Higher-Order Minimizers and Generalized ( ) , ρ F-Convexity in Nonsmooth Vector Optimization over Cones

In this paper, we introduce the concept of a (weak) minimizer of order k for a nonsmooth vector optimization problem over cones. Generalized classes of higher-order cone-nonsmooth (F, ρ)convex functions are introduced and sufficient optimality results are proved involving these classes. Also, a unified dual is associated with the considered primal problem, and weak and strong duality results are established.


Introduction
It is well known that the notion of convexity plays a key role in optimization theory [1] [2].In the literature, various generalizations of convexity have been considered.One such generalization is that of a ρ -convex function introduced by Vial [3].Hanson and Mond [4] defined the notion of an F-convex function.As an extended unification of the two concepts, Preda [5] introduced the concept of a ( ) gave the notion of a locally Lipschitz ( ) , F ρ -convex scalar function of order k [6] and a differentiable ( ) , F ρconvex vector function of order 2 [7].
L. Cromme [8] defined the concept of a strict local minimizer of order k for a scalar optimization problem.This concept plays a fundamental role in convergence analysis of iterative numerical methods [8] and in stability results [9].The definition of a strict local minimizer of order 2 is generalized to the vectorial case by Antczak [7].
Recently, Bhatia and Sahay [10] introduced the concept of a higher-order strict minimizer with respect to a nonlinear function for a differentiable multiobjective optimization problem.They proved various sufficient optimality and mixed duality results involving generalized higher-order strongly invex functions.
The main purpose of this paper is to extend the concept of a higher-order minimizer to a nonsmooth vector optimization problem over cones.The paper is organized as follows.We begin in Section 2 by recalling some known concepts in the literature.We then define the notion of a (weak) minimizer of order k for a nonsmooth vector optimization problem over cones.Thereafter, we introduce various new generalized classes of conenonsmooth ( ) , F ρ -convex functions of higher-order.In Section 3, we study several optimality conditions for higher-order minimizers via the introduced classes of functions.In Section 4, we associate a unified dual to the considered problem and establish weak and strong duality results.( ) ( )

Preliminaries and Definitions
x within a neighbourhood of u.
A function ψ is said to be locally Lipschitz on S if it is locally Lipschitz at each point of S. Definition 2.1.[11] Let The Clarke's generalized gradient of ψ at u is denoted by and is defined as Let : m f S → R be a vector valued function given by ( ) , , , The generalized directional derivative of a locally Lipschitz function The generalized gradient of f at u is the set A functional : R is sublinear with respect to the third variable if, for all ( ) , ; , ; , ; (i) and (ii) together imply ( ) (1) We consider the following nonsmooth vector optimization problem (NVOP) K-minimize ( ) p S → R , K and Q are closed convex cones with nonempty interiors in R m and R p respectively.We assume that i f for each ∈ denote the set of all feasible solutions of (NVOP).The following solution concepts are well known in the literature of vector optimization theory.Definition 2.2.A point 0 x S ∈ , is said to be (i) a weak minimizer (weakly efficient solution) of (NVOP) if for every (ii) a minimizer (efficient solution) of (NVOP) if for every With the idea of analyzing the convergence and stability of iterative numerical methods, L. Cromme [8] introduced the notion of a "strict local minimizer of order k".As a recent advancement on this platform, Bhatia and Sahay [10] defined the concept of a higher-order strict minimizer with respect to a nonlinear function for a differentiable multiobjective optimization problem.We now generalize this concept and give the definition of a higher-order (weak) minimizer with respect to a function ω for a nonsmooth vector optimization problem over cones.
Definition 2.3.A point 0 x S ∈ is said to be (i) a weak minimizer of order − , the definition of a weak minimizer of order k reduces to the definition of a strict minimizer of order k (see [8] [9] [12] [13]). ( x x ω = − , the definition of a (weak) minimizer of order k becomes the definition of a vector strict global (weak) minimizer of order 2 given by Antczak [7]. ( the definition of a weak minimizer of order k reduces to the definition of a strict minimizer of order k given by Bhatia and Sahay [10]. Remark 2.2.(1) Clearly a minimizer of order k for (NVOP) with respect to ω is also a weak minimizer of order k for (NVOP) with respect to the same ω .
(2) A direct implication of the fact that intK β ∈ is that, a (weak) minimizer of order k for (NVOP) with respect to ω is a (weak) minimizer for (NVOP).
(3) Note that if x is a (weak) minimizer of order k for (NVOP) with respect to ω , then for all k >  , it is also a (weak) minimizer of order  for (NVOP) with respect to the same ω .
In the sequel, for a vector function , ; , , , ; We now define various classes of nonsmooth ( ) , F ρ -convex functions of higher-order over cones.

Definition 2.4. A locally Lipschitz function
on S if there exist a sublinear (with respect to the third variable) functional , , ,

If the above relation holds for every u S
∈ then f is said to be K-nonsmooth ( ) , F ρ -convex of order k with respect to ω on S. ( , F ρ -convex function of order k with respect to ω becomes the definition of a vector ( ) tion of order 2 given in [7]. ( , introduced by Nahak and Mohapatra [14]. ( η × → R , the above definition becomes the definition of a higher-order strongly invex function given by Bhatia and Sahay [10].Definition 2.5.A locally Lipschitz function : on S if there exist a sublinear (with respect to the third variable) functional : Clearly, if f is K-nonsmooth ( ) , F ρ -convex of order k with respect to ω , then f is K-nonsmooth ( ) 2 , 0 , 0 ( ) , F ρ -pseudoconvex type I of order 3 with respect to ω at u on S.
However, for 1 x = and ( ) If the above relation holds for every u S ∈ , then f is said to be K-nonsmooth ( ) of order k with respect to ω on S.
We now give an example to show that a K -nonsmooth ( ) , F ρ -pseudoconvex type II function of order k with respect to ω may fail to be a K -nonsmooth ( ) , 1 , 1 ( ) However, for 5 4 x = and ( ) Thus, f is not K-nonsmooth ( ) , F ρ -convex of any order k with respect to ω at u on S.

Definition 2.7. A locally Lipschitz function
: type I of order k with respect to ω at u S ∈ on S if there exist a sublinear (with respect to the third variable) functional : If the above relation holds at every u S ∈ , then f is said to be K-nonsmooth ( )

Optimality
In this section, we obtain various nonsmooth Fritz John type and Karush-Kuhn-Tucker (KKT) type necessary and sufficient optimality conditions for a feasible solution to be a (weak) minimizer of order k for (NVOP).
On the lines of Craven [15] we define Slater-type cone constraint qualification as follows: Definition 3.1.The problem (NVOP) is said to satisfy Slater-type cone constraint qualification at x if, for all ( ) The following inclusion relation is worth noticing.For ( ) .
Since a weak minimizer of order 1 k ≥ for (NVOP) is a weak minimizer for (NVOP), the following non- smooth Fritz John type necessary optimality conditions can be easily obtained from Craven [15].
The necessary nonsmooth KKT type optimality conditions for (NVOP) can be given in the following form.Theorem 3.2.If a vector 0 x S ∈ is a weak minimizer of order k with respect to ω for (NVOP) with n S = R and if Slater-type cone constraint qualification holds at x , then there exist Lagrange multipliers Proof.Assume that 0 x S ∈ is a weak minimizer of order k with respect to ω for (NVOP), then by Theo- ( )( ) ( ) So there exists ( ) Now, since Slater-type cone constraint qualification holds at x , we have for all ( ) , we get 0 t B µ ξ < .In particular, 0 t B µ ξ < .On the contrary (5) implies 0 t t B ξ µ = .This contradiction justifies 0 λ ≠ .Now, we give sufficient optimality conditions for a feasible solution to be a higher-order (weak) minimizer for (NVOP).
Theorem 3.3.Let x be a feasible solution for (NVOP) and suppose there exist vectors ( ) Further, assume that f is K-nonsmooth ( ) , F ρ -convex of order k with respect to ω at x on 0 S and g is , F σ -convex of order k with respect to the same ω at x on 0 then x is a weak minimizer of order k with respect to ω for (NVOP).
Proof.Assume on the contrary that x is not a weak minimizer of order k with respect to ω for (NVOP).
Then, for any intK , there exists a vector 0 x S ∈ such that, As intK ρ ∈ , the above relation holds in particular for β ρ = , so that we have As ( 6) holds, there exist ( ) Since f is K-nonsmooth ( ) , F ρ -convex of order k with respect to ω at x on 0 S , we have Adding ( 8) and ( 10), we get ( ) Also, since g is Q-nonsmooth ( ) , F σ convex of order k with respect to ω at x on 0 S and However, Adding ( 11) and ( 12), we get Using sublinearity of F under the assumption 0 λ > and 0 µ ≥ , we obtain ( ) which on using ( 9) and ( 1), gives ( ) This is impossible as and Q σ ∈ , so that 0 t µ σ ≥ , and norm is a non-negative function.Hence x is a weak minimizer of order k with respect to ω for (NVOP).such that ( 6) and ( 7) hold.Moreover, assume that f is K-nonsmooth ( ) , F ρ -pseudoconvex type I of order k with respect to ω at x on 0 S and g is Q -nonsmooth ( ) , F σ -quasiconvex type I of order k with respect to the same ω at x on 0

S . If intK ρ ∈
and Q σ ∈ , then x is a weak minimizer of order k with respect to ω for (NVOP).
Proof: Let if possible, x be not a weak minimizer of order k with respect to ω for (NVOP).Then, for any intK , there exists 0 x S ∈ such that, Since intK ρ ∈ taking, in particular, β ρ = in the above relation, we obtain As ( 6) holds, there exist ( ) , F ρ -pseudoconvex type I of order k with respect to ω at x on 0 S , ( Now, 0 x S ∈ means ( ) ≤ .This along with (7) gives If 0 µ ≠ , then (15) implies ( ) ( ) If 0 µ = , then also (16) holds.Now, proceeding as in Theorem 3.3, we get a contradiction.Hence, x is a weak minimizer of order k with respect to ω for (NVOP).Theorem 3.5.Assume that all the conditions of Theorem 3.3 (Theorem 3.4) hold with , 0 x is a minimizer of order k with respect to ω for (NVOP).Proof: Let if possible, x be not a minimizer of order k with respect to ω for (NVOP), then for any intK Proceeding on similar lines as in proof of Theorem 3.3 (Theorem3.4)and using (17) we have This leads to a contradiction as in Theorem 3.3 (Theorem 3.4).Hence, x is a minimizer of order k with respect to ω for (NVOP).

As ( )
., , λ µ L is K-nonsmooth ( ) , F ρ -pseudoconvex type II of order k with respect to ω , we have for all Now, since ( ) ( ) Substituting in (28) and then using (1), we get ( ) and norm is a non-negative function.Case (ii): Let 1 δ = , then we have to prove that Since x is feasible for (NVOP) and ( ) , , y λ µ is feasible for (NVUD), we have As g is Q-nonsmooth ( ) .
This contradicts the assumption that 0 hold for all feasible x for (NVOP) and all feasible ( ) , , y λ µ for (NVUD), then x is a weak maximizer of or- der k with respect to ω for (NVUD).

Conclusion
In this paper, we introduced the concept of a higher-order (weak) minimizer for a nonsmooth vector optimization problem over cones.Furthermore, to study the new solution concept, we defined new generalized classes of cone-nonsmooth (F, ρ)-convex functions and established several sufficient optimality and duality results using these classes.The results obtained in this paper will be helpful in studying the stability and convergence analysis of iterative procedures for various optimization problems.

S
⊆ R be a nonempty open subset of n R .Let m K ⊆ R be a closed convex cone with nonempty interior and let intK denote the interior of K.The dual cone K * of K is defined as said to be locally Lipschitz at a point u S ∈ if for some 0 l > , 's generalized directional derivative of ψ at u S ∈ in the direction v and is defined as

Remark 2 . 3 .
(1) If f is a scalar valued function and K + = R , the above definition reduces to the definition of a (locally Lipschitz) ( ) , F ρ -convex function of order k with respect to ω given by Antczak[6].

F 2 . 1 .
ρpseudoconvex type I of order k with respect to the same ω , however the converse may not be true as shown by the following example.Example Consider the following nonsmooth function 2 :

Remark 2 . 4 .
type II of order k with respect to ω at every u S ∈ , then f is said to be K-nonsmooth ( ) , F ρ -quasiconvex type II of order k with respect to ω on S. When f is a differentiable function, R , Definition 2.4 -2.7 take the form of the corresponding definitions given by Bhatia and Sahay[10].

Theorem 3 . 1 .
If a vector 0 x S ∈ is a weak minimizer of order k with respect to ω for (NVOP) with n S = R , then there exist Lagrange multipliers K

Theorem 3 . 4 .
Suppose there exists a feasible solution x for (NVOP) and vectors ,

Theorem 4 . 1 .
(Weak Duality) Let x be feasible for (NVOP) and ( ) , , y λ µ be feasible for (NVUD).If f is K-nonsmooth ( ) , F ρ -convex of order k with respect to ω at y on 0 S and g is Q-nonsmooth ( ) , F σ -convex of order k with respect to the same ω at y on 0

Theorem 4 . 2 .F≥
(Weak Duality) Let x be feasible for (NVOP) and ( ) ρ -pseudoconvex type II of order k with respect to ω at y on 0 , f is K-nonsmooth ( ) , F ρ -pseudoconvex type II of order k with respect to ω at y on 0 S and g is Q-nonsmooth ( ) , F σ -quasiconvex type I of order k with respect to ω at y on 0 the result.Theorem 4.3.(Strong Duality) Let x be a weak minimizer of order k with respect to ω for (NVOP) with n S = R , at which Slater-type cone constraint qualification holds.Then there exist is feasible for (NVUD).Further, if the conditions of Weak Duality Theorem 4.1 (Theorem 4.2) of order 3 at u on S.
m f S → R is said to be K-nonsmooth ( ) , F ρ -pseudoconvextype II of order k with respect to ω at u S m ρ ∈ R such that, for each ( )

,
F ρ -convex function of order k with respect to ω . 2:

,
F σ -quasiconvex type I of order k with respect to ω at y on 0