Optimality Conditions for Generalized Ky Fan Quasi-Inequalities with Applications

Scalarizations and Optimality of Constrained Set-Valued Optimization Using Improvement Sets and Image Space Analysis

“…We observe that, in the case where Y = R, Z = R m and E = R + \{0}, w 2 reduces to another version, which was discussed in [21]. Proof The proofs that w 1 and w 2 are weak separation functions are similar to Proposition 3.1(i) in [22] and Proposition 3.1(i) in [21], respectively. Definition 4.…”

“…, which was studied in [22]. We observe that, in the case where Y = R, Z = R m and E = R + \{0}, w 2 reduces to another version, which was discussed in [21].…”

“…According to (17) and ( 20), we have K g-lev y ( x, ȳ) ∩ H = ∅, which contradicts (16). Proof Since K ( x, ȳ) and H admit a nonlinear regular separation with respect to w 2 , then there exists ( φ, λ) ∈ E × R + such that (22) holds. For any (u, v) ∈ H, by Proposition 4.3(ii), (iii), we obtain φ(u) − λ D (−v) > 0.…”

“…Moreover, the disjunction between the two suitable sets can be proved by verifying that they lie in two disjoint level sets of a suitable separation function. Various linear or nonlinear functions were proposed in [12,13,18,[20][21][22][23][24][25] and then were proved to be weak separation functions or regular weak separation functions or strong separation functions. Recently, Chinaie and Zafarani [26] have proposed two kinds of separation functions, which unify the known linear or nonlinear separation functions.…”

“…Recently, Chinaie and Zafarani [26] have proposed two kinds of separation functions, which unify the known linear or nonlinear separation functions. By virtue of ISA, some authors have investigated regularity, duality and optimality conditions for constrained extremum problems (see [12,13,18,[20][21][22][23][24][26][27][28][29][30][31] and the references therein). It is noteworthy that the authors in [12] used the improvement set and the oriented/signed distance function to introduce a nonlinear vector regular weak separation functions and a nonlinear scalar weak separation function, and obtained Lagrangian-type optimality conditions in the sense of vector separation and scalar separation, respectively.…”