Duality and saddle-points for convex-like vector optimization problems on real linear spaces

Abstract

Vector optimization, duality, saddle-points, vector convexlikeness, weak efficiency, proper efficiency, 90C29, 90C46,

An Inequality Approach to Approximate Solutions of Set Optimization Problems in Real Linear Spaces

“…This is not only mathematically interesting, but deepens the theoretical understanding of approximate minimality in set optimization. It is furthermore in line with the recent increased interest in studying optimality conditions and separation concepts in spaces without a particular topology underneath it, see [12][13][14][15][16][17][18][19][20][21] and the references therein.…”

“…Now let us assume, by contradiction, that A j / ∈ V. Then, there exists some A ∈ A \ U with A A j (A + H A j , respectively), but A j A + H, a contradiction to (13).…”

E-Saddle Point Criteria for E-differentiable Vector Optimization Problems with Inequality and Equality Constraints

“…But probably the most important one is related to the result that a saddle point of the Lagrange function is always a global optimal solution of a constrained optimization problem. Because of the importance of this result, for example, in optimization theory and economics, therefore, many authors have analyzed and studied theory of saddle point criteria for nonconvex multiobjective programming problems (see, for instance, (Adán and Novo, 2005;Antczak, 2003;2005;2015;Bhatia, 2008;Bigi, 2001;Craven, 1990;Ehrgott and Wiecek, 2005;Jiang and Xu, 2010;Kuk et al, 1998;Kumar and Garg, 2015;Maciel et al, 2016;Mishra and Giorgi, 2008;Li and Wang, 1994;Van Rooyen et al, 1994;Tanaka, 1990;1994;Tanino, 1982;Vályi, 1987;Varalakshmi et al, 2010;Yan and Li, 2004;Zeng, 2017). Taninio (1982) proved that solutions of multicriteria optimization problems and corresponding multiplier vectors are saddle points of vector-valued Lagrange functions.…”

ϵ‐Henig Saddle Points and Duality of Set‐Valued Optimization Problems in Real Linear Spaces

“…How to generalize saddle point theorems and duality theorems of set-valued optimization from locally convex spaces to linear spaces is interesting. Adán and Novo [ 5 ] studied saddle points and duality for convexlike vector optimization problems in real linear spaces. In the ϵ -global prober efficiency, Zhou et al [ 6 ] introduced the concept of the ϵ -global prober saddle point and obtained the relationships between the ϵ -global proper saddle points of Lagrangian set-valued maps and the ϵ -global properly efficient element of set-valued optimization problems.…”