Strict Efficiency in Vector Optimization with Nearly Convexlike Set-Valued Maps

and Applied Analysis 3 (3) Δ S (y) < 0 for every y ∈ intA, Δ S (y) = 0 for every y ∈ ∂A, and Δ S (y) > 0 for every y ∈ int S; (4) if S is closed, then it holds that S = {y : Δ S (y) ≤ 0}; (5) if S is a convex, then Δ S is convex; (6) if S is a cone, then Δ S is positively homogeneous; (7) if S is a closed convex cone, then Δ S is nonincreasing with respect to the ordering relation induced on Y. We consider the following parameterized scalar problem: minΔ −C (y − y) s.t. y ∈ F (A) . (P y ) The following theorem characterize the relation between strictly efficient points of (VP) and the parameterized scalar problem (P y ). Theorem 9. Let x ∈ A, y ∈ F(x). Then (x, y) is a strictly efficient minimizer of (VP) if and only if there exists a nondecreasing function φ : R + → R + with φ(0) = 0 and φ(t) > 0 for all t > 0, such that Δ −C (y − y) ≥ φ(‖y − y‖) for all y ∈ F(A). Proof. Since (x, y) is a strictly efficient minimizer of vector optimization problem (VP) can be rephrased as follows: for every ε > 0 there exists δ > 0 such that d −C (y − y) ≥ δ for every y ∈ F(A) with ‖y − y‖ > ε. So suppose that the point (x, y) is a strictly efficientminimizer of (VP) and consider the following functions: φ 0 (ε) = inf {d −C (y − y) | y ∈ F (A) , 󵄩󵄩󵄩󵄩y − y 󵄩󵄩󵄩󵄩 ≥ ε} , φ (ε) = min (φ 0 (ε) , 1) . (12) It is evident that φ is nondecreasing, null at the origin, and positive elsewhere; moreover, for every y ∈ F(A) it holds that Δ −C (y − y) = d −C (y − y) ≥ φ ( 󵄩󵄩󵄩󵄩y − y 󵄩󵄩󵄩󵄩) . (13) If, on the other hand, there exists a nondecreasing function φ with the above properties and such that Δ −C (y −y) ≥ φ(‖y − y‖) for all y ∈ F(A), then it holds that Δ −C (y − y) = d −C (y − y) > 0 for all y ∈ F(A) with y ̸ = y. To show that (x, y) is a strictly efficient minimizer of (VP), for every ε > 0, we can let δ = inf{d −C (y − y) : ‖y − y‖ > ε}, and it implies that the proof is completed. The scalar problem (P y ) is Tikhonov well posed if Δ −C (y − y) > 0 for all y ∈ F(A) with y ̸ = y and y n ∈ F (A) , d −C (y n − y) 󳨀→ 0 󳨐⇒ y n 󳨀→ y. (14) Theorem 10. Let x ∈ A, y ∈ F(x). The (x, y) is a strictly efficient minimizer of (VP) if and only if y is a solution of (P y ) and the scalar problem (P y ) is Tikhonov well posed. Proof. If (x, y) is a strictly efficient minimizer of (VP), then, by Theorem 9, y is the unique solution of (P y ) and there exists a forcing function φ such that Δ −C (y−y) ≥ φ(‖y−y‖), for all y ∈ F(A). Since φ(t n ) → 0 implies t n → 0, hence for any sequence y n ∈ F(A) such that Δ −C (y n − y) → 0, then it must converge to y. Conversely, if the scalar problem (P y ) is Tikhonov well posed, thend −C (y−y) = Δ −C (y−y)holds for everyy ∈ F(A). Thus, we consider the function φ(ε) = inf{d −C (y − y) : ‖y − y‖ ≥ ε}; it holds by the construction that d −C (y−y ≥ φ(y−y), and it is to see that φ is nondecreasing on [0,∞) with φ(0) = 0 and φ(t) > 0 for all t > 0. Hence, again, by theTheorem 9, we get that (x, y) is a strictly efficient minimizer of (VP). 4. Strict Efficiency and Linear Scalarization In association with the vector optimization problem (VP) involving set-valuedmaps, we consider the following linearly scalar optimization problem with a set-valued map: min (φF) (x) s.t. x ∈ A, (LSP φ ) where φ ∈ Y∗ \ {0 Y ∗}. Definition 11. If x ∈ A, y ∈ F(A) and φ (y) ≤ φ (y) , ∀y ∈ F (A) , (15) then x and (x, y) are called a minimal solution and a minimizer of (LSP φ ), respectively. Lemma 12. Let y ∈ S ⊂ Y, and C is a closed convex cone with having a compact base Θ. Then y is a strictly efficient point of S if and only if cl (S + C − y) ∩ −C = {0 Y }. Proof. The definition of strict efficiency of S can be rephrased as follows: for every ε > 0, there exists δ > 0 such that d −C (y− y) > δ for every y ∈ S with ‖y − y‖ > ε. Hence, if there exists sequence y n ∈ S, c n ∈ C, and some c ∈ int C such that y n + c n − y → −c, then y n + c n + c − y → 0; hence, d −C (y − y) → 0. Since y n − y = −(c n + c), c ̸ = 0, and C is pointed, y n − y is outside some small ball around the origin. This shows that y is not the strictly efficient point of S, this contraction shows that cl (S + C − y) ∩ −C = {0 Y }. Conversely, if y is not a strictly efficient point of S, then there exists ε > 0 and a sequence y n ∈ S, and c n ∈ C such that 󵄩󵄩󵄩󵄩yn − y 󵄩󵄩󵄩󵄩 ≥ ε, 󵄩󵄩󵄩󵄩yn + cn − y 󵄩󵄩󵄩󵄩 󳨀→ 0. (16) We write c n = λ n θ n with λ n > 0 and θ n ∈ Θ, then, by (16) and asΘ is compact, there exists α > 0 andN ∈ R+ such that α < λ n . Indeed, by (16), we have c n ∉ (ε/2)B; furthermore, since Θ is compact, thus λ n does not converse to 0, and it implies that there exists a real number α > 0 and N ∈ R+ such that α < λ n for all n ≥ N. Now, we define c󸀠 n = (λ n −α)θ n , n ≥ N. Thus, we obtain