When ε = 0, then the weak ε- efficiency becomes the weak efficiency for (MFP) (see the definition of efficient solution of a multiobjective optimization problem in [21]).
Besides the weak efficiency and efficiency mentioned above, a meaningful solution concept called a strict efficient solution of higher-order (also called a strict minimizer of higher-order) was recently extended by Jiménez in [10] from the strict minimizer of higher-order in scalar optimization given by Auslender in [11] and Ward in [12].
Furthermore, the minimal element is introduced in the paper to distinguish the weak efficiency and efficiency.
Moreover, the partially ordered set and minimal element are used to distinguish the weak efficiency and efficiency.
In order to distinguish the weak efficiency and efficiency, the partially ordered set and minimal element are introduced in the paper.
For fixed x ∈ R n, let S ( x ) denote the weak efficiency set of solutions to the lower level problem ( P x ) : min y D y, s. t.
In this paper, we establish sufficient optimality conditions for the (weak) efficiency to multiobjective fractional programming problems involving generalized ( F, α, ρ, d ) -V-type I functions.
On the other hand, if there exists a feasible point x ∈ X such that f i ( x ) g i ( x ) < f i ( x * ) g i ( x * ) = u i *, then f i ( x ) - u i * g i ( x ) < 0 = f i ( x * ) - u i * g i ( x * ), ∀ i ∈ P, and this is a contradiction to <span class="lh">the weak efficiency of x* for (VFP u *).
Suppose that there exists a feasible point x ∈ X, such that for each i ∈ P, f i ( x ) - u i * g i ( x ) < f i ( x * ) - u i * g i ( x * ) ⇒ f i ( x ) - u i * g i ( x ) < f i ( x * ) - f i ( x * ) g i ( x * ) g i ( x * ) = 0 ⇒ f i ( x ) < f i ( x * ) g i ( x * ) g i ( x ) ⇒ f i ( x ) g i ( x ) < f i ( x * ) g i ( x * ), which contradicts <span class="lh">the weak efficiency of x* for (VFP).
To improve cancer immunotherapy, a better understanding of the weak efficiency of tumor-infiltrating T lymphocytes (TIL) is necessary.
