On a Nonsmooth Vector Optimization Problem with Generalized Cone Invexity

and Applied Analysis 3 Definition 2.2 see 16 . Let ψ : R → R be a locally Lipschitz function, then ψ◦ u;v denotes Clarke’s generalized directional derivative of ψ at u ∈ R in the direction v and is defined as ψ◦ u;v lim sup y→u t→ 0 ψ ( y tv ) − ψ(y) t . 2.4 Clarke’s generalized gradient of ψ at u is denoted by ∂ψ u and is defined as ∂ψ u { ξ ∈ R | ψ◦ u;v ≥ 〈ξ, v〉, ∀v ∈ Rn}. 2.5 Let f : R → R be a vector-valued function given by f f1, f2, . . . , fm , where fi : R → R, i 1, 2, . . . , m. Then f is said to be locally Lipschitz on R if each fi is locally Lipschitz on R. The generalized directional derivative of a locally Lipschitz function f : R → R at u ∈ R in the direction v is given by f◦ u;v { f◦ 1 u;v , f ◦ 2 u;v , . . . , f ◦ m u;v } . 2.6 The generalized gradient of f at u is the set ∂f u ∂f1 u × ∂f2 u × · · · × ∂fm u , 2.7 where ∂fi u i 1, 2, . . . , m is the generalized gradient of fi at u. Every A a1, a2, . . . , am ∈ ∂f u is a continuous linear operator from R to R and Au 〈a1, u〉, 〈a2, u〉, . . . , 〈am, u〉 ∈ R, ∀u ∈ R. 2.8 Lemma 2.3 see 16 . (a) If fi : R → R is locally Lipschitz then, for each u ∈ R, f◦ i u;v max {〈ξ, v〉 | ξ ∈ ∂fi u } , ∀v ∈ R, i 1, 2, . . . , m. 2.9 (b) Let fi i 1, 2, . . . , m be a finite family of locally Lipschitz functions on R, then ∑m i 1 fi is also locally Lipschitz and


Introduction
In optimization theory, convexity plays a key role in many aspects of mathematical programming including sufficient optimality conditions and duality theorems; see 1, 2 .Many attempts have been made during the past several decades to relax convexity requirement; see 3-7 .In this endeavor, Hanson 8 introduced invex functions and studied some applications to optimization problem.Subsequently, many authors further weakened invexity hypotheses to establish optimality conditions and duality results for various mathematical programming problems; see, for example, 9-11 and the references cited therein.
Above all, Yen and Sach 12 introduced cone-generalized invex and cone-nonsmooth invex functions.Giorgi and Guerraggio 13 presented the notions of α-K-invex, α-K pseudoinvex, and α-K quasi-invex functions in the differentiable case and derived optimality and duality results for a vector optimization problem over cones.Khurana 14 extended pseudoinvex functions to differentiable cone-pseudoinvex and strongly cone-pseudoinvex functions.Based on this, Suneja et al. 15 defined cone-nonsmooth quasi-invex, conenonsmooth pseudoinvex, and other related functions in terms of Clarke's 16 generalized directional derivatives and established optimality and duality results for a nonsmooth vector optimization problem.
On the other hand, Noor 17 proposed several classes of α-invex functions and investigated some properties of the α-preinvex functions and their differentials.Mishra et al. 18 defined strict pseudo-α-invex and quasi-α-invex functions.Mishra et al. 19 further introduced the concepts of nonsmooth pseudo-α-invex functions and established a relationship between vector variational-like inequality and nonsmooth vector optimization problems by using the nonsmooth α-invexity.
In the present paper, by using Clarke's generalized gradients of locally Lipschitz functions we are concerned with a nonsmooth vector optimization problem with cone constraints and introduce several generalized invex functions over cones namely K-αgeneralized invex, K-α-nonsmooth invex, and other related functions, which, respectively, extend some corresponding concepts of 12, 13, 15, 17 .Some sufficient optimality conditions for this problem are obtained by using the above defined concepts.Furthermore, a Mond-Weir type dual is formulated and a few weak and converse duality results are established.We generalize and extend some results presented in the literatures on this topic.

Preliminaries and Definitions
Throughout this paper, let η : R n × R n → R n and α : R n × R n → R \ {0} be two fixed mappings.int K and K denote the interior and closure of K ⊆ R m , respectively.We always assume that K is a closed convex cone with int K / ∅.
The positive dual cone K of K is defined as The strict positive dual cone K of K is given by The following property is from 20 , which will be used in the sequel.
Clarke's generalized gradient of ψ at u is denoted by ∂ψ u and is defined as Let f : R n → R m be a vector-valued function given by f f 1 , f 2 , . . ., f m , where f i : R n → R, i 1, 2, . . ., m.Then f is said to be locally Lipschitz on R n if each f i is locally Lipschitz on R n .The generalized directional derivative of a locally Lipschitz function f : R n → R m at u ∈ R n in the direction v is given by The generalized gradient of f at u is the set where ∂f i u i . ., m be a finite family of locally Lipschitz functions on R n , then m i 1 f i is also locally Lipschitz and

2.10
Definition 2.4 see 17 .A function h : R n → R is said to be α-invex function at u ∈ R n with respect to α and η, if there exist functions α and η such that, for every x ∈ R n , we have

2.11
In this paper, we consider the following vector optimization problem with cone constraints: where f : R n → R m , g : R n → R p are locally Lipschitz functions on R n and K, Q are closed convex cones with nonempty interiors in R m and R p , respectively.Denote X {x ∈ R n : −g x ∈ Q} the feasible set of problem VP .
For each λ ∈ K and μ ∈ Q , we suppose that λf λ • f and μg μ • g are locally Lipschitz.Now, we present the concepts of solutions for problem VP in the following sense.
Based on the lines of Yen and Sach 12 and Noor 17 , we define the notions as follows.
Definition 2.6.Let f : R n → R m be a locally Lipschitz function.f is said to be K-α-generalized invex at u ∈ R n , if there exist functions α and η such that for every x ∈ R n and A ∈ ∂f u , where
Lemma 2.9.If f is K-α-generalized invex at u with respect to α and η, then f is K-α-nonsmooth invex at u with respect to the same α and η.
Proof.Since f is K-α-generalized invex at u, then there exist α and η such that for every By Lemma 2.3, for each i ∈ {1, 2, . . ., m}, we choose a i ∈ ∂f i u such that

2.19
Then A a 1 , . . ., a m ∈ ∂f u and Hence, f is K-α-nonsmooth invex at u with respect to the same α and η.
The following example shows that converse of the above lemma is not true.
Definition 2.11.f is said to be K-α-nonsmooth quasi-invex at u ∈ R n , if there exist functions α and η such that for every x ∈ R n ,

2.25
Definition 2.12.f is said to be K-α-nonsmooth pseudo-invex at u ∈ R n , if there exist functions α and η such that for every x ∈ R n ,

2.26
Definition 2.13.f is said to be strict K-α-nonsmooth pseudo-invex at u ∈ R n , if there exist functions α and η such that for every x ∈ R n ,

2.27
Definition 2.14.f is said to be strong K-α-nonsmooth pseudo-invex at u ∈ R n , if there exist functions α and η such that for every x ∈ R n ,
Remark 2.17.If α x, u ≡ 1 for all x, u ∈ R n , then the above definitions reduce to the corresponding definitions 15 .If f is differentiable, then K-α-generalized invex and K-αnonsmooth pseudo-invex functions reduce to α-K-invex and α-K pseudo-invex functions 13 , respectively.

Optimality Criteria
In this section, we establish a few sufficient optimality conditions for problem V P by using the above defined functions.Theorem 3.1.Let f be K-α-generalized invex and g be Q-α-generalized invex at u ∈ X with respect to the same α and η.We assume that there exist λ ∈ K , λ / 0, μ ∈ Q such that Then u is a weak minimum of VP .
Proof.By contradiction, we assume that u is not a weak minimum of VP .Then there exists a feasible solution x of VP such that From 3.1 , it follows that there exist s ∈ ∂ λf u and t ∈ ∂ μg u such that Summing 3.3 and 3.5 , we have −α x, u Aη x, u ∈ int K, ∀A ∈ ∂f u .

3.13
By virtue of 3.2 and x ∈ X, the above inequality implies which is a contradiction to 3.12 .Therefore, u is a weak minimum of VP .
Theorem 3.2.Let f be K-α-generalized invex and g be Q-α-generalized invex at u ∈ X with respect to the same α and η.We assume that there exist λ ∈ K , μ ∈ Q such that 3.1 and 3.2 hold.Then u is a minimum of VP .
Proof.Assume contrary to the result that u is not a minimum of VP .Then there exists x ∈ X such that

3.16
From 3.1 , it follows that there exist s ∈ ∂ λf u and t ∈ ∂ μg u such that s t 0.

3.20
Next proceeding on the same lines as in the proof of Theorem 3.1, we obtain a contradiction.Thus, u is a minimum of VP .
Theorem 3.3.Let f be K-α-nonsmooth pseudo-invex and g be Q-α-nonsmooth quasi-invex at u ∈ X with respect to the same α and η.We assume that there exist λ ∈ K , λ / 0, μ ∈ Q such that 3.1 and 3.2 hold.Then u is a weak minimum of VP .
Proof.It follows from 3.1 that there exist s ∈ ∂ λf u and t ∈ ∂ μg u such that s t 0.

3.21
Suppose that u is not a weak minimum of VP .Then there exists x ∈ X such that

3.33
Hence, sη x, u ≥ 0, 3.34 which is in contradiction with 3.26 .Therefore, u is a weak minimum of VP .
The following example illustrates the above theorem.
Example 3.4.Consider the vector optimization problem VP where K { x, y y ≥ −x, y ≥ x}, Q { x, y − x ≤ y ≤ x, x ≥ 0}, and f i , g i : R → R, i 1, 2 are defined as

3.35
Let α : R × R → R \ {0} and η : R × R → R be defined as α x, u 2 and η x, u x 2u 3 , respectively.It is easily testified that f and g are K-α-nonsmooth pseudo-invex and K-αnonsmooth quasi-invex at u 0, respectively.The feasible set of VP is given by X −∞, 0 .It is also easy to verify ∂f 0 which imply that 3.1 and 3.2 hold.Therefore, by Theorem 3.3, u 0 is a weak minimum of VP .
Theorem 3.5.Let f be strong K-α-nonsmooth pseudo-invex and g be Q-α-nonsmooth quasi-invex at u ∈ X with respect to the same α and η.We assume that there exist λ ∈ K , λ / 0, μ ∈ Q such that 3.1 and 3.2 hold.Then u is a strong minimum of VP .
Proof.From 3.1 , it follows that there exist s ∈ ∂ λf u and t ∈ ∂ μg u such that s t 0.

3.37
Assume that u is not a strong minimum of VP .Then there exists x ∈ X such that

3.39
Next proceeding on the same lines as in the proof of Theorem 3.3, we get a contradiction.Hence u is a strong minimum of VP .
Theorem 3.6.Let f be strict K-α-nonsmooth pseudo-invex and g be Q-α-nonsmooth quasi-invex at u ∈ X with respect to the same α and η.We assume that there exist λ ∈ K , λ / 0, μ ∈ Q such that 3.1 and 3.2 hold.Then u is a minimum of VP .
Proof.From 3.1 , it follows that there exist s ∈ ∂ λf u and t ∈ ∂ μg u such that s t 0.

3.40
By contradiction, assume that u is not a minimum of VP .Then there exists x ∈ X such that Since f is strict K-α-nonsmooth pseudo-invex at u, we have Next as in Theorem 3.3 we arrive at a contradiction.Therefore, u is a minimum of VP .

Duality
In relation to VP , we consider the following Mond-Weir type dual problem: Now, we establish weak and converse duality results.Proof.The proof of the above theorem is very similar to the proof of Theorem 3.1, except that for this case we use the feasibility of y, λ, μ for VD instead of the relations 3.1 and 3.2 .

Theorem 4 . 4
Converse duality .Let y ∈ X and y, λ, μ ∈ G. Assume that f is K-α-generalized invex and g is Q-α-generalized invex at y with respect to the same α and η.Then y is a weak minimum of VP .
Definition 2.2 see 16 .Let ψ : R n → R be a locally Lipschitz function, then ψ • u; v denotes Clarke's generalized directional derivative of ψ at u ∈ R n in the direction v and is defined as

Lemma 2.3 see 16
. (a) If f i : R n → R is locally Lipschitz then, for each u ∈ R n , Theorem 4.1 Weak duality .Let x ∈ X and y, λ, μ ∈ G.If f is K-α-nonsmooth pseudo-invex and g is Q-α-nonsmooth quasi-invex at y with respect to the same α and η, then Converse duality .Let y ∈ X and y, λ, μ ∈ G. Assume that f is K-α-nonsmooth pseud-invex and g is Q-α-nonsmooth quasi-invex at y with respect to the same α and η.Then y is a weak minimum of VP ., thus t μB * , where B * ∈ ∂g y .Hence, μB * η x, y > 0, where B * ∈ ∂g y .4.39By x ∈ X and y, λ, μ ∈ G, we have μg x ≤ 0 ≤ μg y .4.40By the similar argument to that of Theorem 4.1, we can prove that Proof.Since y, λ, μ ∈ G, from VD , it follows that there exist s ∈ ∂ λf y and t ∈ ∂ μg y such that s t 0.4.2Bycontradiction,weassume that f y − f x ∈ int K.Since f is K-α-nonsmooth pseudo-invex at y, we have−α x, y f • y; η x, y ∈ int K. Theorem 4.2 Weak duality .Let x ∈ X and y, λ, μ ∈ G.If f is K-α-generalized invex and g is Q-α-generalized invex at y with respect to the same α and η, then Proof.Since y, λ, μ ∈ G, from VD , it follows that there exist s ∈ ∂ λf y and t ∈ ∂ μg y such that s t 0. 4.32Assume contrary to the result that y is not a weak minimum of VP .Then there exists x ∈ X such that f y − f x ∈ int K.4.33Since f is K-α-nonsmooth pseudo-invex at y, we have −α x, y f • y; η x, y ∈ int K.