On directionally differentiable multiobjective programming problems with vanishing constraints

In this paper, a class of directionally differentiable multiobjective programming problems with inequality, equality and vanishing constraints is considered. Under both the Abadie constraint qualification and the modified Abadie constraint qualification, the Karush–Kuhn–Tucker type necessary optimality conditions are established for such nondifferentiable vector optimization problems by using the nonlinear version Gordan theorem of the alternative for convex functions. Further, the sufficient optimality conditions for such directionally differentiable multiobjective programming problems with vanishing constraints are proved under convexity hypotheses. Furthermore, vector Wolfe dual problem is defined for the considered directionally differentiable multiobjective programming problem vanishing constraints and several duality theorems are established also under appropriate convexity hypotheses.


Introduction
Multiobjective optimization problems, also known vector optimization problems or multicriteria optimization problems, are extremum problems involving more than one objective function to be optimized. Many real-life problems can be formulated as multiobjective programming problems which include human decision making, economics, financial investment, portfolio, resource allocation, information transfer, engineering design, mechanics, control theory, etc. During the past five decades, the field of multiobjective programming, has grown remarkably in different directional in the setting of optimality conditions and duality theory. One of the classes of nondifferentiable multicriteria optimization problems studied in the recent past is the class of directionally differentiable vector optimization problems for which many authors have established the aforesaid fundamental results in optimization theory (see, for example, (Ahmad, 2011;Antczak, 2002Antczak, , 2009Arana-Jiménez et al., 2013;Dinh et al., 2005;Ishizuka, 1992;Kharbanda et al., 2015;Mishra & Noor, 2006;Mishra et al., 2008Mishra et al., , 2015Slimani & Radjef, 2010;Ye, 1991) and others).
Recently, a special class of optimization problems, known as the mathematical programming problems with vanishing constraints, was introduced by Achtziger and Kanzow (2008), which serves as a unified frame work for several applications in structural and topology optimization. Since optimization problems with vanishing constraints, in their general form, are quite a new class of mathematical programming problems, only very few works have been published on this subject so far (see, for example, (Achtziger et al. 2013;Antczak 2022;Dorsch et al. 2012;Dussault et al. 2019;Guu et al. 2017;Kanzow 2008, 2009;Hoheisel et al. 2012;Hu et al. 2014Hu et al. , 2020Izmailov and Solodov 2009;Khare and Nath 2019;Mishra et al. 2015Mishra et al. , 2016Thung 2022). However, to the best our knowledge there are no works on optimality conditions for (convex) directionally differentiable multiobjective programming problems with vanishing constraints in the literature.
The main purpose of this paper is, therefore, to develop optimality conditions for a new class of nondifferentiable multiobjective programming problems with vanishing constraints. Namely, this paper represents the study concerning both necessary and sufficient optimality conditions for convex directionally differentiable vector optimization problems with inequality, equality and vanishing constraints. Considering the concept of a (weak) Pareto solution, we establish Karush-Kuhn-Tucker type necessary optimality conditions which are formulated in terms of directional derivatives. In proving the aforesaid necessary optimality conditions, we use a nonlinear version of the Gordan alternative theorem for convex functions and also the Abadie constraint qualification. Further, we illustrate the case that the necessary optimality conditions may not hold under the aforesaid constraint qualification. Therefore, we introduce the V C-Abadie constraint qualification and, under this weaker constraint qualification in comparison to that classical one, we present the Karush-Kuhn-Tucker type necessary optimality conditions for the considered directionally differentiable multiobjective programming problem. Further, we prove the sufficiency of the aforesaid necessary optimality conditions for such nondifferentiable vector optimization problems under appropriate convexity hypotheses. The optimality results established in the paper are illustrated by the example of a convex directionally differentiable multiobjective programming problem with vanishing constraints. Furthermore, for the considered directionally differentiable vector optimization problem with vanishing constraints, we define its vector Wolfe dual problem and we prove several duality theorems also under convexity hypotheses.

Preliminaries
In this section, we provide some definitions and results that we shall use in the sequel. The following convention for equalities and inequalities will be used throughout the paper.
For any x = (x 1 , x 2 , ..., x n ) T , y = (y 1 , y 2 , ..., y n ) T in R n , we define: (i) x = y if and only if x i = y i for all i = 1, 2, ..., n; (ii) x < y if and only if x i < y i for all i = 1, 2, ..., n; (iii) x y if and only if x i y i for all i = 1, 2, ..., n; (iv) x ≤ y if and only if x y and x = y.
Throughout the paper, we will use the same notation for row and column vectors when the interpretation is obvious.

Definition 2.1
The affine hull of the set C of points x 1 , ..., x k ∈ C is defined by Definition 2.2 (Hiriart-Urruty & Lemaréchal, 1993) The relative interior of the set C (denoted by relint C) is defined as where B (x, r ) := y ∈ R n : x − y r is the ball of radius r around x with respect to some norm on R n . (Rockafellar, 1970) The definition of the relative interior of a nonempty convex set C can be reduced to the following:

Remark 2.3
holds for all x, u ∈ C and any λ ∈ [0, 1]. It is said that ϕ is said to be strictly convex on C if the inequality holds for all x, u ∈ C, x = u, and any λ ∈ (0, 1).

Definition 2.5
We say that a mapping ϕ : X → R defined on a nonempty set X ⊆ R n is directionally differentiable at u ∈ X into a direction v ∈ R n if the limit Theorem 2.8 (Giorgi, 2002) Let C ⊂ R n be a a nonempty convex set, F : C → R k , : C → R m be convex functions and : R n → R q be a linear function. Let us assume that there exists x 0 ∈ r elint C such that j (x 0 ) < 0, j = 1, ..., m, and s (x 0 ) 0, s = 1, ..., q. Then, the system ⎧ ⎨ admits no solutions if and only if there exists a vector (λ, θ, β) Definition 2.9 The cone of sequential linear directions (also known as the sequential radial cone) to a set Q ⊂ R n at x ∈ Q is the set denoted by Z (Q; x) and defined by Definition 2.10 The tangent cone to a set Q ⊂ R n at x ∈ cl Q is the set denoted by T (Q; x) and defined by where cl Q denotes the closure of Q.
Note that the aforesaid cones are nonempty, T (Q; x) is closed, it may not be convex and Z (Q; x) ⊂ T (Q; x).

Multiobjective programming with vanishing constraints
In the paper, we consider the following constrained multiobjective programming problem (MPVC) with vanishing constraints defined by .., r }, are real-valued functions and C ⊆ R n is a nonempty open convex set. For the purpose of simplifying our presentation, we will next introduce some notations which will be used frequently throughout this paper. Let = {x ∈ C : 0, t ∈ T } be the set of all feasible solutions for (MPVC). Further, we denote by J (x) := j ∈ J : g j (x) = 0 the set of inequality constraint indices that are active at x ∈ and by J < (x) = { j ∈ {1, ..., m} : g j (x) < 0} the set of inequality constraint indices that are inactive at x ∈ . Then, J (x) ∪ J < (x) = J .
Before studying optimality in multiobjective programming, one has to define clearly the well-known concepts of optimality and solutions in multiobjective programming problem. The (weak) Pareto optimality in multiobjective programming associates the concept of a solution with some property that seems intuitively natural.
As it follows from the definition of (weak) Pareto optimality, x is nonimprovable with respect to the vector cost function f . The quality of nonimprovability provides a complete solution if x is unique. However, usually this is not the case, and then one has to find the entire exact set of all Pareto optimality solutions in a multiobjective programming problem. Now, for any feasible solution x, let us denote the following index sets Further, let us divide the index set T + (x) into the following index subsets: Similarly, the index set T 0 (x) can be partitioned into the following three index subsets: Before proving the necessary optimality conditions for the considered directionally differentiable multiobjective programming problem with vanishing constraints, we introduce the Abadie constraint qualification for this multicriteria optimization problem.
In order to introduce the aforesaid constraint qualification, for x ∈ , we define the sets Q l (x), l = 1, ..., p, and Q (x) as follows Now, we give the definition of the almost linearizing cone for the considered multiobjective programming problem (MPVC) with vanishing constraints. It is a generalization of the almost linearizing cone introduced by Preda and Chitescu (1999) for a directionally differentiable multiobjective optimization problem with inequality constraints only.

Definition 3.3
The almost linearizing cone L ( , x) to the set at x ∈ is defined by Now, we prove the result which gives the formulation of the almost linearizing cone to the sets Q l (x), l = 1, ..., p.
Proposition 3.4 Let x ∈ be a Pareto solution in the considered multiobjective programming problem (MPVC) with vanishing constraints. Then, the linearizing cone to the set to each set Q l (x), l = 1, ..., p, at x, denoted by L Q l (x) ; x , is given by Proof Let us assume that x ∈ is a Pareto solution in the considered multiobjective programming problem (MPVC) with vanishing constraints. Then, by the definitions of the almost linearizing cone and index sets, we get Note that, by Lemma 2.7, one has Then, by the definition of index sets, ( 7 ) gives Combining ( 6 )- ( 8 ), we get ( 5 ). This completes the proof of this proposition.
Remark 3.5 Note that the almost linearizing cone to Q (x) at x ∈ Q (x) is given by Indeed, by ( 5 ), we get ( 9 ). In other words, the formulation of L (Q (x) ; x) is given by Now, we prove that it is a convex cone. Let v 1 , v 2 ∈ L (Q (x) ; x) and α ∈ [0, 1]. By convexity assumption, it follows that Now, we prove the closedness of L (Q (x) ; x). In order to prove this property, we take a . This means that the set L (Q (x) ; x) is closed.

Remark 3.7
Based on the result established in the above proposition, we conclude that also L Q l (x) ; x , l = 1, ..., p, are also closed convex cones.
Note that, in general, the converse inclusion of ( 11 ) does not hold. Therefore, in order to prove the necessary optimality condition for efficiency in (MPVC), we give the definition of the Abadie constraint qualification. Definition 3.9 It is said that the Abadie constraint qualification holds at x ∈ for (MPVC) iff ( 2 2 ) Remark 3.10 By ( 11 ), ( 22 ) means that the Abadie constraint qualification (ACQ) holds at Now, we state a necessary condition for efficiency in (MPVC).

Theorem 3.11 Let x ∈ be an efficient solution in (MPVC) and, for each
has no solution v ∈ R n .
Proof We proceed by contradiction. Suppose, contrary to the result, that there exists l 0 ∈ {1, ..., p} such that the system has a solution v ∈ R n . Then, by ( 8 ), the system Hence, x + α k v ∈ C and, moreover, By the definition of indexes sets, one has g j ( . Therefore, by the continuity of g j , j ∈ J < (x), H t , t ∈ T + (x), G t , t ∈ T +− (x), at x, there exists k 0 ∈ N such that, for all k > k 0 , Thus, we conclude by ( 40 ) On the other hand, it follows from the assumption that x ∈ is an efficient solution in (MPVC). Hence, by Definition 3.1, there exists a number δ > 0 such that there is no Hence, since x + α k v ∈ ∩ B (x; δ) and (39) holds, by (48) and (49), we conclude that, for all k ∈ N , the inequality holds. Then, by Definition 2.5, the inequality above implies that the inequality holds, which is a contradiction to (28). Hence, the proof of this theorem is completed.

Remark 3.12
As follows from the proof of Theorem 3.11, if the system (23)- (27) has no solution v ∈ R n , then, for each l = 1, ..., p, the system has no solution v ∈ R n .
Let us define the functions F = F 1 , ..., F p : R n → R p , = 1 , ..., We are now in a position to formulate the Karush-Kuhn-Tucker necessary optimality conditions for a feasible solution x to be an efficient solution in (MPVC) under the Abadie constraint qualification (ACQ).

Theorem 3.13 (Karush-Kuhn-Tucker Type Necessary Optimality Conditions)
. Let x ∈ be an efficient solution in the considered multiobjective programming problem (MPVC) with vanishing constraints. We also assume that f i , i ∈ I , g j , j ∈ J , h s , s ∈ S, H t , t ∈ T , G t , t ∈ T , are directionally differentiable functions at

continuous functions at x and, moreover, the Abadie constraint qualification (ACQ) is satisfied at x for (MPVC). If there exists
0, then there exist Lagrange multipliers λ ∈ R p , μ ∈ R m , ξ ∈ R q , ϑ H ∈ R r and ϑ G ∈ R r such that the following conditions hold.
Note that, in general, the Abadie constraint qualification may not be fulfilled at an efficient solution in (MPVC) if T 00 (x) = ∅.
Based on the definition of the index sets, we substitute the constraint H t G t (x) 0, t ∈ T by the constraints in which the index sets depend on x.
Then, we define the following vector optimization problem derived from (MPVC), some of the constraints of which depends on the optimal point x: In order to introduce the modified Abadie constraint qualification, for x ∈ , we define the sets Q l (x), l = 1, ..., p, and Q (x) as follows Then, the almost linearizing cone for the sets Q l (x) is defined by Hence, the almost linearizing cone for the set Q (x) is given as follows Remark 3.14 Note that the only difference between L (Q (x) ; x) and L Q (x) ; x is that we add the inequality G + t (x; v) 0, ∀t ∈ T 00 (x) in L Q (x) ; x in comparison to L (Q (x) ; x). In particular, we always have the relation (73)

Proposition 3.15 Let x be a feasible solution in (MPVC). Then
Proof By Proposition 3.8, it follows that Moreover, as it follows from the proof of Proposition 3.8, one has Thus, (76) and (72) Combining (75)-(79), we get (74). Now, we are ready to introduce the modified Abadie constraint qualification which we name the VC-Abadie constraint qualification.
Definition 3.16 Let x ∈ be an efficient solution in (MPVC). Then, the VC-Abadie constraint qualification (VC-ACQ) holds at x for (MPVC) iff ( 8 0 ) Now, we define the Abadie constraint qualification for (MP(x)) and we show that then the VC-Abadie constraint qualification (VC-ACQ) holds at x for (MPVC), even in a case in which the Abadie constraint qualification (ACQ) is not satisfied.
Definition 3.17 Let x ∈ be a (weakly) efficient solution in (MPVC). Then, the modified Abadie constraint qualification (MACQ) holds at x for (MP(x)) iff We now give the sufficient condition for the VC-Abadie constraint qualification to be satisfied at an efficient solution in (MPVC).

Lemma 3.18 Let x ∈ be an efficient solution in (MPVC). If the modified Abadie constraint qualification (MACQ) holds at x for (MP(x)), then the VC-Abadie constraint qualification (VC-ACQ) holds at x for (MPVC).
Proof Assume that x ∈ is an efficient solution in (MPVC) and, moreover, the modified Abadie constraint qualification (MACQ) holds at x for (MP(x)). Then, by Definition 3.17, it follows that Hence, (84) implies Then, (83) gives Thus, by (82), (85) and (86), we get as was to be shown.
Since the VC-Abadie constraint qualification (VC-ACQ) is weaker than the Abadie constraint qualification (ACQ), the necessary optimality conditions (59)-(65) may not hold. Therefore, in the next theorem, we formulate the Karush-Kuhn-Tucker necessary optimality conditions for a feasible solution x to be an efficient solution in (MPVC) under the VC-Abadie constraint qualification (VC-ACQ).

Theorem 3.19 (Karush-Kuhn-Tucker Type Necessary Optimality Conditions). Let x ∈
be an efficient solution in the considered multiobjective programming problem (MPVC) with vanishing constraints. We also assume that f i , i ∈ I , g j , j ∈ J , h s , s ∈ S, H t , t ∈ T , G t , t ∈ T , are directionally differentiable functions at

continuous functions at x and, moreover, the VC-Abadie constraint qualification (VC-ACQ) is satisfied at x for (MPVC). If there exists
0, then there exist Lagrange multipliers λ ∈ R p , μ ∈ R m , ξ ∈ R q , ϑ H ∈ R r and ϑ G ∈ R r such that the following hold.
Now, we prove the sufficiency of the Karush-Kuhn-Tucker optimality conditions for the considered multiobjective programming problem (MPVC) with vanishing constraints under appropriate convexity hypotheses.

Theorem 3.20 Let x be a feasible solution in (MPVC) and the Karush-Kuhn-Tucker type necessary optimality conditions (59)-(65) be satisfied at x for (MPVC) with
Lagrange multipliers λ ∈ R k + , μ ∈ R m + , ξ ∈ R q , ϑ H ∈ R r and ϑ G ∈ R r . Further, we assume that f i , i ∈ I ,

convex on . Then x is a weak Pareto solution in (MPVC).
Proof We proceed by contradiction. Suppose, contrary to the result, that x is not a weak Pareto solution in (MPVC). Thus, by Definition 3.1, there exists x ∈ such that By assumption, f is convex at x on . Hence, by Proposition 2.6, (94) yields Since λ ≥ 0, the inequalities (95) give From x, x ∈ and the definition of J (x), it follows that By assumption, , are convex on . Then, by Proposition 2.6, (97)-(100) imply, respectively, Taking into account that Combining (96) and (106)- (109), we get that the inequality holds, contradicting the Karush-Kuhn-Tucker type necessary optimality condition (59). This means that x is a weak Pareto solution in (MPVC).
In order to prove the sufficient optimality conditions for a feasible solution x to be a Pareto solution in (MPVC), stronger convexity assumptions are needed imposed on the objective functions.

Theorem 3.21 Let x be a feasible solution in (MPVC) and the Karush-Kuhn-Tucker type necessary optimality conditions (59)-(65) be satisfied at x for (MPVC) with Lagrange multipliers
, are convex on . Then x is a Pareto solution in (MPVC).

Remark 3.22
In Theorem 3.21, all objective functions f i , i ∈ I , are assumed to be strictly convex on in order to prove that x ∈ is a Pareto solution in (MPVC). However, as it follows from the proof of the aforesaid theorem, it is sufficient if we assume in Theorem 3.21 that at least one the objective function f i , i ∈ I , is strictly convex on , but Lagrange multiplier λ i associated to such an objective function f i should be greater than 0.

Remark 3.23
If x is such a feasible solution at which the Karush-Kuhn-Tucker type necessary optimality conditions (87)-(93) in place of (59)-(65), then also the functions G t , t ∈ T 00 (x), should be assumed to be convex on in the sufficient optimality conditions. Now, we illustrate the results established in the paper by an example of a convex directionally differentiable multiobjective programming problem with vanishing constraints.

Example 3.24 Consider a directionally differentiable multiobjective programming problem with vanishing constraints defined by
Then, by definition, we have Further, by Definition 2.9 and the definition of the almost linearizing cone (see (5), (10)), we have, respectively, Note that the Abadie constraint qualification (ACQ) is not satisfied at x = (0, 0) for (MPVC1) since the relation L (Q (x) ; x) ⊂ 2 l=1 Z Q l (x) ; x is not satisfied. But the VC-Abadie constraint qualification (VC-ACQ) holds at x = (0, 0) for (MPVC1) since the relation L Q (x) ; x ⊂ 2 l=1 Z Q l (x) ; x is satisfied. As it follows even from this example, the VC-Abadie constraint qualification (VC-ACQ) is weaker than the Abadie constraint qualification (ACQ). Moreover, the Karush-Kuhn-Tucker type necessary optimality conditions (87)-(93) are fulfilled at x with Lagrange multipliers λ 1 = 1 2 , λ 2 = 1 4 , ϑ H 1 = 1 4 , ϑ G 1 = 1 4 . Further, note that the functions constituting (MPVC1) are convex on and the objective function f 1 is strictly convex on . Hence, by Theorem 3.21, x = (0, 0) is a Pareto solution in (MPVC1). Note that the optimality conditions established in the literature (see, for example, (Achtziger et al., 2013;Dorsch et al., 2012;Dussault et al., 2019;Hoheisel & Kanzow, 2008, 2007, 2009Hoheisel et al., 2012;Izmailov & Solodov, 2009)) are not applicable for the considered multiobjective programming problem (MPVC1) with vanishing constraints since the results established in the above mentioned works have been proved for scalar optimization problems with vanishing constraints. Moreover, the results presented in Guu et al. (2017) and Mishra et al. (2015) have been established for differentiable multiobjective programming problems with vanishing constraints only and, therefore, they are not useful for finding (weak) Pareto solutions in such nondifferentiable vector optimization problems as the directionally differentiable multiobjective programming problem (MPVC1) with vanishing constraints.

Wolfe duality
In this section, for the considered vector optimization problem (MPVC) with vanishing constraints, we define its vector Wolfe dual problem. Then we prove several duality results between problems (MPVC) and (WDVC) under convexity assumption imposed on the functions constituting them.

Remark 4.1
In the Wolfe dual problem (WDVC(x)) given above, the significance of w t and θ t is the same as v t and β t in Theorem 1 (Achtziger and Kazanov (2008)). Now, on the line Hu et al. (2020), we define the following vector dual problem in the sense of Wolfe related to the considered multicriteria optimization problem (MPVC) with vanishing constraints by where the set of all feasible solutions in (WDVC) is defined by = x∈ (x). Further, let us define the set Y by Y = y ∈ X : y, λ, μ, ξ, ϑ H , ϑ G , w, θ ∈ . Theorem 4.2 (Weak duality): Let x and y, λ, μ, ξ, ϑ H , ϑ G , w, θ be any feasible solutions for (MPVC) and (WDVC), respectively. Further, we assume that one of the following hypotheses is fulfilled: Proof We proceed by contradiction. Suppose, contrary to the result, that Hence, by definition of the Lagrange function L, the aforesaid inequality gives Using the first constraint of (WDVC), we get that the following inequality holds, contradicting (169). This completes the proof of this theorem under hypothesis B).

Conclusions
This paper represents the study concerning the new class of nonsmooth vector optimization problems, that is, directionally differentiable multiobjective programming problems with vanishing constraints. Under the Abadie constraint qualification, the Karush-Kuhn-Tucker type necessary optimality conditions have been established for such nondifferentiable vector optimization problems in the terms of the right directional derivatives of the involved functions. The nonlinear Gordan alternative theorem has been used in proving these aforesaid necessary optimality conditions. However, the Abadie constraint qualification may not hold for such multicriteria optimization problems and therefore, the aforesaid necessary optimality conditions may not hold. Therefore, we have introduced the modified Abadie constraint qualification for the considered multiobjective programming problem with vanishing constraints. Then, under the modified Abadie constraint qualification, which is weaker than the standard Abadie constraint qualification, we prove weaker necessary optimality conditions of the Karush-Kuhn-Tucker type for such nondifferentiable vector optimization problems with vanishing constraints. The sufficiency of the Karush-Kuhn-Tucker necessary optimality conditions have also been proved for the considered directionally differentiable multiobjective programming problem with vanishing constraints under appropriate convexity hypotheses. Furthermore, for the considered directionally differentiable multiobjective programming problems with vanishing constraints, its vector Wolfe dual problem has been defined on the line Hu et al. (2020). Then several duality theorems have been established between the primal directionally differentiable multiobjective programming problems with vanishing constraints and its vector Wolfe dual problem under convexity hypotheses. Thus, the above mentioned optimality conditions and duality results have been derived for a completely a new class of directionally differentiable vector optimization problems in comparison to the results existing in the literature, that is, for directionally differentiable multiobjective programming problems with vanishing constraints. Hence, the results established in the literature generally for scalar differentiable extremum problems with vanishing constraints have been generalized and extended to directionally differentiable multiobjective programming problems with vanishing constraints.
It seems that the techniques employed in this paper can be used in proving similarly results for other classes of nonsmooth mathematical programming problems with vanishing constraints. We shall investigate these problems in the subsequent papers.

Conflict of interest No potential conflict of interest was reported by the author.
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