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Erratum to “Huard type second-order converse duality for nonlinear programming” [Appl. Math. Lett. 18 (2005) 205–208]
Applied Mathematics Letters 20 (2007) 241 www.elsevier.com/locate/aml
Erratum
Erratum to “Huard type second-order converse duality for nonlinear programming” [Appl. Math. Lett. 18 (2005) 205–208] X.M. Yang a,∗ , X.Q. Yang b , K.L. Teo c a Department of Mathematics, Chongqing Normal University, Chongqing 400047, China b Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China c Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6102, Australia
Received 14 March 2006; accepted 16 March 2006
Abstract T.R. Gulati and Divya Agarwal pointed out that the statement: α > 0 and θ = rα∗ imply θ > 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 θ = rα∗ is obtained letting α = 0. Here is a simple proof that θ > 0. c 2005 Elsevier Ltd. All rights reserved.
We proved that α > 0. If θ = 0, then from θ p∗ + β = 0 (see (15) in [1]), we have β = 0. Thus, (α − r ∗ θ )[∇ y ∗ T g(x ∗ ) + ∇ 2 (y ∗ T g(x ∗ )) p∗ ] + 12 (βr ∗ )T ∇(∇ 2 ( f (x ∗ ))) + ∇ 2 (y ∗ T g(x ∗ )) p ∗ = 0 (see (18) in [1]) yields α[∇ y ∗ T g(x ∗ ) + ∇ 2 (y ∗ T g(x ∗ )) p ∗ ] = 0 which contradicts the assumption (A2) and α > 0. Hence, θ > 0. References [1] X.M. Yang, X.Q. Yang, K.L. Teo, Huard type second-order converse duality for nonlinear programming, Appl. Math. Lett. 18 (2005) 205–208.
DOI of original article: 10.1016/j.aml.2004.04.008.
∗ Corresponding author.
E-mail address: [email protected] (X.M. Yang). c 2005 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.03.015
Erratum
Erratum to “Huard type second-order converse duality for nonlinear programming” [Appl. Math. Lett. 18 (2005) 205–208] X.M. Yang a,∗ , X.Q. Yang b , K.L. Teo c a Department of Mathematics, Chongqing Normal University, Chongqing 400047, China b Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China c Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6102, Australia
Received 14 March 2006; accepted 16 March 2006
Abstract T.R. Gulati and Divya Agarwal pointed out that the statement: α > 0 and θ = rα∗ imply θ > 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 θ = rα∗ is obtained letting α = 0. Here is a simple proof that θ > 0. c 2005 Elsevier Ltd. All rights reserved.
We proved that α > 0. If θ = 0, then from θ p∗ + β = 0 (see (15) in [1]), we have β = 0. Thus, (α − r ∗ θ )[∇ y ∗ T g(x ∗ ) + ∇ 2 (y ∗ T g(x ∗ )) p∗ ] + 12 (βr ∗ )T ∇(∇ 2 ( f (x ∗ ))) + ∇ 2 (y ∗ T g(x ∗ )) p ∗ = 0 (see (18) in [1]) yields α[∇ y ∗ T g(x ∗ ) + ∇ 2 (y ∗ T g(x ∗ )) p ∗ ] = 0 which contradicts the assumption (A2) and α > 0. Hence, θ > 0. References [1] X.M. Yang, X.Q. Yang, K.L. Teo, Huard type second-order converse duality for nonlinear programming, Appl. Math. Lett. 18 (2005) 205–208.
DOI of original article: 10.1016/j.aml.2004.04.008.
∗ Corresponding author.
E-mail address: [email protected] (X.M. Yang). c 2005 Elsevier Ltd. All rights reserved. 0893-9659/$ - see front matter doi:10.1016/j.aml.2006.03.015