Second-order optimality and duality in vector optimization over cones

In this paper, we introduce the notion of a second-order coneconvex function involving second-order directional derivative. Also, second-order cone-pseudoconvex, second-order cone-quasiconvex and other related functions are defined. Second-order optimality and Mond-Weir type duality results are derived for a vector optimization problem over cones using the introduced classes of functions.


Introduction
Generalized convexity notions have always been a significant aid in the progress of optimization theory.In 1981 Hanson [6] generalized convexity to invexity.Kaul and Kaur [11] named differentiable invex functions as η-convex and further generalized them to η-pseudoconvex and η-quasiconvex functions.Craven [5] extended the concept of convex functions to cone-convex functions.Recently Ivanov [7,8,9] illuminated the following definition of a second-order directional derivative obtained by solving the Taylor expansion formula of a function with respect to the second-order term.Definition 1.1 ([7, 8, 9]) Let S be a nonempty open subset of R n and f : S → R be a differentiable function.The second-order directional derivative f ′′ (x, d) of f at the point x ∈ S in the direction d ∈ R n is defined as an element of R given by If f ′′ (x, d) exists and is finite, then f is said to be second-order directionally differentiable at the point x ∈ S in the direction d ∈ R n and f ′′ (x, d) is called its second-order directional derivative.The function f is said to be second-order directionally differentiable on S if the derivative f ′′ (x, d) exists for each x ∈ S and every direction d ∈ R n .

SECOND-ORDER OPTIMALITY AND DUALITY IN VECTOR OPTIMIZATION OVER
Ivanov [8] used the above definition of second-order directional derivative to introduce a second-order invex function in the following manner: The function f is said to be second-order invex (or 2-invex) at x ∈ S if there exist vector-valued functions η, ξ : S × S → R n such that for all y ∈ S, the second-order directional derivative f ′′ (x, ξ(x, y)) exists and If the above inequality holds for all x, y ∈ S, then f is called second-order invex on S.
In this paper, we unify the notions of η-convex functions [11] and second-order invex functions [8] to define a new class of second-order cone-(η, ξ)-convex functions.We also define related classes of second-order conepseudoconvex and second-order cone-quasiconvex functions.
Second-order optimality conditions for vector optimization problems have been widely studied in the past, mainly due to their usefulness in sensitivity analysis of optimal solutions and convergence analysis of various algorithms.Several researchers, like Andreani [1], Ben-Tal [2], Burke [4], Kawasaki [12], have considered secondorder optimality conditions in terms of Hessians of the involved functions.However, various kinds of second-order directional derivatives have also been introduced to enable the development of second-order optimality conditions in the absence of second-order differentiability (see for example Ben-Tal, Zowe [3], Ivanov [7,8,9], Studniarski [15], Yang [16] and the references therein).
We employ the introduced classes of functions to obtain second-order necessary and sufficient Karush-Kuhn-Tucker (KKT) type conditions for a vector optimization problem over cones in terms of the second-order directional derivatives of the functions involved.Furthermore, a second-order Mond-Weir type dual is associated to the considered problem and weak and strong duality results are established.

Second-order cone-convexity and related concepts
In this section, we introduce the following new classes of second-order cone-(η, ξ)-convex functions.
Let S be a nonempty open subset of R n and f = (f 1 , . . ., f m ) t : S → R m be a differentiable vector-valued function.

Definition 2.1
The function f is said to be second-order K-(η, ξ)-convex at x ∈ S on S if there exist vector-valued functions η, ξ : S × S → R n such that for all x ∈ S, the second-order directional derivative f ′′ (x, ξ(x, x)) exists and (ii) If ξ ≡ 0 then Definition 2.1 becomes the definition of K-invexity given by Yen and Sach [17].
(iii) If f is a scalar valued function, K = R + and ξ ≡ 0 then Definition 2.1 becomes the definition of η-convexity given by Kaul and Kaur [11].
(iv) If ξ ≡ 0 and η(x, y) = x − y, x, y ∈ S, then the above definition reduces to the definition of cone-convex functions introduced in [5].
To justify the introduction of second-order K-(η, ξ)-convex functions, we give an example of a function which is second-order and η is an ) / ∈ K .

Definition 2.4
The function f is said to be second-order K-(η, ξ)-pseudoconvex at x ∈ S on S if there exist vector-valued functions η, ξ : S × S → R n such that for all x ∈ S, the second-order directional derivative f ′′ (x, ξ(x, x)) exists and Remark 2.5 (i) If f is twice differentiable and ξ = η, then the above definition becomes the definition of a Ksecond order pseudoinvex function with respect to η at x for p = η(x, x) ∈ R n introduced by Mishra and Lai [14].
(iii) If f is a scalar valued function, K = R + and ξ ≡ 0 then Definition 2.4 reduces to the definition of η-pseudo convexity given by Kaul and Kaur [11].It is clear that every second-order K-(η, ξ)-convex function is K-(η, ξ)-pseudoconvex.However the converse may not be true as shown by the following example.

SECOND-ORDER OPTIMALITY AND DUALITY IN VECTOR OPTIMIZATION OVER CONES
Remark 2.7 In the above example, f is a vector-valued function which is not twice differentiable.Hence, f is not K-second order pseudoinvex as defined by Mishra and Lai [14].Further, if ξ ≡ 0, then f is not K-pseudoinvex in the sense of Khurana [13].Therefore Definition 2.4 widens the field of applications of generalized convex functions.

Definition 2.8
The function f is said to be second-order K-(η, ξ)-strictly pseudoconvex at x ∈ S on S if there exist vector-valued functions η, ξ : S × S → R n such that for all x ∈ S, f ′′ (x, ξ(x, x)) exists and Definition 2.9 The function f is said to be second-order K-(η, ξ)-quasiconvex at x ∈ S on S if there exist vector-valued functions η, ξ : S × S → R n such that for all x ∈ S, f ′′ (x, ξ(x, x)) exists and If f is second-order K-(η, ξ)-convex (pseudoconvex, strictly pseudoconvex, quasiconvex) at every x ∈ S on S then f is said to be second-order K-(η, ξ)-convex (pseudoconvex, strictly pseudoconvex, quasiconvex) on S.
We shall study the following vector optimization problem (VOP) over cones: where f : S → R m and g : S → R p are differentiable functions and K ⊆ R m , Q ⊆ R p are closed convex cones with nonempty interior.Let S 0 = {x ∈ S : −g(x) ∈ Q} denote the set of feasible solutions to (VOP).

Definition 2.10
Let K ⊆ R m be a closed convex pointed cone with nonempty interior and let int K denote the interior of K.The positive dual cone K * and the strict positive dual cone K s * of K, are respectively defined as

Definition 2.11
A point x ∈ S 0 is said to be (i) a weak minimum of (VOP) if for every x ∈ S 0 ,

Second-order necessary conditions over cones
We now prove second-order necessary optimality conditions for the problem (VOP) in terms of second-order directional derivatives.

Theorem 3.1
Let x be a weak minimum of (VOP).If We assert that the system If possible, let there be a solution ( d1 , d2 ) ∈ R n × R n of (3).Then, and and −[∇g(x) d1 + lim and where lim Since S is a nonempty open set, we can find s 0 > 0 such that for all s ∈ (0, s 0 ), x + s d2 ∈ S, and which is a contradiction as x is a weak minimum of (VOP).Hence the system (3) has no solution Therefore by the Alternative Theorem given in Jeyakumar [10], there exist λ ∈ K * , µ ∈ Q * not both zero such that Taking Hence, from (4) we obtain Now we give an example to illustrate the result obtained in Theorem 3.1.Clearly x = (0, 0) is a weak minimum for the problem, ) ) , then for all We now introduce the following second-order Slater-type constraint qualification over cones.

Definition 3.3
The problem (VOP) is said to satisfy second-order Slater-type cone-constraint qualification at x if g is Q-(η, ξ)convex at x and there exists x ∈ S such that −g(x) ∈ int Q.

Theorem 3.4
Let x be a weak minimum of (VOP) at which second-order Slater-type cone-constraint qualification holds.If 1) and (2) hold.
Since second-order Slater-type cone-constraint qualification holds at x, g is Q-(η, ξ)-convex at x and there exists x ∈ S such that −g(x) ∈ int Q.We have to prove that λ ̸ = 0.
Let if possible λ = 0, then µ ̸ = 0 and from (1) we get Also since g is Q-(η, ξ)-convex at x there exist vector-valued functions η, ξ : S × S → R n such that for all x ∈ S, the second-order directional derivative g ′′ (x, ξ(x, x)) exists and Using (2) and ( 5) Theorem 3.5 Let x be a weak minimum of (VOP) at which second-order Slater-type cone-constraint qualification holds.If Proof Since all the conditions of Theorem 3.4 hold, there exist λ ∈ K * \ {0} and µ ∈ Q * such that (1) holds along with µ t g(x) = 0.
Taking d 2 = 0 in (1), we have As the above inequality holds for all d 1 ∈ R n , we get Again, taking d 1 = 0 in (1), we obtain Or, This completes the proof.

Second-order sufficient optimality conditions over cones
We now provide several second-order sufficient conditions for the existence of a weak minimum or minimum for (VOP).
a scalar valued function and K = R + , the above definition reduces to second-order invexity introduced by Ivanov[8].