# 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) The first sentence in Definition 2.2, we delete "i{1,2,···,p}".

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

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

to

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

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

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

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

to

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

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

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

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

to

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

(7) The second sentence in the proof of Theorem 2.1, we replace. ′′ c i > 0, i = 1, ,p ′′ to ′′ $c\in int{R}_{+}^{p}$′′.

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

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

to

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

(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 int Rp" to "d i > 0, i = 1,···,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 int Rp" to "d i > 0, i = 1,···, 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 int Rp," to "d i > 0, i = 1,···, 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,···, p." to "i = 1,···,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

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

to

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

## 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

## Author information

Authors

### Corresponding author

Correspondence to Do Sang Kim.

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

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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 