Let (hat{y}in Y) be a properly efficient solution in sense Geoffrion definition.
Theorem 4.3 Let ( y ¯ be a properly efficient solution in problem (MWDP) and y ¯ ∈ Γ ( Ω t 0, t 1 ).
Let (hat{y}) be a properly efficient solution of (1) and let ((y^, alpha )in (mathbb {R}^{p}_)^{a#}) be a positive weighted vector for obtaining (hat{y}).
Let ( y ¯ be a properly efficient solution in the Wolfe dual problem (WDP) and y ¯ ∈ Γ ( Ω t 0, t 1 ).
Let (hat{y}) be a properly efficient solution of (1) and let (y^) be a positive weighted vector for obtaining (hat{y}).
Let (E Rtimes[0,1]to R) be a mapping such that (E t,lambda)=lambdamin{ t,lambda} ), (t in R), (lambdain[0,1]), and (x^) be a properly efficient solution for problem (P).
(b) A feasible solution x0 is said to be a properly efficient solution of (P) if it is an efficient solution of (P), and there exists a real number M > 0 such that for all i ∈ {1,..., m}, x ∈ S, and f i (x) < f i (x0), f i ( x 0 ) - f i ( x ) ≦ M ( f j ( x ) - f j ( x 0 ) ) .
[1] The feasible element (hat{y}in mathbb {R}^{p}) is said to be a properly efficient element in Benson's sense of (1) if begin{aligned} {rm cl}({rm cone}(Y+mathbb {R}^{p}_-hat{y}))cap (-mathbb {R}^{p}_backslash {0_{mathbb {R}^{p}}})=emptyset. end{aligned}.
A feasible solution x0 is said to be a properly efficient solution of (P) if it is an efficient solution of (P), and there exists a real number M > 0 such that for all i ∈ {1,..., m}, x ∈ S, and f i (x) < f i (x0), f i ( x 0 ) - f i ( x ) ≦ M ( f j ( x ) - f j ( x 0 ) ). for some j ∈ {1,..., m} such that f j (x) > f j (x0).
(iii) is said to be a local properly -efficient solution of problem (2.1), if there exists a neighborhood of such that.
