Sentence examples for efficient element of the from inspiring English sources

end{aligned}Kasimbeyli in [11] showed that if there exists a pair ((y^,alpha )in C^{a#}) such that begin{aligned} langle y^, hat{y}rangle + alpha Vert hat{y}Vert le langle y^, yrangle + alpha Vert yVert, end{aligned}for all (yin Y) then (hat{y}) is a properly efficient element of the optimization problem 1.

The vector criterion is the most commonly used in problem (P), which is looking for efficient elements of the set (F(S =bigcup_{xin S} F x)).

Some properties of second-order tangent derivatives are discussed, using which second-order necessary optimality conditions are established for a point pair to be a Henig efficient element of a set-valued optimization problem, and in the expressions the second-order tangent derivatives of the objective function and the constraint function are separated.

Just applying these properties, we established second-order necessary optimality conditions for a point pair to be a Henig efficient element of a set-valued optimization problem where the second-order tangent derivatives of the objective function and constraint function are separated.

Suppose that ((overline{x}, overline{y})) is not a S-super efficient element of (VP).

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

Therefore, ((overline {x},overline{y})) is a S-super efficient element of (VP).

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

The point pair (langle overline{x},overline{y}rangle) is called a S-super efficient element of (VP).

If there exists (overline{T}in L^(Z, Y)) such that ((overline {x},overline{y})) is a S-super efficient element of ((mathrm{UVP})_{overline{T}}), then ((overline{x},overline{y})) is a S-super efficient element of (VP).

Suppose that ((hat{x},hat{y})) is a Henig efficient element of (P) and (g(hat{x} in-D).