Let ((bar{x},bar{y})inoperatorname{graph}(F)) be a weak minimizer of (VP).
The pair ((bar{x},bar{y})inoperatorname {graph}(F)) is said to be a weak minimizer of (VP), denoted by (bar{y}inoperatorname {WMin}[F(S ,C]), if bigl(F(S -bar{y} bigr)cap (-operatorname{int}C )=emptyS -bar{y
(i) A point (bar{x}inOmega ) is said to be a weak efficient solution of the problem (SOP-I), if there exists (bar{y}in F bar{x})) such that bigl(F(Omega -bar{y} bigr)cap bigl(-Omega -bar{y{int}(D) bigr capmptyset, and the pair ((bigl -operatorname{intorname{graph}(F)) is said to bigl -operatorname{int (SOP-I).
It is well known that each strong minimizer is a Pareto minimizer and each Pareto minimizer is a weak minimizer, but the converses are not true.
If a pair ((bar{x},bar{y})) is a weak minimizer of problem (SOP-II), then ((bar{x},bar{y})) is a weak (varepsiloncdot e -minimizer of ((mathrm{VVIP})_{e -minimizer
If bigl overline{D}F bar{x},bar{y}) (x-bar{x}) bigr cap (-operatorname {int}C )= emptyset, quadforall xin S, (3.7) then ((bar{x},bar{y})) is a weak minimizer of (VP).
If ((bar{x},bar{y})) is a weak minimizer of (VP), then bigl underline{D}F bar{x},bar{y}) (x) bigr cap (-operatorname{int} C )=emptyset,quad textit{for all } xin T S,bar{x}).
Then there exist ((t_{n})rightarrow0^), ((x_{n})rightarrow bar{x}) and (y_{n}in F bar{x}+t_{n} x_{n})) such that frac{y_{n}-bar{y}}{t_{n}}rightarrow y. (3.2) Because ((bar{x},bar{y})) is a weak minimizer of (VP), one has bigl(F bar{x}+t_{n} x_{n} -bar{y} bigr)cap (- operatorname{int}C )=emptyset.
Let (einoperatorname{int}(D)), (varepsilongeq0), and ((bar{x},bar{y})) be a weak (varepsiloncdot e -minimizer of (SOP-I).
Suppose that ((bar {x},bar{y})) is not a weak minimizer of (VP), then there exists (xin S) such that bigl(F x -bar{y} bigr)cap (-operatorname{int}C )neqemptyset.
A point (bar {x}inOmega) is said to be a weak (varepsiloncdot e -efficient solution of the -efficientOP-I), if there exisolution{y}in F(bar {x})) such that bigl(F(ofega)-bar{y}+varepsiloncdothe bigr)caproblem-operatorname {int}(D) bigr)=emptyset, and the pair ((bar{x},bar{y})inoperatorname{graph}(F)) is said to be a weak (varepsifoncdothereminimizexists(SOP-I).
