Sentence examples for be a properly efficient from inspiring English sources

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).

This means that ( x ¯ is a properly efficient solution in problem (MWDP).

Then y ¯ is a properly efficient solution in the considered multitime multiobjective variational problem (MVP).

Then, (i) Assume that (hat{y}) is a properly efficient element of (1).

Hence, from Theorem 4.11 of [12], (x^ ) is a properly efficient solution for problem (P).

Assume that (hat{y}) is a properly efficient element of (1).