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# Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization

The Original Article was published on 25 August 2011

We wish to indicate the following corrections to our original paper [1].

(1) The first sentence in Definition 2.2, we delete "iâˆˆ{1,2,Â·Â·Â·,p}".

(2) The first sentence in Definition 2.2, we replace

${f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)â‰®{f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)$

to

$f\left(x\right)+s\left(x|D\right)â‰®f\left({x}^{0}\right)+s\left({x}^{0}|D\right).$

(3) The second sentence in Definition 2.3, we delete "iâˆˆ{1,2,Â·Â·Â·,p}".

(4) The second sentence in Definition 2.3, we replace

${f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)â‰®{f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)+{c}_{i}||x-{x}^{0}|{|}^{m}$

to

$f\left(x\right)+s\left(x|D\right)â‰®f\left({x}^{0}\right)+s\left({x}^{0}|D\right)+c||x-{x}^{0}|{|}^{m}.$

(5) The second sentence in Definition 2.4, we delete "iâˆˆ{1,2,Â·Â·Â·,p}".

(6) The second sentence in Definition 2.4, we replace

$\mathrm{â€²}\mathrm{â€²}{f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)â‰®{f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)+{c}_{i}||x-{x}^{0}|{|}^{m}$

to

$f\left(x\right)+s\left(x|D\right)â‰®f\left({x}^{0}\right)+s\left({x}^{0}|D\right)+c||x-{x}^{0}|{|}^{m}.\mathrm{â€²}\mathrm{â€²}.$

(7) The second sentence in the proof of Theorem 2.1, we replace. â€²â€² c i > 0, i = 1, â‹¯,p â€²â€² to â€²â€² $câˆˆint{R}_{+}^{p}$â€²â€².

(8) The second sentence in the proof of Theorem 2.1, we replace

${f}_{i}\left(x\right)+s\left(x|{D}_{i}\right)â‰®{f}_{i}\left({x}^{0}\right)+s\left({x}^{0}|{D}_{i}\right)+{c}_{i}||x-{x}^{0}|{|}^{m}$

to

$f\left(x\right)+s\left(x|D\right)â‰®f\left({x}^{0}\right)+s\left({x}^{0}|D\right)+c||x-{x}^{0}|{|}^{m}.$

(9) In equation (3.8), we replace "c i " to "d i ".

(10) The eighth sentence in the proof of Theorem 3.3, we replace "where c = ae," to "where d = ae".

(11) The ninth sentence in the proof of Theorem 3.3, we replace "c âˆˆ int Rp" to "d i > 0, i = 1,Â·Â·Â·,p,".

(12) The ninth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".

(13) The tenth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".

(14) The tenth sentence in the proof of Theorem 3.3, we replace "c i " to "d i ".

(15) In equation (4.8), we replace "c i " to "d i ".

(16) The tenth sentence in the proof of Theorem 4.1, we replace "where c = ae," to "where d = ae".

(17) The eleventh sentence in the proof of Theorem 4.1, we replace "c âˆˆ int Rp" to "d i > 0, i = 1,Â·Â·Â·, p,".

(18) The eleventh sentence in the proof of Theorem 4.1, we replace "c i " to "d i ".

(19) The twelfth sentence in the proof of Theorem 4.1, we replace "c âˆˆ int Rp," to "d i > 0, i = 1,Â·Â·Â·, p,".

(20) The twelfth sentence in the proof of Theorem 4.1, we replace "c i " to "d i ".

(21) The twelfth sentence in the proof of Theorem 4.1, we replace "i = 1,Â·Â·Â·, p." to "i = 1,Â·Â·Â·,p,".

(22) The fourth sentence in the proof of Theorem 4.2, we replace "c i " to "c".

(23) The fourth sentence in the proof of Theorem 4.2, we replace

$\begin{array}{c}{f}_{i}\left({x}^{0}\right)+{\left({x}^{0}\right)}^{T}{w}_{i}^{0}+{c}_{i}||u-{x}^{0}||{}^{m}\\ â‰®{f}_{i}\left(u\right)+{u}^{T}{w}_{i},\phantom{\rule{0.3em}{0ex}}i=1,â‹\dots â‹\dots â‹\dots ,p.\end{array}$

to

$\begin{array}{c}f\left({x}^{0}\right)+{\left({x}^{0}\right)}^{T}{w}^{0}+c||u-{x}^{0}||{}^{m}\\ â‰®f\left(u\right)+{u}^{T}w.\hfill \end{array}$

## References

1. Bae , Kim : Optimality and Duality Theorems in Nonsmooth Multiobjective Optimization. Fixed Point Theory and Applications 2011, 2011: 42. 10.1186/1687-1812-2011-42

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Correspondence to Do Sang Kim.

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The online version of the original article can be found at 10.1186/1687-1812-2011-42

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Deok Bae, K., Kim, D.S. Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization. Fixed Point Theory Appl 2012, 28 (2012). https://doi.org/10.1186/1687-1812-2012-28

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• DOI: https://doi.org/10.1186/1687-1812-2012-28