Nonlinear separation concerning E-optimal solution of constrained multi-objective optimization problems

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

“…Clearly, x t ∈ U for sufficiently small t > 0 and it follows from Remark 2.5 that x t ∈ S. According to y ∈ F( x) + E, ȳ ∈ F(x t ) + E and Condition (iii), we deduce that y ∈ F(x t ) + E, that is, x t ∈ F −1 (y − E). Therefore, (12) holds for sufficiently small t > 0. The combination of ( 11) and ( 12) contradicts the fact that ( x, ȳ) is a strict local optimal point of ϕ-SVOP.…”

“…The E-optimal solution unifies some known exact and approximate solutions in vector optimization problems. So far, some applications of improvement sets in vector optimization are investigated (see [1][2][3][4][5][6][7][8][9][10][11][12][13] and the references therein). It is worth noticing that Zhao et al [5] used the improvement set to introduce E-optimal solution and weak E-optimal solution of the constrained setvalued optimization problem (for short, SVOP) and established scalarization theorems and Lagrange multiplier theorems of weak E-optimal solution under the assumption of near E-subconvexlikeness.…”

“…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.…”

“…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. In [13], Chen et al also defined a vector-valued regular weak separation function and a scalar weak separation function via the improvement set and obtained some optimality conditions in terms of Eoptimal solution for multiobjective optimization with general constraints.…”