(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 ) ) .
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).
If also all the hypotheses of Theorem 5.1 are satisfied, then ( x ¯ is a properly efficient solution in (WDP).
[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}.
If also all the hypotheses of Theorem 4.1 are satisfied, and λ ¯ > 0, then ( x ¯ is a properly efficient solution in (MWDP).
(14) If (f_{i} ), (i=1,2,ldots,k), and (g_{i} ), (iin I(x^ )), are differentiable (Embox[0,1]) convex functions at (x^ in M), then (x^ ) is a properly efficient solution for problem (P).
Because (hat{y}) is a properly efficient solution, for any (yin Y) and for any (iin {1,2,ldots,p}) with (y_{i}
On the other hand, it is clear that this scalarization generalizes the linear scalarization, setting (alpha =0) in the problem 18. [11, 12] Assume that (hat{y}) is a properly efficient element of (1).
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).
