Erratum to: Optimality and duality theorems in nonsmooth multiobjective optimization

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

(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

$$f_i\left(x\right)+s\left(x|D_i\right)\nless f_i\left(x^0\right)+s\left(x^0|D_i\right)+c_i\left|\right|x-x^0|{|}^m$$

to

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

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

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

$$\prime \prime f_i\left(x\right)+s\left(x|D_i\right)\nless f_i\left(x^0\right)+s\left(x^0|D_i\right)+c_i\left|\right|x-x^0|{|}^m$$

to

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

(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

$$f_i\left(x\right)+s\left(x|D_i\right)\nless f_i\left(x^0\right)+s\left(x^0|D_i\right)+c_i\left|\right|x-x^0|{|}^m$$

to

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

(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 R^p" 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 R^p" 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 R^p," 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

$$\begin{array}{c}{f}_i\left(x^0\right)+{\left(x^0\right)}^T{w}_i^0+c_i\left|\right|u-x^0\left|\right|{}^m\\ \nless f_i\left(u\right)+u^T{w}_i,i=1,\cdot \cdot \cdot ,p.\end{array}$$

to

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