Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2014, Article ID 496149, 12 pages http://dx.doi.org/10.1155/2014/496149
Research Article Multiobjective Fractional Programming Involving Generalized Semilocally V-Type I-Preinvex and Related Functions Hachem Slimani1 and Shashi Kant Mishra2 1 2
LaMOS Research Unit, Computer Science Department, University of Bejaia, 06000 Bejaia, Algeria Department of Mathematics, Faculty of Science, Banaras Hindu University, Varanasi 221005, India
Correspondence should be addressed to Hachem Slimani;
[email protected] Received 12 January 2014; Revised 5 April 2014; Accepted 5 April 2014; Published 5 May 2014 Academic Editor: Ram U. Verma Copyright Β© 2014 H. Slimani and S. K. Mishra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study a nonlinear multiple objective fractional programming with inequality constraints where each component of functions occurring in the problem is considered semidifferentiable along its own direction instead of the same direction. New Fritz John type necessary and Karush-Kuhn-Tucker type necessary and sufficient efficiency conditions are obtained for a feasible point to be weakly efficient or efficient. Furthermore, a general Mond-Weir dual is formulated and weak and strong duality results are proved using concepts of generalized semilocally V-type I-preinvex functions. This contribution extends earlier results of Preda (2003), Mishra et al. (2005), Niculescu (2007), and Mishra and Rautela (2009), and generalizes results obtained in the literature on this topic.
1. Introduction Because of many practical optimization problems where the objective functions are quotients of two functions, multiobjective fractional programming has received much interest and has grown significantly in different directions in the setting of efficiency conditions and duality theory these later years. The field of multiobjective fractional optimization has been naturally enriched by the introductions and applications of various types of convexity theory, with and without differentiability assumptions, and in the framework of symmetric duality, variational problems, minimax programming, continuous time programming, and so forth. More specifically, works in the area of nonsmooth setting can be found in Chen [1], Kim et al. [2], Kuk et al. [3], Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], Preda [7], and Soleimani-damaneh [8]. Efficiency conditions and duality models for multiobjective fractional subset programming problems are studied by Preda et al. [9], Verma [10], and Zalmai [11β13]. Higher order duality in multiobjective fractional programming is discussed in Gulati and Geeta [14] and Suneja et al. [15]. Solving nonlinear multiobjective fractional programming problems by a modified objective function
method is the subject matter of Antczak [16]. Further works on multiobjective fractional programming are established by Chinchuluun et al. [17], J.-C. Liu and C.-Y. Liu [18], Mishra et al. [19], Verma [20], Zhang and Wu [21], and others. The common point in all of these developments is the convexity theory that does not stop extending itself in different directions with new variants of generalized convexity and various applications to nonlinear programming problems in different settings. The concept of invexity introduced by Hanson [22] is a generalization of convexity which has received much interest these later years, and many advances in the theory and practice have been established using this concept and its extensions. In practice, recently Dinuzzo et al. [23] have obtained some kernel function in machine learning which is not quasiconvex (and hence also neither convex nor pseudoconvex), but it is invex. Nickisch and Seeger [24] have studied a multiple kernel learning problem and have used the invexity to deal with the optimization which is nonconvex. The concept of semilocally convex functions was introduced by Ewing [25] and was further extended to semilocally quasiconvex, semilocally pseudoconvex functions by Kaul and Kaur [26, 27]. Other generalizations of semilocally convex functions and their properties were investigated in
2 Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], Preda [7, 28], Preda and Stancu-Minasian [29], Preda et al. [30], and Stancu-Minasian [31]. In Preda [7], necessary and sufficient efficiency conditions for a nonlinear fractional multiple objective programming problem are obtained involving π-semidifferentiable functions. Furthermore, a general dual was formulated and duality results were proved using concepts of generalized semilocally preinvex functions. Thus, results of Preda [28], Preda and Stancu-Minasian [29], and Preda et al. [30] were generalized. Mishra et al. [5] extended the issues of Preda [7] to the case of semilocally type I and related functions, generalizing results of Preda [7] and Stancu-Minasian [31]. Niculescu [6] extended the work of Mishra et al. [5] by using concepts of generalized π-semilocally type I-preinvex functions. Mishra and Rautela [4] extended the works of Mishra et al. [5] and Preda [7] to the case of semilocally type I univex and related functions. By considering the invexity with respect to different (ππ )π (each function occurring in the studied problem is considered with respect to its own function ππ instead of the same function π), Slimani and Radjef [32β34] have obtained necessary and sufficient optimality/efficiency conditions and duality results for nonlinear scalar and (nondifferentiable) multiobjective problems. Ahmad [35] has considered a nondifferentiable multiobjective problem and, by using generalized univexity with respect to different (ππ )π , he has obtained efficiency conditions and duality results. Μ Arana-Jimenez et al. [36] have used the concept of semidirectionally differentiable functions introduced in [34] to derive characterizations of solutions and duality results by means of generalized pseudoinvexity for nondifferentiable multiobjective programming. In this paper, motivated by the work of Slimani and Radjef [34], we define semilocally vector-type I problems, where each component of the objective and constraint functions is semidifferentiable along its own direction instead of the same direction. Then we consider necessary and sufficient efficiency conditions for a nonlinear fractional multiple objective programming problem involving semidifferentiable functions. Furthermore, we formulate a general Mond-Weir dual and prove duality results using concepts of generalized semilocally V-type I-preinvex and related functions. Thus, we extend the works of Mishra and Rautela [4], Mishra et al. [5], Niculescu [6], and Preda [7] and generalize results obtained in the literature on this topic.
2. Preliminaries and Definitions The following conventions for equalities and inequalities will be used. If π₯ = (π₯1 , . . . , π₯π ) and π¦ = (π¦1 , . . . , π¦π ) β Rπ , then π₯ = π¦ β π₯π = π¦π , π = 1, . . . , π; π₯ < π¦ β π₯π < π¦π , π = 1, . . . , π; π₯ β¦ π¦ β π₯π β¦ π¦π , π = 1, . . . , π; π₯ β€ π¦ β π₯ β¦ π¦ and π₯ =ΜΈ π¦. π π π We also denote by Rβ§ (resp., Rβ₯ or R> ) the set of vectors π¦ β Rπ with π¦ β§ 0 (resp., π¦ β₯ 0 or π¦ > 0). Let π· β Rπ be a set and π : π· Γ π· β Rπ a vector application. We say that π· is π-invex at π₯0 β π· if π₯0 +
International Journal of Mathematics and Mathematical Sciences ππ(π₯, π₯0 ) β π· for any π₯ β π· and π β [0, 1]. We say that the set π· is π-invex if π· is π-invex at any π₯0 β π·. Definition 1 (see [7]). We say that the set π· β Rπ is an πlocally star-shaped set at π₯0 , π₯0 β π·, if, for any π₯ β π·, there exists 0 < ππ (π₯, π₯0 ) β¦ 1 such that π₯0 + ππ(π₯, π₯0 ) β π· for any π β [0, ππ (π₯, π₯0 )]. Definition 2 (see [7]). Let π : π· β Rπ be a function, where π· β Rπ is an π-locally star-shaped set at π₯0 β π·. We say that π is π-semidifferentiable at π₯0 if (ππ)+ (π₯0 , π(π₯, π₯0 )) exists for each π₯ β π·, where +
(ππ) (π₯0 , π (π₯, π₯0 )) = lim+ πβ0
1 [π (π₯0 + ππ (π₯, π₯0 )) β π (π₯0 )] π
(1)
(the right derivative of π at π₯0 along the direction π(π₯, π₯0 )). If π is π-semidifferentiable at any π₯0 β π·, then π is said to be π-semidifferentiable on π·. Note that a function which is π-semidifferentiable at π₯0 is not necessarily directionally differentiable at this point (see [34]). Slimani and Radjef [34] have considered the functions whose directional derivatives exist and are finite in some directions (not necessarily in all directions) and they called them semidirectionally differentiable functions. This class of functions, where no assumptions on continuity are needed, is an extension of locally Lipschitz functions. Definition 3 (see [37]). A function π : π· β Rπ is a convexlike function if, for any π₯, π¦ β π· and 0 β¦ π β¦ 1, there exists π§ β π· such that π (π§) β¦ ππ (π₯) + (1 β π) π (π¦) .
(2)
We consider the following multiobjective fractional optimization problem: (VFP) Minimize subject to
π (π₯) π (π₯) π (π₯) ,..., π ), =( 1 π (π₯) π1 (π₯) ππ (π₯) βπ (π₯) β¦ 0,
(3)
π = 1, 2, . . . , π,
where ππ , ππ , βπ : π· β R, π β N = {1, 2, . . . , π}, π β πΎ = {1, 2, . . . , π} with π· is a nonempty subset of Rπ , and ππ (π₯) β§ 0, ππ (π₯) > 0 for all π₯ β π· and each π β N. Let π = (π1 , . . . , ππ), π = (π1 , . . . , ππ), and β = (β1 , . . . , βπ ). We put π = {π₯ β D : β(π₯) β¦ 0} as the set of all feasible solutions of VFP. For π₯0 β π·, we denote by π½(π₯0 ) the set {π β πΎ : βπ (π₯0 ) = 0}, where π½ = |π½(π₯0 )| is the cardinal of Μ 0 ) (resp., π½(π₯0 )) the set {π β πΎ : βπ (π₯0 ) < set π½(π₯0 ), and by π½(π₯ Μ 0 ) βͺ π½(π₯0 ) = πΎ 0 (resp., βπ (π₯0 ) > 0) }. We have π½(π₯0 ) βͺ π½(π₯ and if π₯0 β π, π½(π₯0 ) = 0. For such optimization problems, minimization means obtaining (weakly) efficient solutions in the following sense.
International Journal of Mathematics and Mathematical Sciences Definition 4. A point π₯0 β π is said to be a weakly efficient solution of the problem VFP, if there exists no π₯ β π such that ππ (π₯) ππ (π₯0 ) < , ππ (π₯) ππ (π₯0 )
βπ β N.
(4)
Definition 5. A point π₯0 β π is said to be an efficient solution of the problem VFP, if there exists no π₯ β π such that for some π β N ππ (π₯) ππ (π₯)
0,
πβπ½0
π=1
Lemma 13. Suppose that
(πππ ) (π₯0 , ππ (π₯, π₯0 )) < 0,
π
π=1
To prove necessary conditions for VFP, we need to prove the following lemma.
πβπ½π
βπ₯ β π,
(16) π = 1, 2, . . . , πΌ.
If the second (implied) inequality in (16) is strict (π₯ =ΜΈ π₯0 ), we say that VFP is semilocally extendedly quasi strictly partially pseudo-V-type I-preinvex at π₯0 with respect to {(ππ )πβN , (ππ )πβN , (ππ )πβπ½0 } and {(ππ )πβπ½π , π = 1, 2, . . . , πΌ}. If, for all π β N, ππ = ππ , we say that VFP is semilocally extendedly quasi (strictly) partially pseudo-V-type I-preinvex at π₯0 with respect to {(ππ )πβN , (ππ )πβπ½0 } and {(ππ )πβπ½π , π = 1, 2, . . . , πΌ}. Remark 12. In Definition 11 if π½0 = 0 and πΌ = 1, then the concept of semilocally extendedly quasi (strictly) partially pseudo-V-type I-preinvexity reduces to the concept of semilocally quasi (strictly) pseudo-V-type I-preinvexity given in Definition 8.
Proof. Let π₯0 β π be a local weakly efficient solution for VFP and suppose there exists π₯Μ β π such that inequalities (20)β (22) are true. Μ π) = ππ (π₯0 + ππ(π₯, Μ π₯0 )) β ππ (π₯0 ). For π β N, let πππ (π₯0 , π₯, We observe that this function vanishes at Μ π) β πππ (π₯0 , π₯, Μ 0)] π = 0 and limπ β 0+ πβ1 [πππ (π₯0 , π₯, Μ π₯0 )) β ππ (π₯0 )] = = limπ β 0+ πβ1 [ππ (π₯0 + ππ(π₯, Μ π₯0 )) β¦ (πππ )+ (π₯0 , ππ (π₯, Μ π₯0 )) < 0 using (πππ )+ (π₯0 , π(π₯, (17) and (20). Μ π) < 0 if π is in It follows that, for all π β N, πππ (π₯0 , π₯, some open interval (0, πΏππ ), πΏππ > 0. Thus, for all π β N, Μ π₯0 )) < ππ (π₯0 ) , ππ (π₯0 + ππ (π₯,
π β (0, πΏππ ) .
(23)
Similarly, by using (18) with (21) and (19) with (22), we get Μ π₯0 )) > ππ (π₯0 ) , ππ (π₯0 + ππ (π₯,
π β (0, πΏππ ) , βπ β N,
Μ π₯0 )) < βπ (π₯0 ) = 0, βπ (π₯0 + ππ (π₯, π β (0, πΏβπ ) , βπ β π½ (π₯0 ) , (24) where, for all π β N, πΏππ > 0 and, for all π β π½(π₯0 ), πΏβπ > 0.
International Journal of Mathematics and Mathematical Sciences
5
Μ 0 ), βπ (π₯0 ) < 0 and βπ is continuous Now, since, for π β π½(π₯ at π₯0 , then there exists πΏπ > 0 such that Μ π₯0 )) < 0, βπ (π₯0 + ππ (π₯,
Μ 0 ) and there exist (ii) βπ is continuous at π₯0 for π β π½(π₯ vector functions ππ : π Γ π· β Rπ , ππ : π Γ π· β Rπ , π β N, and ππ : π Γ π· β Rπ , π β π½(π₯0 ), which satisfy at π₯0 with respect to π : π Γ π· β Rπ inequalities (17)β(19);
π β (0, πΏπ ) , βπ β π½Μ (π₯0 ) . (25)
Let πΏ0 = min{πΏππ , π β N, πΏππ , π β N, πΏβπ , π β π½(π₯0 ), πΏπ , π β Μ 0 )}. Then π½(π₯ Μ π₯0 )) β ππΏ0 (π₯0 ) , (π₯0 + ππ (π₯,
π β (0, πΏ0 ) ,
(26)
where ππΏ0 (π₯0 ) is a neighborhood of π₯0 . Now, for all π β (0, πΏ0 ), we have Μ π₯0 )) < ππ (π₯0 ) , ππ (π₯0 + ππ (π₯,
π β N,
(27)
Μ π₯0 )) > ππ (π₯0 ) , ππ (π₯0 + ππ (π₯,
π β N,
(28)
Μ π₯0 )) < 0, βπ (π₯0 + ππ (π₯,
π β πΎ.
(29)
Μ π₯0 )) β ππΏ0 (π₯0 ) β© π, for By (26) and (29), we get (π₯0 + ππ(π₯, all π β (0, πΏ0 ). Using (27) and (28), for π
(π₯) = (π1 (π₯)/π1 (π₯), . . ., ππ(π₯)/ππ(π₯)), we get Μ π₯0 )) < π
(π₯0 ) , π
(π₯0 + ππ (π₯,
(30)
which contradicts the assumption that π₯0 is a (local) weakly efficient solution for VFP. Hence, there exists no π₯ β π satisfying the system (20)β(22). Thus the lemma is proved. The following lemma given by Hayashi and Komiya [39] will be used. Lemma 14. Let π be a nonempty set in Rπ and let π : π β Rπ be a convex-like function. Then either π (π₯) < 0 βππ π π πππ’π‘πππ π₯ β π
(31)
(iii) for all π β N, ππ , ππ (for all π β π½(π₯0 ), βπ ) is semidifferentiable at π₯0 along the direction ππ , ππ (ππ ) and let πΏ(π₯) = [(πππ )+ (π₯0 , ππ (π₯, π₯0 )), β(πππ )+ (π₯0 , ππ (π₯, π₯0 )), π β N, (πβπ )+ (π₯0 , ππ (π₯, π₯0 )), π β π½(π₯0 )] β R2π+π½ be a convex-like function of π₯ on π. Then there exists (π, π, πΏ) β R2π+π½ such that (π₯0 , π, π, πΏ) β₯ satisfies π
π
+
+
βππ (πππ ) (π₯0 , ππ (π₯, π₯0 )) β βπ π (πππ ) (π₯0 , ππ (π₯, π₯0 )) π=1
π=1
+
(33)
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0, πβπ½(π₯0 )
βπ₯ β π.
Proof. If the conditions (i) and (ii) are satisfied, then, by Lemma 13, system (20)β(22) has no solution for π₯ β π. Since, by hypothesis (iii), πΏ(π₯) = [(πππ )+ (π₯0 , ππ (π₯, π₯0 )), β(πππ )+ (π₯0 , ππ (π₯, π₯0 )), π β N, (πβπ )+ (π₯0 , ππ (π₯, π₯0 )), π β π½(π₯0 )] β R2π+π½ is a convex-like function of π₯ on π, therefore, by Lemma 14, there exists π = (π, π, πΏ) β R2π+π½ such that β₯ relation (33) is satisfied. Remark 16. As particular case of Theorem 15, if ππ = ππ = π, βπ β N, ππ = π, and βπ β π½(π₯0 ) (i.e., if we consider that all of the functions ππ , ππ , π β N, and βπ , π β π½(π₯0 ), are semidifferentiable at π₯0 along the same direction π), we obtain Theorem 14 of Preda [7], Lemma 5 of Mishra et al. [5], Lemma 5 of Niculescu [6], and Lemma 2.5 of Mishra and Rautela [4]. Now, we define a constraint qualification given as follows.
or ππ π (π₯) β§ 0 πππ πππ π₯ β π, πππ π πππ π β Rπ β₯,
(32)
but both alternatives are never true (here the symbol π denotes the transpose of matrix). Preda [7], Mishra et al. [5], Niculescu [6], and Mishra and Rautela [4] have given necessary conditions for π₯0 β π to be a weakly efficient solution for VFP by taking the functions ππ , ππ , π β N, and βπ , π β π½(π₯0 ), semidifferentiable along the same direction π(π₯, π₯0 ). Now we give necessary efficiency criteria by considering each function ππ , ππ , π β N (resp., βπ , π β π½(π₯0 )) semidifferentiable along its own direction ππ (π₯, π₯0 ), ππ (π₯, π₯0 ), π β N (resp., ππ (π₯, π₯0 ), π β π½(π₯0 )). In the next theorem, we obtain Fritz John type necessary efficiency conditions. Theorem 15 (Fritz John type necessary efficiency conditions). Suppose that (i) π₯0 is a (local) weakly efficient solution for VFP;
Definition 17. Let π₯0 be a feasible point of ππΉπ and let ππ : π Γ π β Rπ , π β N, ππ : π Γ π β Rπ , π β π½(π₯0 ) be vector functions. (i) The function β is said to satisfy the semiconstraint qualification at π₯0 β π with respect to (ππ )πβπ½(π₯0 ) , if there exist π₯ β π such that +
(πβπ ) (π₯0 , ππ (π₯, π₯0 )) < 0,
βπ β π½ (π₯0 ) .
(34)
(ii) The function β is said to satisfy the semiconstraint qualification at π₯0 β π with respect to ((ππ )πβN , (ππ )πβN , (ππ )πβπ½(π₯0 ) ), if there exist π₯ β π such that +
(πππ ) (π₯0 , ππ (π₯, π₯0 )) > 0, +
(πβπ ) (π₯0 , ππ (π₯, π₯0 )) < 0,
βπ β N, βπ β π½ (π₯0 ) .
(35)
To prove Karush-Kuhn-Tucker type necessary efficiency conditions for VFP, we need to prove the following result.
6
International Journal of Mathematics and Mathematical Sciences
Theorem 18. Suppose that (i) π₯0 is a (local) weakly efficient solution for the following problem: πππππππ§π
(π1 (π₯) , . . . , ππ (π₯)) ,
π π’πππππ‘ π‘π
βπ (π₯) β¦ 0,
π = 1, 2, . . . , π,
(36)
where π = (π1 , . . . , ππ) : π· β Rπ; Μ 0 ) and there exist (ii) βπ is continuous at π₯0 for π β π½(π₯ vector functions ππ : π Γ π· β Rπ , π β N, and ππ : π Γ π· β Rπ , π β π½(π₯0 ), which satisfy at π₯0 with respect to π : π Γ π· β Rπ the following inequalities: +
Lemma 19 (see [7]). If π₯0 is a (local) weakly efficient solution for VFP, then π₯0 is a (local) weakly efficient solution for (VFPπ0 ), where π0π = ππ (π₯0 )/ππ (π₯0 ), π β N. Using this lemma and Theorem 18, we can derive KarushKuhn-Tucker type necessary efficiency conditions for the problem VFP. Theorem 20 (Karush-Kuhn-Tucker type necessary efficiency conditions). Suppose that (i) π₯0 is a (local) weakly efficient solution for VFP; Μ 0 ) and there exist (ii) βπ is continuous at π₯0 for π β π½(π₯ vector functions ππ : π Γ π· β Rπ , π β N, and ππ : π Γ π· β Rπ , π β π½(π₯0 ), which satisfy at π₯0 with respect to π : π Γ π· β Rπ inequalities (17) and (19) with
+
(πππ ) (π₯0 , π (π₯, π₯0 )) β¦ (πππ ) (π₯0 , ππ (π₯, π₯0 )) , βπ₯ β π, βπ β N, +
+
(πβπ ) (π₯0 , π (π₯, π₯0 )) β¦ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) ,
(37)
+
+
(πππ ) (π₯0 , π (π₯, π₯0 )) β§ (πππ ) (π₯0 , ππ (π₯, π₯0 )) ,
βπ₯ β π, βπ β π½ (π₯0 ) ;
βπ₯ β π, βπ β N;
(iii) for all π β N, ππ (for all π β π½(π₯0 ), βπ ) is semidifferentiable at π₯0 along the direction ππ (ππ ) and let πΏ 1 (π₯) = [(πππ )+ (π₯0 , ππ (π₯, π₯0 )), π β N, (πβπ )+ (π₯0 , ππ (π₯, π₯0 )), π β π½(π₯0 )] β Rπ+π½ be a convex-like function of π₯ on π; (iv) the function β satisfies the semiconstraint qualification at π₯0 β π with respect to (ππ )πβπ½(π₯0 ) . π½ Then there exist π β Rπ β₯ and πΏ β Rβ§ such that (π₯0 , π, πΏ) satisfies π
(iii) for all π β N, ππ , ππ (for all π β π½(π₯0 ), βπ ) is semidifferentiable at π₯0 along the direction ππ (ππ ) and let πΏ 2 (π₯) = [[(πππ )+ (π₯0 , ππ (π₯, π₯0 )) β π0π (πππ )+ (π₯0 , ππ (π₯, π₯0 ))], π β N, (πβπ )+ (π₯0 , ππ (π₯, π₯0 )), π β π½(π₯0 )] β Rπ+π½ be a convex-like function of π₯ on π, where π0 = (π01 , . . . , π0π), π0π = ππ (π₯0 )/ππ (π₯0 ), π β N; (iv) the function β satisfies the semiconstraint qualification at π₯0 β π with respect to (ππ )πβπ½(π₯0 ) . π½ Then there exist π β Rπ β₯ and πΏ β Rβ§ such that 0 (π₯0 , π, π , πΏ) satisfies
+
βππ (πππ ) (π₯0 , ππ (π₯, π₯0 )) π=1
(38)
+
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0, πβπ½(π₯0 )
βπ₯ β π.
+
β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0,
βπ₯ β π,
πβπ½(π₯0 )
(39)
which contradicts semiconstraint qualification (34). Hence π =ΜΈ 0. π For each π = (π 1 , . . . , π π) β Rπ + , where R+ denotes the π positive orthant of R , we consider
(VFPπ ) Minimize
(π1 (π₯) β π 1 π1 (π₯) , . . . , ππ (π₯) β π πππ (π₯)) ,
subject to βπ (π₯) β¦ 0, π = 1, 2, . . . , π. The following lemma can be proved without difficulty.
π
+
+
βππ [(πππ ) (π₯0 , ππ (π₯, π₯0 )) β π0π (πππ ) (π₯0 , ππ (π₯, π₯0 ))] π=1
Proof. In the same line as in the proof of Theorem 15, we prove that there exists π = (π, πΏ) β Rπ+π½ such that relation β₯ (38) is satisfied. Now it is enough to prove that π =ΜΈ 0. We proceed by contradiction. If π = 0, then πΏ =ΜΈ 0 and (38) takes the following form:
(40)
(41)
+
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0, πβπ½(π₯0 )
(42) βπ₯ β π.
Proof. Let π₯0 be a (local) weakly efficient solution for VFP. According to Lemma 19 we have that π₯0 is a (local) weakly efficient solution for (VFPπ0 ). Now applying Theorem 18 to π½ problem (VFPπ0 ), we get that there exist π β Rπ β₯ and πΏ β Rβ§ such that relation (42) is satisfied, and the theorem is proved. In the Karush-Kuhn-Tucker type necessary efficiency condition (42) of Theorem 20, the functions ππ and ππ are considered semidifferentiable at π₯0 along the same direction ππ , π β N. To obtain a necessary condition with different directions ππ and ππ , we need to use the second variant of the semiconstraint qualification given in Definition 17. Theorem 21 (Karush-Kuhn-Tucker type necessary efficiency conditions). Suppose that the hypotheses (π), (ππ), and (πππ) of Theorem 15 are satisfied and the function β satisfies the
International Journal of Mathematics and Mathematical Sciences semiconstraint qualification at π₯0 β π with respect to ((ππ )πβN , (ππ )πβN , (ππ )πβπ½(π₯0 ) ). Then there exist π β Rπ β₯, π β π½ , and πΏ β R such that (π₯ , π, π, πΏ) satisfies relation (33). Rπ 0 β§ β§
7 Multiplying (6) by ππ , π β N, and (8) by πΏπ , π β π½(π₯0 ), then summing the obtained relation and using (44), we get π
βππ [ππ (π₯) β ππ (π₯0 )] β β πΏπ βπ (π₯0 ) π=1 πβπ½(π₯0 )
Proof. Based on Theorem 15, we obtain the existence of (π, π, πΏ) β R2π+π½ such that (π₯0 , π, π, πΏ) satisfies relation β₯ (33). Now it is enough to prove that π =ΜΈ 0. We proceed by contradiction. If π = 0, then (π, πΏ) =ΜΈ 0 and (33) takes the following form:
π
+
(47)
β§ βππ (πππ ) (π₯0 , ππ (π₯, π₯0 )) π=1
+
π
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0. πβπ½(π₯0 )
+
β β π π (πππ ) (π₯0 , ππ (π₯, π₯0 )) π=1
(43) +
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0,
βπ₯ β π,
From the above inequality and the fact that βπ (π₯0 ) = 0, βπ β π½(π₯0 ), it follows that
πβπ½(π₯0 )
π
which contradicts the semiconstraint qualification (ii) of Definition 17. Hence π =ΜΈ 0.
4. Sufficient Efficiency Criteria In this section, we present some Karush-Kuhn-Tucker type sufficient efficiency conditions for a feasible solution to be efficient or weakly efficient for VFP under various types of generalized semilocally V-type I-preinvex assumptions. Theorem 22. Let π₯0 β π and suppose that there exist (2π + π½) vector functions ππ : π Γ π· β Rπ , ππ : π Γ π· β Rπ , π β N and ππ : πΓπ· β Rπ , π β π½(π₯0 ), such that VFP is semilocally V-type I-preinvex at π₯0 with respect to (ππ )πβN , (ππ )πβN , and π½ π (ππ )πβπ½(π₯0 ) . If there exist vectors π β Rπ β₯ , π β Rβ§ , and πΏ β Rβ§ such that π
+
βππ (πππ ) (π₯0 , ππ (π₯, π₯0 )) π=1
(44)
+
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0, πβπ½(π₯0 ) +
(πππ ) (π₯0 , ππ (π₯, π₯0 )) β¦ 0,
βπ₯ β π,
βπ₯ β π, βπ β N,
(45)
then π₯0 is a weakly efficient solution for VFP. Proof. Suppose that π₯0 is not a weakly efficient solution of VFP. Then there exists a feasible solution π₯ β π of VFP such that
βππ [ππ (π₯) β ππ (π₯0 )] β§ 0.
(48)
π=1
Since π β₯ 0, from (48) we obtain that there exists π0 β N such that ππ0 (π₯) β§ ππ0 (π₯0 ) .
(49)
By (45) and (7) it follows that ππ (π₯) β¦ ππ (π₯0 ) ,
βπ β N.
(50)
Now using (49), (50), and π β§ 0, π > 0, we obtain ππ0 (π₯) ππ0 (π₯)
β§
ππ0 (π₯0 ) ππ0 (π₯0 )
,
(51)
which is a contradiction to (46). Thus π₯0 is a weakly efficient solution for VFP and the theorem is proved. Remark 23. As particular case of Theorem 22, if ππ = ππ = π, βπ β N, and ππ = π, βπ β π½(π₯0 ), we obtain Theorem 1 of Mishra et al. [5]. Theorem 24. Let π₯0 β π and suppose that there exist (2π + π½) vector functions ππ : π Γ π· β Rπ , ππ : π Γ π· β Rπ , π β N, and ππ : πΓπ· β Rπ , π β π½(π₯0 ), such that VFP is semilocally V-type I-preinvex at π₯0 with respect to (ππ )πβN , (ππ )πβN , and π (ππ )πβπ½(π₯0 ) . If there exist vectors π β Rπ β₯ , π β Rβ§ (π π = π½ ππ (π₯0 )/ππ (π₯0 ), π β N), and πΏ β Rβ§ such that π
+
βππ [(πππ ) (π₯0 , ππ (π₯, π₯0 )) π=1
ππ (π₯) ππ (π₯0 ) , < ππ (π₯) ππ (π₯0 )
βπ β N.
(46)
Since VFP is semilocally V-type I-preinvex at π₯0 with respect to (ππ )πβN , (ππ )πβN , and (ππ )πβπ½(π₯0 ) , we get that inequalities (6), (7), and (8) are true for π₯ = π₯.
+
βπ π (πππ ) (π₯0 , ππ (π₯, π₯0 ))] +
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0, πβπ½(π₯0 ) then π₯0 is a weakly efficient solution for VFP.
(52) βπ₯ β π,
8
International Journal of Mathematics and Mathematical Sciences
Proof. Suppose that π₯0 is not a weakly efficient solution of VFP. Then there exists a feasible solution π₯ β π of VFP such that ππ (π₯) ππ (π₯0 ) < , ππ (π₯) ππ (π₯0 )
βπ β N,
that is, ππ (π₯) < π π ππ (π₯) ,
βπ β N.
(53) (54)
Since VFP is semilocally V-type I-preinvex at π₯0 with respect to (ππ )πβN , (ππ )πβN , and (ππ )πβπ½(π₯0 ) , we get that inequalities (6), (7), and (8) are true for π₯ = π₯. Using these inequalities, π β₯ 0, π β§ 0, πΏ β§ 0, and (52), we get π
π
π=1
π=1
βππ [ππ (π₯) β ππ (π₯0 )] β βππ π π [ππ (π₯) β ππ (π₯0 )] β β πΏπ βπ (π₯0 ) πβπ½(π₯0 ) π
Remark 26. As a particular case of Theorem 24, if ππ = ππ = π, βπ β N, and ππ = π, βπ β π½(π₯0 ), we obtain Theorem 2 of Mishra et al. [5]. Preda [7], Mishra et al. [5], Niculescu [6], and Mishra and Rautela [4] have given sufficient efficiency conditions for a feasible solution to be weakly efficient under various types of (generalized) π-semilocally (type I-) preinvex (univex) assumptions. In the following theorem, we give sufficient efficiency conditions for a feasible solution to be efficient involving generalized semilocally V-type I-preinvex functions. Theorem 27. Let π₯0 β π and suppose that there exist (2π + π½) vector functions ππ : π Γ π· β Rπ , ππ : π Γ π· β Rπ , π β N, ππ : π Γ π· β Rπ , π β π½(π₯0 ), and vectors π½ π π β Rπ β₯ , π β Rβ§ (π π = ππ (π₯0 )/ππ (π₯0 ), π β N), and πΏ β Rβ§ such that relation (52) is satisfied. If any one of the following conditions holds, (a) VFP is semilocally strictly pseudo quasi-V-type Ipreinvex at π₯0 with respect to (ππ )πβN , (ππ )πβN , and (ππ )πβπ½(π₯0 ) and for π, π, and πΏ;
(55)
+
β§ βππ [(πππ ) (π₯0 , ππ (π₯, π₯0 )) π=1
(b) VFP is semilocally quasi strictly pseudo-V-type Ipreinvex at π₯0 with respect to (ππ )πβN , (ππ )πβN , and (ππ )πβπ½(π₯0 ) and for π, π, and πΏ,
+
βπ π (πππ ) (π₯0 , ππ (π₯, π₯0 ))] +
+ β πΏπ (πβπ ) (π₯0 , ππ (π₯, π₯0 )) β§ 0. πβπ½(π₯0 )
then π₯0 is an efficient solution for VFP. Proof. Suppose that π₯0 is not an efficient solution of VFP. Then there exists a feasible solution π₯ β π of VFP such that for some π β N
Therefore, π
βππ [(ππ (π₯) β π π ππ (π₯)) β (ππ (π₯0 ) β π π ππ (π₯0 ))] π=1
(56) β β πΏπ βπ (π₯0 ) β§ 0. πβπ½(π₯0 )
ππ (π₯) ππ (π₯)
0; (b) the problem VFP is semilocally strictly extendedly pseudo partially quasi-V-type I-preinvex at π¦ with respect to {(ππ )πβN , (ππ )πβπ½0 } and {(ππ )πβπ½π , π = 1, 2, . . . , πΌ} and for π, π, and πΏ; (c) the problem VFP is semilocally extendedly quasi strictly partially pseudo-V-type I-preinvex at π¦ with respect to {(ππ )πβN , (ππ )πβπ½0 } and {(ππ )πβπ½π , π = 1, 2, . . . , πΌ} and for π, π, and πΏ.
(77)
ππ0 (π₯) β π π0 ππ0 (π₯) < 0 ππππ πππ π0 β N.
(78)
Proof. By condition (a), since the problem VFP is semilocally extendedly pseudo partially quasi-V-type I-preinvex at π¦ with respect to {(ππ )πβN , (ππ )πβπ½0 } and {(ππ )πβπ½π , π = 1, 2, . . . , πΌ} and for π, π, and πΏ, then +
(83)
πΏπ½π0 βπ½0 (π₯) β¦ 0.
(84)
Combining (82), (83), (84), and (72), we get ππ (π₯) β π π ππ (π₯) + πΏπ½π0 βπ½0 (π₯) β¦ ππ (π¦) β π π ππ (π¦) + πΏπ½π0 βπ½0 (π¦)
for each π β N,
(85)
ππ0 (π₯) β π π0 ππ0 (π₯) + πΏπ½π0 βπ½0 (π₯) < ππ0 (π¦) β π π0 ππ0 (π¦) + πΏπ½π0 βπ½0 (π¦)
for some π0 β N. (86)
Since ππ > 0 for any π β N, by (85), (86) and (74), we obtain π
π
πβπ½0
πβπ½0
where by using the reverse implication in (79) it follows that π
+
+
β ππ [(πππ ) (π¦, ππ (π₯, π¦)) β π π (πππ ) (π¦, ππ (π₯, π¦))]
π=1
+
+ β πΏπ (πβπ ) (π¦, ππ (π₯, π¦)) < 0.
π =1πβπ½π
(79)
β§ βππ [ππ (π¦) β π π ππ (π¦)] + β πΏπ βπ (π¦) ,
π=1
+
β β πΏπ (πβπ ) (π¦, ππ (π₯, π¦)) > 0,
πβπ½0
πβπ½0
< βππ [ππ (π¦) β π π ππ (π¦)] + β πΏπ βπ (π¦) ,
πΌ
+
+ β πΏπ (πβπ ) (π¦, ππ (π₯, π¦)) β§ 0 π
(87)
π
(88)
Now, from (88) and (71) we obtain
π=1
σ³¨β βππ [ππ (π₯) β π π ππ (π₯)] + β πΏπ βπ (π₯)
πβπ½0
πβπ½0
+
βππ [(πππ ) (π¦, ππ (π₯, π¦)) β π π (πππ ) (π¦, ππ (π₯, π¦))]
π=1
ππ0 (π₯) β π π0 ππ0 (π₯) < 0 for some π0 β N.
π=1
ππ (π₯) β π π ππ (π₯) β¦ 0 ππππππβ π β N,
π=1
(82)
βππ [ππ (π₯) β π π ππ (π₯)] + β πΏπ βπ (π₯)
Then the following cannot hold:
π
ππ (π₯) β π π ππ (π₯) β¦ 0 for each π β N,
(89)
which is a contradiction to (81). The proofs of parts (b) and (c) are very similar to the proof of part (a), except that, for part (b), since π β₯ 0, then inequality (87) becomes nonstrict (β¦) and it follows that inequalities (88) and (89) remain true and strict (), respectively.
International Journal of Mathematics and Mathematical Sciences For part (c), inequality (81) becomes strict (