Separation Functions and Optimality Conditions in Vector Optimization

Weak separation functions constructed by Gerstewitz and topical functions with applications in conjugate duality

“…The Gerstewitz function has been applied for introducing different separation functions in literatures like [50,51]. Here, we want to further exploit the capability of the Gerstewitz function for constructing separation functions.…”

Approximate Benson efficient solutions for set-valued equilibrium problems

“…It is widely used in investment decision-making, quantitative economy, optimal control, and engineering technology. Because of the universality and unity of the problems involved and the profundity of solving them, vector equilibrium has become a hot issue in the field of nonlinear analysis and operational research [1][2][3][4][5][6]. In Banach spaces, Feng et al [1] established Kuhn-Tucker-like conditions for weakly efficient solutions of vector equilibrium problems with constraints by using the Gerstewitz's functional, and obtained sufficient conditions of weakly efficient solutions under the assumption of generalized invexity.…”

“…In Banach spaces, Feng et al [1] established Kuhn-Tucker-like conditions for weakly efficient solutions of vector equilibrium problems with constraints by using the Gerstewitz's functional, and obtained sufficient conditions of weakly efficient solutions under the assumption of generalized invexity. You et al [2] established Lagrangian-type sufficient optimality conditions for general constrained vector optimization problems by applying Gerstewitz's function, and, under suitable restriction qualifications, by virtue of Clarke subdifferentials, they obtained Karush-Kuhn-Tucker necessary conditions. Luu et al [3] derived necessary conditions for efficient solutions to vector equilibrium problems with equality and inequality constraints.…”

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

“…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.…”