈ R m such that Hence, Combining this and the assumption that h is a Λ -lower Lipschitz function at u 0 ∈ domh, we get the desired conclusion by applying Proposition 2.4. 2 Let us consider the following problem: where f i with i ∈ M and g t with t ∈ T are real valued functions defined on a linear space X , T is as above, and S is a nonempty subset of X . Denote by A the feasible set of (MP). Set We need the following function (see [15]): The augmented Lagrange vector function associated to (MP) is defined by The perturbation function ϕ i : Hence, the perturbation vector function ϕ : We need the following lemmas. They are quoted from the ones in [16].

Lemma 2.7
For each i ∈ M and for every x ∈ S , it holds that Definition 2.8 For the problem (MP), a point z ∈ A is said to be: The efficient solution set of (MP) is denoted by E(MP). Assume that E(MP) ̸ = ∅ .
We note that the following duality scheme is also a generalization of the duality scheme given in [2]. The dual problem for (MP) is formulated as follows (see [16]): and

Definition 2.9
For the problem (MD), a pair (Λ,k) ∈ F m is said to be: The set of efficient solutions of (MD) is called the efficient solution set of (MD) and is denoted by E(MD). In the next section, we establish a connection between weak-stability and the saddle point theorem for (MP).

Weak-stability of (MP) and related properties
For the function ϕ defined by (2.5), let us consider the following vector function G defined as follows: for each i ∈ M . We recall the notation of weak-stable introduced in [16].
Then, for any i ∈ M , we get By Lemma 2.6, we get We get Since The following theorem is developed from Theorem 6 in [2].

. Then (MP) is weakstable if and only if both the following conditions hold:
i) There exist Λ ∈ L , L ∈ R m + , U Λ (0), and x * ∈ S such that ii) There exists vectors p ∈ R m + and q ∈ R m such that

3)
Proof If (MP) is weak-stable, then G is weak-subdifferentiable at u = 0 and G i (0) is finite for each i ∈ M .
By Proposition 2.4, there exists Λ ∈ L such that the function G is Λ -locally lower Lipschitz at u = 0 and there exist vectors p ∈ R m + and q ∈ R m such that Hence, On the other hand, since G is Λ-lower locally Lipschitz at u = 0 , there exist a Λ -neighborhood U Λ (0) and a vector L ∈ R m + such that

Note that
Then there exists

Conversely, suppose that conditions i) and ii) hold. Note that
Thus, G(u) is Λ -locally lower Lipschitz at u = 0 . On the other hand, since (3.3) holds for all x ∈ S , by taking the infimum over S , we get Thus, by applying Proposition 2.4, G is weak-subdifferentiable at u = 0 , and (MP) is weak-stable. 2

Theorem 3.4 For the problem (MP), suppose that (inf
Proof Suppose that there exist Λ ∈ L , L ∈ R m + , and x * ∈ S such that (3.5) holds. By (3.1), we obtain Moreover, since ϕ(x * , 0) ≥ G(0), the inequality above implies that The function G is Λ -lower Lipschitz at u = 0 . From this it can be deduced that the problem (MP) is weak-stable by Proposition 2.4.
Conversely, suppose that (MP) is weak-stable. Hence, the function G is Λ -lower Lipschitz at u = 0 , i.e. there exists Λ ∈ R m + such that Since 0). The desired conclusion is derived 2

Saddle point theorems
Let us consider the vector Lagrange function L given by (2.2), where Λ ∈ L and k ∈ R m + .

Definition 4.1 A pair (x,Λ) ∈ S × L − is called a saddle point of the vector function L if the following
inequalities are satisfied for some k ∈ R m + : That is to say that there exist no λ i ∈ R (T ) − and x ∈ S, such that: with at least one strict inequality, and there exists no x ∈ S such that with at least one strict inequality.
First, we claim thatx ∈ A. Assume to the contrary thatx / ∈ A . Then there exists t 0 ∈ T such that g t0 (x) > 0 . For i ∈ M , choose λ i * such that λ i * t0 < 0 and λ * t =λ i t for all t ∈ T \ {t 0 } . Since λ i * t0 < 0 is chosen arbitrarily, by letting λ i * t0 → −∞ , we get such that the left-hand side of (4.2) tends to +∞ , a contradiction.
Next, we prove thatx ∈ E(MP). It needs to be proved that there is no Suppose to the contrary that there exists x ∈ A such that f (x) ≤ D * f (x) . Then (4.4) with at least one strict inequality.
On the other hand, This is equivalent to Adding side by side (4.4), (4.5), and (4.6), we can see that there exists x ∈ A such that (4.3) holds, a contradiction.
To complete the proof, we prove that (Λ, k) is an efficient solution of (MD). It needs to be proved that there exists no (Λ * , k) such that Suppose to the contrary that there exists (Λ * , k) such that (4.7) holds. There exists i ∈ M such that or in other words, Thus, there exists x ∈ S such that Thus, or in other words, a contradiction to (4.2). Hence, for every i ∈ M , we get Thus, (4.8) By Lemma 2.7, from the above inequality, we get We claim that (4.1b) holds. Suppose to the contrary that there exists x ′ ∈ S such that Combining this with (4.9) and (4.8), we deduce that a contradiction. To complete the proof, we need to show that (4.1a) holds. Suppose there exists By (4.9), we get This implies that Since inequality (4.11) holds for all (λ i , k i ) ∈ F, i ∈ M , we get a contradiction to (4.10). 2 i) There exist Λ ∈ L , L ∈ R m + , U Λ (0), and x * ∈ S such that ii) There exists vectors p ∈ R m + and q ∈ R m such that Then the function L has a saddle point.
, then the function L has a saddle point.
Proof Suppose that there exists Λ ∈ L such that function G is Λ -lower Lipschitz at u = 0 . By Corollary 2.5,

Example
It is easy to check that the feasible set of (MP 1 ) is A = [0, 1] andx = 0 is an efficient solution of (MP 1 ).
Suppose to the contrary that there exists λ i ∈ R (T ) − such that i) is satisfied with a strict inequality. Then with some i ∈ {1, 2}. Note that λ i ∈ R (T ) − . It is easy to check that This and (4.14) allow us to deduce that −⟨λ i , (−1) t ⟩ > 0, a contradiction Next, assume to the contrary that there exists x ∈ S such that ii) is satisfied with a strict inequality. Then we get f i (0) + ω((−1) t , 0, k i ) > f i (x) + ω(tx − 1) t , 0, k i ), with at least an index i ∈ {1, 2}. Note thatx = 0 is an efficient solution of (MP 1 ). Sinceλ i = 0 , by (2.1), we get T (λ i ) = ∅. Hence, from the inequality above, it is deduced that If x ∈ S \ A then f i (x) > 0 = f i (0) , a contradiction. If x ∈ A, it contradicts the fact thatx = 0 is an efficient solution of (MP 1 ). The proof is completed.