Erratum to “Huard type second-order converse duality for nonlinear programming” [Appl. Math. Lett. 18 (2005) 205–208]

T.R. Gulati and Divya Agarwal pointed out that the statement: α>0α>0 and View the MathML sourceθ=αr∗ imply θ>0θ>0, on line 4 of page 208 of [X.M. Yang, X.Q. Yang and K.L. Teo, Appl. Math. Lett. 18 (2005) 205–208], is erroneous since View the MathML sourceθ=αr∗ is obtained letting α=0α=0. Here is a simple proof that θ>0θ>0. We proved that α>0α>0. If θ=0θ=0, then from θp∗+β=0θp∗+β=0 (see (15) in [1]), we have β=0β=0. Thus, View the MathML source(α−r∗θ)[∇y∗Tg(x∗)+∇2(y∗Tg(x∗))p∗]+12(βr∗)T∇(∇2(f(x∗)))+∇2(y∗Tg(x∗))p∗=0 (see (18) in [1]) yields α[∇y∗Tg(x∗)+∇2(y∗Tg(x∗))p∗]=0α[∇y∗Tg(x∗)+∇2(y∗Tg(x∗))p∗]=0 which contradicts the assumption (A2) and α>0α>0. Hence, θ>0θ>0.