On comparison of ℓ-stable vector optimization results

Abstract

ABSTRACT. Two papers with similar optimality conditions were published by I. Ginchev, and by D. Bednařík and K. Pastor independently in Nonlinear Analysis in 2011. We compare these results. We also show the equivalence of two definitions of -stability of vector functions. Moreover, we generalize another result given by I. Ginchev and V. I. Ivanov [J. Math. Anal. Appl. 340 (2008), 646-657] and compare it with the previously mentioned optimality conditions.

Derivatives of Hadamard type in scalar constrained optimization

“…If the functions f and g i , i ∈ I(x 0 ), are -stable at x 0 , then also the function L is -stable at x 0 by Lemma 3 from [12]. Now, we can formulate the following sufficient optimality condition for the problem (6) when the considered functions are -stable at the feasible point.…”

“…The properties of scalar or vector functions that are -stable at some point functions and their applications in optimization were studied e. g. in [1,2,3,4,5,6,7,8,9,12,13,15,16,24,25,28,30,31]. Let us remind that a second-order sufficient optimality condition for the constrained scalar problem for -stable at some point function in terms of Dini derivatives was introduced in [28].…”

Necessary conditions for vector optimization in infinite dimension

“…The properties of −stable at some point functions were studied e. g. in [1,4,5,6,7,8,11,12,14,15,28,29,30] for both scalar and vector functions. Among the others, the sufficient second-order optimality condition for problem (1) was stated indepedently in [14] and [8] in terms of −stable at some point functions.…”

“…The obtained results were surpassed in 2011, when I. Ginchev [14] and D. Bednařík with K. Pastor [8] published indepedently the following equivalent result (Theorem 1.1). We recall that the equivalence was shown in [11].…”