E-Saddle Point Criteria for E-differentiable Vector Optimization Problems with Inequality and Equality Constraints

Corresponding Author: Najeeb Abdulaleem Department of Mathematics, Hadhramout University, P.O. BOX: (50511-50512), Al-Mahrah, Yemen Email: nabbas985@gmail.com Abstract: In this study, the class of E-differentiable vector optimization problems with both inequality and equality constraints is considered. For such (not necessarily) differentiable multicriteria optimization problems, two E-Lagrange functions and their -saddle points are defined. Then the so-called -saddle point criteria are established for such (not necessarily) differentiable multiobjective programming problems with Edifferentiable -convex functions.


Introduction
In recent years, saddle point criteria for vector optimization problems, also called multiobjective programming problems or multicriteria optimization problems, have attracted much interest. There are several reasons for studying the characterization of a (weak) Pareto solution for multicriteria optimization problems via the saddle point criteria. But probably the most important one is related to the result that a saddle point of the Lagrange function is always a global optimal solution of a constrained optimization problem. Because of the importance of this result, for example, in optimization theory and economics, therefore, many authors have analyzed and studied theory of saddle point criteria for nonconvex multiobjective programming problems (see, for instance, (Adán and Novo, 2005;Antczak, 2003;2005;2015;Bhatia, 2008;Bigi, 2001;Craven, 1990;Ehrgott and Wiecek, 2005;Jiang and Xu, 2010;Kuk et al., 1998;Kumar and Garg, 2015;Maciel et al., 2016;Mishra and Giorgi, 2008;Li and Wang, 1994;Van Rooyen et al., 1994;Tanaka, 1990;1994;Tanino, 1982;Vályi, 1987;Varalakshmi et al., 2010;Yan and Li, 2004;Zeng, 2017). Taninio (1982) proved that solutions of multicriteria optimization problems and corresponding multiplier vectors are saddle points of vector-valued Lagrange functions. Used a scalarization method, Tanaka (1990) characterized generalized saddle points of vector-valued Lagrangians. Li and Wang (1994) derived several conditions for the existence of a Lagrange multiplier or a weak saddle point of vector optimization problems and proved relationships between them. Bigi (2001) proved criteria for saddle points of nondifferentiable multiobjective programming problems under convexity hypotheses. Ehrgott and Wiecek (2005) studied relations between saddle points and Pareto points for both convex and nonconvex vector optimization problems used scalarizations. Antczak (2003;2005) proved saddle point criteria for the considered multiobjective programming problem by using the introduced nonlinear approximation methods. Bhatia (2008) established the equivalence between a mixed saddle point of order m defined for the vector-valued partial Lagrange function and higher order minima of the considered nondifferentiable multiobjective programming problem. Kumar and Garg (2015) introduced the definition of a socalled mixed saddle point of a vector-valued Lagrange function defined for the considered nonsmooth multicriteria optimization problem and proved the equivalence between a Pareto solution and a mixed saddle point under the concept of generalized (V, ρ)-invexity.
Recently, under generalized notions of convexity, Maciel et al. (2016) investigated the relations between a Fritz John critical point of the considered differentiable multiobjective programming problem and a saddle point of the vector-valued Lagrange function.
One of generalized convexity notions introduced into optimization theory is the concept of E-convexity which was defined by Youness (1999). The concept of Econvexity is based on the effecting of an operator E : R n → R n on the sets and the domain on which functions are defined. Recently, the definition of an Edifferentiable function was introduced by Megahed et al. (2013) in which, based on the effect of an operator E : R n → R n , a (not necessarily) differentiable function is transformed to a differentiable function (in the usual sense).
Up to now, the most of papers dealing with the characterization of Pareto optimality by saddle point criteria is restricted to differentiable multiobjective programming problems. Therefore, in the present paper, we consider the class of E-differentiable multiobjective programming problems with both inequality and equality constraints in which the functions involved are (not necessarily) differentiable. For such (not necessarily) differentiable vector optimization problems involving Edifferentiable functions, characterizations of their saddle points are presented. In the used approach, for the original E-differentiable multicriteria optimization problem, its related E-vector optimization problem is constructed in which the involved functions are differentiable (in the usual sense). Then, so-called scalar and vector-valued E-Lagrange functions and their Esaddle points are defined for the considered multiobjective programming problem. Further, the socalled E-saddle point criteria are established for such (not necessarily) differentiable multiobjective programming problems under E-convexity hypotheses. Then, the equivalence is proved between (weak) E-Pareto solutions and E-saddle points of scalar and vectorial E-Lagrange functions defined in the considered E-differentiable multiobjective programming problem with E-convex functions via the similar result established for its associated E-vector optimization problem. Hence, tools of differentiable analysis are used in proving saddle point criteria for (not necessarily) differentiable multiobjective programming problems.

Preliminaries
In this section, we provide some notations and results that we shall use in the sequel.
Let R n be the n-dimensional Euclidean space and n R + be its nonnegative orthant. The following convention for equalities and inequalities will be used in the paper. Namely, for any vectors x = (x 1 , x 2 ,...,x n ) T and y = (y 1 , y 2 ,...,y n ) T in R n , we define x = y if and only if x i = y i for all i = 1,2,..., n; x ≧ y if and only if x i ≧ y i for all i = 1,2,...,n; x ≧ y if and only if x i > y i for all i = 1,2,...,n; x ≥ y if and only if x ≧ y and x ≠ y.
Definition 1 (Youness, 1999) A set M ⊆ R n is said to be E-convex (with respect to an operator E: R n → R n ) if and only if the following relation: holds for all x, u ∈ M and any λ ∈ [0,1].
Definition 2 (Youness, 1999) Let E: R n → R n and M be a nonempty E-convex subset of R n . f: M → R is said an E-convex function on M if and only if the inequality: is satisfied for all x, u ∈ M and any λ ∈ [0,1].
Definition 3 (Megahed et al., 2013) Let E : R n → R n , M be an open E-convex set, f : M → R be a (not necessarily) differentiable function at a given point u. It is said that f is an E-differentiable function at u if and only if f ∘ E is a differentiable function at u (in the usual sense) and, moreover:

Proposition 4
Let E : R n → R n , M be an E-convex subset of R n and f : R n → R be an E-convex (a strictly E-convex) function on M and u ∈ M. Moreover, we assume that the function f is E-differentiable at u. Then, the inequality: holds for all x ∈ M, (E(x)) ≠ (E(u)).
In the paper, we consider the following (not necessarily differentiable) vector optimization problem (VP) with both inequality and equality constraints defined as follows: Let Ω denote the set of all feasible solutions of the problem (VP), that is, the set: Further, the set of inequality constraint indices that are active at a feasible solution x is denoted by ( ) J x and it is defined by ( ) ( ) For such multicriteria optimization problems, the key optimality concept is that of a Pareto (a weak Pareto) solution which is defined as follows.
Definition 5 x ∈ Ω is said to be a weak Pareto solution (a weakly efficient solution) of the problem (VP) if and only if there is no another feasible point x such that: Definition 6 x ∈ Ω is said to be a Pareto solution (an efficient solution) of the problem (VP) if and only if there is no another feasible point x such that: Let E: R n → R n be a given one-to-one and onto operator. Throughout the paper, we shall assume that the functions involved in the vector optimization problem (VP) are E-differentiable at any its feasible solution. Now, for the original vector optimization problem (VP), we define its associated differentiable multicriteria optimization problem as follows: minimize subject to ( ( )) ( ( ( )) ,..., ( ( ))) ( ) 0, 1, , , (VP ) .
Lemma 8 (Antczak and Abdulaleem, 2018) Let z ∈ Ω be a weak Pareto solution (a Pareto solution) of the considered multicriteria optimization problem (VP). Then, there exists

Remark 10
As it follows from Lemma 9, if Let E x ∈ Ω be a weak Pareto solution of the E-vector optimization problem (VP E ) (and, thus, ( ) E x be a weak E-Pareto solution of the considered multiobjective programming problem (VP)). Further, assume that the objective functions f i , i ∈ I, the constraint functions g j , j ∈ J and h t , t ∈ T, are E-differentiable at x and the E-Abadie Constraint Qualification (ACQ E ) is satisfied at x . Then there exist Lagrange multipliers We now give the definitions of a Karush-Kuhn-Tucker point for the E-vector optimization problem (VP E ) and an E-Karush-Kuhn-Tucker point for the considered multiobjective programming problem (VP).

Definition 12
We say that ( )

Scalar E-saddle Point Criteria
Now, we introduce the definition of the scalar Lagrange function : for the E-vector optimization problem (VP E ) as follows: Then, we give the definition of a saddle point of the Lagrangian L E defined for (VP E ).

Definition 14
We say that ( ) We now define a scalar Lagrange function Ω× × × → for the considered multiobjective programming problem (VP) as follows: For the scalar Lagrange function L defined above, we now give the definition of its E-saddle point.

Definition 15
Let x be a feasible solution of the E-vector optimization problem (VP E ). We say that First, we prove the necessary condition for a saddle point of the Lagrangian L E which is defined for the Evector optimization problem (VP E ).

Theorem 16
is a saddle point of the scalar Lagrange function L E . Thus, by the condition a) in Definition 14 and (7), the inequality: holds for all m R µ + ∈ and ξ ∈ R s . From the feasibility of Then, for µ = 0, (10) gives: By two inequalities above, it follows that: Now, suppose, contrary to the result, that E x ∈ Ω is not a weak Pareto solution of (VP E ). Hence, by Definition 5, there exists (11) and (12) yield: Hence, by definition of the Lagrange function L E (see (8)), the inequality Now, by using the result established in Theorem 16, we prove the similar result for the original multicriteria optimization problem (VP). Namely, we derive the necessary condition for a saddle point criteria of the Lagrangian L defined for (VP). In other words, we prove the necessary condition for an E-saddle point of the scalar Lagrange function L defined in (VP).

Theorem 17
be an E-saddle point of the scalar Lagrangian L defined in the considered vector optimization problem (VP). Then

Proof
We assume that is an E-saddle point of L. Then, the conditions (i) and (ii) in Definition 15 are fulfilled. For any z ∈ Ω, by Lemma 7, there exists x ∈ Ω E such that z = E(x). Then, by (8), the conditions (i) and (ii) can be rewritten, respectively: and: By (7), inequalities (13) and (14) imply, respectively: Note that (15) and (16)  x ∈ Ω is a weak Pareto solution of (VP E ). Hence, by Lemma 9 (see also Remark 10), ( ) E x is an E-Pareto solution of (VP). Thus, the proof of this theorem is completed.
Note that the result proved in Theorem 17 can also be formulated in the following way.

Theorem 18
Let x be a feasible solution of the the Emultiobjective programming problem (VP E ) such that is a saddle point of the scalar Lagrange function L defined for considered multicriteria optimization problem (VP). Then ( ) E x is a weak E-Pareto solution of the problem (VP). Further, if 0 λ > , then ( ) E x is an E-Pareto solution of (VP). Now, under appropriate E-differentiable E-convexity hypotheses, we establish the sufficient condition for a saddle point of the Lagrange function L E defined in the E-vector optimization problem (VP E ) which we use in proving the sufficient condition for an E-saddle point of the Lagrange function L defined for the considered multiobjective programming problem (VP).
We now prove the condition (ii) in Definition 14. From the assumption, f i , i ∈ I, g j , j ∈ J, h t , , are E-convex at x on Ω E . Then, by Proposition 4 and the KKT optimality condition (6), the inequalities: hold for all x ∈ Ω E . Hence, we add both sides of each inequality (20)-(23). Thus, by the KKT necessary optimality condition (4), we get that the inequality: is satisfied for all x ∈ Ω E . Then, by (7), the inequality: holds for all x ∈ Ω E . Hence, by (19) and (24)

Corollary 20
Let E x ∈ Ω be a (weak) Pareto solution of the Evector optimization problem (VP E ) and all assumptions of Theorem 19 be satisfied. Then, there exist Lagrange multipliers By using Theorem 19 and Corollary 20, we now prove the sufficient condition for an E-saddle point of the Lagrange function L in the problem (VP).

Theorem 21
Let x be a feasible point of the E-vector optimization problem (VP E ) such that is an E-KKT point of the considered multiobjective programming problem (VP). Assume, moreover, that each objective function f i , i ∈ I, is E-convex at x on Ω E , each constraint function g j , j ∈ J, is E-convex at x on Ω E , each constraint function h t , ( ) is an E-saddle point of the Lagrangian L defined for (VP).

Proof
By assumption, x is a feasible solution of the Evector optimization problem (VP E ) such that hold. By (7), it follows that the inequalities: hold, respectively. Since E : R n → R n is an one-to-one and onto operator, this means that, for any x ∈ Ω E , there existas z ∈ Ω, such that z = E(x). Hence, (25) and (26) yield, respectively:

∈ Ω × × ×
is an E-saddle point of the Lagrange function L defined for (VP).
As it follows from the above proof, the definition of an E-saddle point of the Lagrange function L can be reformulated as follows.

Definition 22
Let x be a feasible solution of the E-vector optimization problem (VPE). We say that is an E-saddle point for the considered multiobjective programming problem (VP) if: .
Now, we illustrate the main result established in this section by an example of an E-differentiable multiobjective programming problem.

Example 23
Consider the following nonconvex nondifferentiable vector optimization problem: be an one-to-one and onto mapping defined by ( ) . Now, for the considered nonconvex nondifferentiable multicriteria optimization problem (VP1), we construct its associated E-vector optimization problem (VP1 E ) as follows:

Vector E-saddle Point Criteria
In this section, we introduce a definition of the vector-valued Lagrange function defined for the constrained E-vector optimization problem (VP E ). Namely, this Lagrange function is defined as follows: where, e = [1,...,1] ∈ R p and, moreover,

Definition 24
We say that ( ) Now, for the considered multiobjective programming problem (VP), we define its vector-valued Lagrange function and for all ξ ∈ R s . From the feasibility of x in the problem (VP E ), it follows that the inequality: holds for all m R µ + ∈ . In the similar way as in the proof of Theorem 16, we get: Now, suppose, contrary to the result, that E x ∈ Ω is not a Pareto solution of the problem (VP E ). Then, there exists E x ∈ Ω ɶ such that: By assumption, the Lagrange multiplier λ is assumed to satisfy 0 λ > . Thus: Using the feasibility of x and x of (VP E ) together with (31), we get: Hence, by (29), (34) implies that the inequality: holds, which is a contradiction to the condition ii) in Definition 24.

Remark 27
Note that it is sufficient to assume in Theorem 26 that 0 λ ≥ in order to prove that x is a weak Pareto optimal of (VP E ).
Using the result established in Theorem 28, we prove the necessary condition for to be a vector E-saddle point of the Lagrange function L p in (VP).

Theorem 28
Let x be a feasible solution of the E-vector optimization problem is an E-saddle point of the vector-valued Lagrange function L p defined for the considered multiobjective programming problem (VP).
Then ( ) E x is a weak E-Pareto solution of (VP). If we assume extra that Lagrange multiplier is an E-Pareto solution in (VP).

Proof
Assume that x is a feasible solution of the E-vector optimization problem is an E-saddle point of the vector-valued Lagrange function L p . Then, by Definition 25, the conditions (i) and (ii) are fulfilled. For any z ∈ Ω, by Lemma 7, there exists x ∈ Ω E such that z = E(x). Then, by (30), the following relations: and: hold for all m R µ + ∈ , ξ ∈ R s and any E(x) ∈ Ω. By (35), (36) and (29), we conclude that the conditions (a) and (b) in Definition 24 are satisfied. Since all hypotheses of Theorem 26 are fulfilled, x ∈ Ω is a (weak) Pareto solution of (VP E ). Hence, by Lemma 9 (see also Remark 10), ( ) E x is a (weak) E-Pareto solution of (VP). This completes the proof of this theorem. Now, we prove the sufficient condition for Assume, moreover, that each objective function

Proof
First, we prove the inequality (a) in Definition 24. By assumption, ( ) Hence, from the feasibility of x , we have that the inequalities: hold for all ξ = (ξ 1 ,...,ξ s ) ∈ R s . By E x ∈ Ω and the E-KKT necessary optimality condition (5), it follows that the inequalities: . Combining (37) and (38), we get, for all m R µ + ∈ and ξ ∈ R s : Then, by (29), the inequality: holds for all m R µ + ∈ and ξ ∈ R s . We now establish the condition (b) in Definition 24. We prove it by means of contradiction. Suppose that there exists x ∈ Ω E such that Then, by (29), we have: , are E-convex on Ω E , by Proposition 4, the inequalities: hold, respectively. Then, we multiply inequalities (41)-(44) by corresponding Lagrange multipliers, respectively. After that, we add the resulting inequalities and obtain: , we get that the inequality: contradicting (40). This completes the proof of this theorem.
From Theorem 29, it follows directly the following result.

Corollary 30
Let E x ∈ Ω be a (weak) Pareto solution of the Evector optimization problem (VP E ) and the KKT necessary optimality conditions (4)- (6)

Theorem 31
Let x be a feasible solution of the E-vector optimization problem (VP E ) such that is an E-KKT point of the considered multicriteria optimization problem (VP). Assume, moreover, that each objective function hold. By (29), (45) and (46) yield, respectively: and: , Let E : R 2 → R 2 be an operator defined by ( ) ( ) For the considered vector optimization problem (VP2), we define its associated Emulticriteria optimization problem (VP2 E ) as follows:

Conclusion
In this paper, to the best of our knowledge, for the first time, the saddle point criteria have been derived for the class of (not necessarily) differentiable multiobjective programming problems with both inequality and equality constraints in which the involved functions are E-differentiable. For the considered Edifferentiable multiobjective programming problem, its associated E-vector optimization problem has been constructed in which the involved functions are differentiable in the usual sense. Further, two types of the so-called E-saddle point criteria have been derived for the original (not necessarily) differentiable multiobjective programming problem using the similar results established under E-convexity hypotheses for its associated E-vector optimization problem in which the involved functions are differentiable in the usual sense. The E-saddle point criteria presented in the paper have been illustrated by examples of nondifferentiable multiobjective programming problems with E-differentiable functions. This kind of studies seems to be important because the methods used to characterize saddle point criteria for (not necessarily) differentiable multicriteria optimization problems are based on the similar results established for differentiable vector optimization problems.
The results established in this paper warrant further research in several directions. In particular, it would be of great interest to examine how the results of this paper can be generalized and extended for other classes E-differentiable multiobjective programming problems. This may be the topic of some of our forthcoming papers.