1 Introduction

Optimality conditions and duality results in multiobjective programming problems have attracted many researchers in recent years. The concepts of weak efficient solution, efficient solution and properly efficient solution have played an important role in the analysis of these types of multiobjective optimization problems. Recently, much attention has been paid to other types of solution concepts, one of them is higher order strict minimizer [1]. This concept plays a role in stability results [2] and in the convergence analysis of iterative numerical methods [3]. In [4], Ward discussed the strict minimizer of order σ for a single objective programming problem. Jimenez [5] extended the notion of Ward [4] to introduce the notion of local efficient solution of a multiobjective programming problem and characterized it under tangent cone. Jimenez and Novo [6, 7] discussed optimality conditions for a multiobjective optimization problem. Gupta et al. [8] presented the equivalent definition of higher order strict local efficient solution for a multiobjective programming problem. The notion of Ward [4] was further extended for global strict minimizer in [9].

Agarwal et al. [10] presented the optimality and duality results for multiobjective optimization problems involving locally Lipschitz functions and type I invexity. In [11], Bae et al. formulated nondifferentiable multiobjective programming problem and discussed duality results under generalized convexity. Bae and Kim [12], and Kim and Bae [13] derived optimality conditions and duality theorems for a nondifferentiable multiobjective programming problem with support function. Recently, optimality conditions and duality for a strict minimizer of nonsmooth multiobjective optimization problems with normal cone were derived in [14].

In this paper, we consider the following nondifferentiable multiobjective problem:

$$\begin{aligned}& (\mathrm{MP})\quad \mbox{Minimize } f(x) + s(x\vert D) = \bigl[f_{1}(x) + s(x\vert D_{1}), f_{2}(x) + s(x\vert D_{2}), \ldots, f_{k}(x) + s(x\vert D_{k}) \bigr] \\& \hphantom{(\mathrm{MP})\quad}\quad \mbox{subject to }x \in{X} = \bigl\{ x \in S :g_{j}(x) \leqq0, j = 1, 2, \ldots, m \bigr\} , \end{aligned}$$

where \(f : X \rightarrow R^{k}\) and \(g : X \rightarrow R^{m}\) are locally Lipschitz functions and X is a convex set in \(R^{n}\). \(D_{i}\) is a compact convex set of \(R^{n}\).

The paper is organized as follows. In Section 2, we recall some known concepts in the literature and then introduce the concept of strong invexity of order σ for a locally Lipschitz function and its generalizations. Section 3 deals with several sufficient optimality conditions for higher order minimizers via introduced classes of functions. In Section 4, we establish the Mond-Weir type duality results, and conclusion is discussed in Section 5.

2 Notations and prerequisites

Throughout the paper, \(\bigtriangledown g(x)\) will denote the \(m \times n\) Jacobian matrix of g at x. For \(\bar{x} \in X\), \(I = \{ j: g_{j}(\bar{x}) = 0 \}\) and \(g_{I}\) will denote the vector of active constraints at x̄. The index sets \(K = \{1,2,\ldots,k\}\) and \(M = \{1,2, \ldots, m\}\).

Definition 2.1

[15]

Let D be a compact convex set in \(R^{n}\). The support function \(s(\cdot\vert D)\) is defined by

$$s(x\vert D) = \max\bigl\{ x^{T} y : y \in D\bigr\} . $$

The support function \(s(\cdot\vert D)\) has a subdifferential. The subdifferential of \(s(\cdot\vert D)\) at x is given by

$$\partial s(x\vert D) = \bigl\{ z \in D: z^{T} x = s(x \vert D)\bigr\} . $$

The support function \(s(\cdot\vert D)\) is convex and everywhere finite, that is, there exists \(z \in D\) such that

$$s(y \vert D) \geqq s(x \vert D) + z^{T}(y - x)\quad \mbox{for all } y \in D. $$

Equivalently,

$$z^{T} x = s(x \vert D). $$

A function \(f: R^{n} \rightarrow R\) is said to be locally Lipschitz at \(\bar{x} \in R^{n}\) if there exist scalars \(\delta> 0\) and \(\epsilon> 0\) such that

$$\bigl\vert f\bigl(x^{1}\bigr) - f\bigl(x^{2}\bigr) \bigr\vert \leqq \delta\bigl\Vert x^{1} - x^{2} \bigr\Vert \quad \mbox{for all } x^{1}, x^{2} \in\bar{x} + \epsilon B, $$

where \(\bar{x} + \epsilon B\) is the open ball of radius ϵ about x̄.

The generalized directional derivative [16] of a locally Lipschitz function f at x in the direction v, denoted by \(f^{\circ}(x; v)\), is as follows:

$$f^{\circ} (x; v) = \lim_{y \rightarrow x}\sup_{t \downarrow0} \frac {f(y+tv)- f(y)}{t}. $$

The generalized gradient [17] of f at x is denoted by

$$\partial f(x) = \bigl\{ \xi\in R^{n} : f^{\circ} (x; v) \geqq \xi^{t} v \mbox{ for all } v \in R^{n} \bigr\} . $$

We now consider the following multiobjective problem:

$$\begin{aligned}& (\mathrm{P})\quad \mbox{Minimize }f(x) = \bigl[f_{1}(x) , f_{2}(x) , \ldots, f_{k}(x) \bigr] \\& \hphantom{(\mathrm{P})\quad}\quad \mbox{subject to }x \in{X}. \end{aligned}$$

Since the objectives in such problems generally conflict with one another, an optimal solution is chosen from the set of strict minimizer solutions in the following sense.

Definition 2.2

[5]

A point \(\bar{x} \in X\) is a strict minimizer for (P) if there exists \(\epsilon> 0\) such that

$$f(x) \nless f(\bar{x})\quad \mbox{for all }x \in B(\bar{x}, \epsilon)\cap X, $$

that is, there exists no \(x \in B(\bar{x}, \epsilon) \cap X\) such that

$$f (x) < f(\bar{x}). $$

Let \(\sigma\geq1\) be an integer throughout the paper.

Definition 2.3

[9]

A point \(\bar{x} \in X\) is a local strict minimizer of order σ for (P) if there exist \(\epsilon> 0\) and a constant \(c \in \operatorname{int} R_{+}^{k}\) such that

$$f(x) \nless f(\bar{x}) + c \| x - \bar{x} \|^{\sigma}\quad \mbox{for all }x \in B(\bar{x}, \epsilon)\cap X. $$

The notion of a local strict minimizer reduces to the global sense if the ball \(B(\bar{x}, \epsilon)\) is replaced by the whole space \(R^{n}\).

Bhatia and Sahay [17] introduced the following notion of a strict minimizer of order σ with respect to a nonlinear function for the multiobjective programming problem.

Definition 2.4

A point \(\bar{x} \in X\) is a local strict minimizer of order σ for (P) with respect to a nonlinear function \(\psi: X \times X \rightarrow R^{n}\) if there exists a constant \(c \in \operatorname{int} R_{+}^{k}\) such that

$$f(x) \nless f(\bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\quad \mbox{for all }x \in B(\bar{x}, \epsilon)\cap X. $$

Definition 2.5

A point \(\bar{x} \in X\) is a strict minimizer of order σ for (P) with respect to a nonlinear function \(\psi: X \times X \rightarrow R^{n}\) if there exists a constant \(c \in \operatorname{int} R_{+}^{k}\) such that

$$f(x) \nless f(\bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\quad \mbox{for all }x \in X. $$

We now introduce the higher order strong invexity and its generalizations for nonsmooth locally Lipschitz functions.

Let \(f : S \rightarrow R\) be a locally Lipschitz function on S.

Definition 2.6

f is said to be strongly invex of order σ with respect to η, ψ on S if there exists a constant \(c > 0\) such that for all \(x, \bar{x} \in S\),

$$f (x) - f(\bar{x}) \geqq\xi^{T} \eta(x, \bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\mbox{for all }\xi\in\partial f( \bar{x}). $$

Definition 2.7

f is said to be strongly pseudo-invex type I of order σ with respect to η, ψ on S if there exists a constant \(c > 0\) such that for all \(x, \bar{x} \in S\),

$$\xi^{T} \eta(x,\bar{x}) \geqq0\quad \mbox{for some }\xi\in\partial f( \bar{x})\quad \mbox{implies}\quad f (x) \geqq f(\bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}. $$

Or equivalently

$$f (x) < f(\bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\quad \mbox{implies} \quad \xi^{t} \eta(x, \bar{x}) < 0. $$

Definition 2.8

f is said to be strongly pseudo-invex type II of order σ with respect to η, ψ on S if there exists a constant \(c > 0\) such that for all \(x, \bar{x} \in S\),

$$\xi^{T} \eta(x, \bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\geqq0\quad \mbox{for some }\xi\in\partial f(\bar{x})\quad \mbox{implies} \quad f(x) \geqq f(\bar{x}). $$

Definition 2.9

f is said to be strongly quasi-invex type I of order σ with respect to η, ψ on S if there exists a constant \(c > 0\) such that for all \(x, \bar{x} \in S\),

$$f(x) \leqq f(\bar{x}) \quad \mbox{implies} \quad \xi^{T} \eta(x, \bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\leqq0\quad \mbox{for all }\xi\in\partial f(\bar{x}). $$

Definition 2.10

f is said to be strongly quasi-invex type II of order σ with respect to η, ψ on S if there exists a constant \(c > 0\) such that for all \(x, \bar{x} \in S\),

$$f (x) \leqq f(\bar{x}) + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\quad \mbox{implies} \quad \xi^{T} \eta(x, \bar{x}) \leqq0\quad \mbox{for all }\xi\in \partial f(\bar{x}). $$

3 Karush-Kuhn-Tucker type sufficiency

In this section, we discuss various Karush-Kuhn-Tucker type sufficient optimality conditions for a feasible solution to be a strict minimizer of order σ of (MP).

Theorem 3.1

Let \(f_{i}(\cdot) + (\cdot)^{T} w_{i}\), \(i \in K\) be strongly invex of order σ and \(g_{j}\), \(j \in I\) be strongly quasi-invex type I of order σ with respect to the same η and ψ. If there exist \({\bar{\lambda}_{i}} \geqq0\), \(i = 1, 2, \ldots, k\), \({\bar{\mu}_{j}} \geqq0\), \(j = 1, 2, \ldots, m\) and \(\bar{w}_{i} \in D_{i}\), \(i \in K\) satisfying

$$\begin{aligned}& 0 \in\sum_{i = 1}^{k} {\bar{ \lambda}_{i}}\bigl({\partial f_{i}}(\bar{x}) + \bar{w}_{i}\bigr) + \sum_{j = 1}^{m}{ \bar{\mu}_{j}} {\partial g_{j}}(\bar{x}), \end{aligned}$$
(1)
$$\begin{aligned}& \bar{x}^{T} \bar{w}_{i} = s(\bar{x} \vert D_{i}), \quad i \in K, \end{aligned}$$
(2)
$$\begin{aligned}& \bar{\mu}_{j} g_{j}(\bar{x}) = 0,\quad j \in M, \end{aligned}$$
(3)
$$\begin{aligned}& \bar{\lambda}^{T} e = 1,\quad e =(1, 1, \ldots, 1) \in R^{k}, \end{aligned}$$
(4)

then x̄ is a strict minimizer of order of σ with respect to ψ of (MP).

Proof

Let J = \(\{j: g_{j}(\bar{x}) < 0 \}\). Therefore \(I \cup J = M\). Also \(\bar{\mu}\geqq0\), \(g(\bar{x}) \leqq0\) and \(\bar{\mu}_{j} g_{j}(\bar{x}) = 0\), \(j \in M\) implies \(\bar{\mu}_{j} = 0\).

Condition (1) implies that there exist \(\bar{\xi}_{i} \in\partial f_{i}(\bar{x})\) and \(\bar{\zeta}_{i} \in\partial g_{j}(\bar{x})\) satisfying

$$ 0 = \sum_{i = 1}^{k} \bar{ \lambda}_{i} ( \bar{\xi}+ w_{i}) + \sum _{j \in I} \bar{\mu}_{j} \bar{\zeta}_{j}. $$
(5)

Now suppose that x̄ is not a strict minimizer of order σ with respect to ψ for (MP). Then, for \(c_{i} > 0\), \(i = 1, 2, \ldots, k\), there exists some \(x \in X\) such that

$$f_{i}(x) + s(x \vert D_{i}) < f_{i}(\bar{x}) + s(\bar{x} \vert D_{i}) + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma},\quad i \in K. $$

Since \(x^{T} w_{i} \leqq s(x \vert D_{i})\) and \((\bar{x})^{T} w_{i} = s(x \vert D_{i})\),

$$\begin{aligned} f_{i}(x) + x^{T} w_{i} \leqq& f_{i}(x) + s(x \vert D_{i}) \\ < & f_{i}(\bar{x}) + s(\bar{x} \vert D_{i}) + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}, \quad i \in K \\ =& f_{i}(\bar{x}) + \bar{x}^{T} w_{i} + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma},\quad i \in K. \end{aligned}$$

Using \(\bar{\lambda}_{i} \geqq0\) and \(\bar{\lambda}^{T} e = 1\), we get

$$ \sum_{i = 1}^{k} {\bar{\lambda}_{i}} \bigl(f_{i}(x) + x^{T} w_{i}\bigr) < \sum _{i = 1}^{k} {\bar{\lambda}_{i}} \bigl(f_{i}(\bar{x}) + \bar{x}^{T} w_{i}\bigr) + \sum_{i = 1}^{k} {\bar{\lambda}_{i}} c_{i}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}. $$
(6)

The strong invexity of \(f_{i}(\cdot) + (\cdot)^{T} w_{i}\), \(i \in K\) of order σ with respect to η and ψ,

$$\bigl(f_{i}(x) + x^{T} w_{i}\bigr) - \bigl(f_{i}(\bar{x}) + \bar{x}^{T}w_{i}\bigr) \geqq \bigl\langle \xi_{i} + w_{i}, \eta(x, \bar{x})\bigr\rangle + c_{i}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}, \quad i = 1, 2, \ldots, k. $$

For \(\bar{\lambda}_{i} \geqq0\), we obtain

$$\begin{aligned}& \sum_{i = 1}^{k} {\bar{\lambda}_{i}} \bigl(f_{i}(x) + x^{T} w_{i}\bigr) - \sum _{i = 1}^{k} {\bar{\lambda}_{i}} \bigl(f_{i}(\bar{x}) + \bar{x}^{T} w_{i}\bigr) \\& \quad \geqq \sum_{i = 1}^{k} {\bar{ \lambda}_{i}}\bigl\langle \xi_{i} + w_{i}, \eta(x, \bar{x})\bigr\rangle + \sum_{i =1}^{k} \bar{\lambda}_{i} c_{i}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}. \end{aligned}$$
(7)

As \(x \in X\), we have

$$g_{j}(x) \leqq g_{j}(\bar{x}),\quad j \in I. $$

The strongly quasi-invex type I of \(g_{j}\), \(j \in I\) of order σ with respect to η and ψ gives

$$\bigl\langle \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle + \beta_{j}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\leqq 0,\quad \beta_{j}>0, j \in I. $$

The above inequality along with \(\bar{\mu}_{j} \geqq0\), \(j \in I\) yields

$$\sum_{j \in I} \bigl\langle \bar{\mu}_{j} \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle +\sum _{j \in I}\bar{\mu}_{j} \beta_{j}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\leqq 0. $$

As \(\bar{\mu}_{j} = 0\) for \(j \in J\), we have

$$ \sum_{j = 1}^{m} \bigl\langle \bar{ \mu}_{j} \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle +\sum _{j \in I}\bar{\mu}_{j} \beta_{j}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\leqq 0. $$
(8)

Adding (7), (8) and using (5), we get

$$\sum_{i = 1}^{k} {\bar{\lambda}_{i}} \bigl(f_{i}(x) + x^{T} w_{i}\bigr) - \sum _{i = 1}^{k} {\bar{\lambda}_{i}} \bigl(f_{i}(\bar{x}) + \bar{x}^{T} w_{i}\bigr) \geqq\alpha\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}, $$

where \(\alpha= \sum_{i = 1}^{k} {\bar{\lambda}_{i}}c_{i} + \sum_{j \in I} {\bar{\mu}_{j}}\beta_{j}\). This implies that

$$\sum_{i = 1}^{k} {\bar{\lambda}_{i}} \bigl[\bigl(f_{i}(x) + x_{i}^{T} w_{i} \bigr) - \bigl(f_{i}(\bar{x}) + \bar{x}_{i}^{T} w_{i}\bigr) - a_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\bigr], $$

where \(a = \alpha e\), since \(\bar{\lambda}^{T} e = 1\), which contradicts (5). Hence x̄ is a strict minimizer of order σ with respect to ψ for (MP). □

Remark 3.1

If \(g_{j}\), \(j \in I\) are strongly invex of order σ with respect to ψ on S, then the above Theorem 3.1 holds.

Theorem 3.2

Let \(f_{i}(\cdot) + (\cdot)^{T} w_{i}\), \(i = 1, 2, \ldots, k\), be strongly pseudo-invex type I of order σ and \(g_{j}\), \(j \in I\) be strongly quasi-invex type I of order σ with respect to the same η and ψ. If conditions (1)-(4) are satisfied, then x̄ is a strict minimizer of order σ of (MP).

Proof

Condition (1) implies that there exist \(\bar{\xi}_{i} \in \partial f_{i}(\bar{x})\) and \(\bar{\zeta}_{i} \in\partial g_{j}(\bar{x})\) satisfying

$$ 0 = \sum_{i = 1}^{k} \bar{ \lambda}_{i} ( \bar{\xi}+ w_{i}) + \sum _{j \in I} \bar{\mu}_{j} \bar{\zeta}_{j}. $$
(9)

Now suppose that x̄ is not a strict minimizer of order σ with respect to ψ for (MP).Then, for \(c_{i} > 0\), \(i = 1, 2, \ldots, k\), there exists some \(x \in X\) such that

$$f_{i}(x) + s(x \vert D_{i}) < f_{i}(\bar{x}) + s(\bar{x} \vert D_{i}) + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma},\quad i \in K. $$

Since \(x_{i}^{T} w_{i} \leqq s(x \vert D_{i})\) and \(\bar{x}_{i}^{T} w_{i} = s(x \vert D_{i})\),

$$\begin{aligned} f_{i}(x) + x^{T} w_{i} \leqq& f_{i}(x) + s(x \vert D_{i}) \\ < & f_{i}(\bar{x}) + s(\bar{x} \vert D_{i}) + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}, \quad i \in K \\ =& f_{i}(\bar{x}) + \bar{x}^{T} w_{i} + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma},\quad i \in K. \end{aligned}$$

As \(f_{i}(\cdot) + (\cdot)^{T} w_{i}\), \(i = 1, 2, \ldots, k\), are strongly pseudo-invex type I of order σ with respect to η and ψ,

$$\bigl\langle \xi_{i} + w_{i}, \eta(x, \bar{x})\bigr\rangle < 0, \quad i \in K . $$

For \(\bar{\lambda}_{i} \geqq0\) and \(\lambda^{T}e = 1\), we obtain

$$ \sum_{i = 1}^{k} \bar{\lambda}_{i} \bigl\langle \xi_{i} + w_{i}, \eta(x, \bar{x})\bigr\rangle < 0. $$
(10)

As \(x \in X\), we have

$$g_{j}(x) \leqq g_{j}(\bar{x}),\quad j \in I. $$

The strongly quasi-invex type I of \(g_{j}\), \(j \in I\) of order σ with respect to η and ψ yields

$$\bigl\langle \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle + \beta_{j}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\leqq 0,\quad \beta_{j}>0, j\in I. $$

The above inequality along with \(\bar{\mu}_{j} \geqq0\), \(j \in I\) yields

$$\sum_{j \in I} \bigl\langle \bar{\mu}_{j} \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle +\sum _{j \in I}\bar{\mu}_{j} \beta_{j}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\leqq 0. $$

As \(\bar{\mu}_{j} = 0\) for \(j \in J\), we have

$$ \sum_{j = 1}^{m} \bigl\langle \bar{ \mu}_{j} \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle +\sum _{j \in I}\bar{\mu}_{j} \beta_{j} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\leqq 0. $$
(11)

On adding (10) and (11), we obtain

$$\eta^{t}(x, \bar{x}) \Biggl[ \sum_{i = 1}^{k} \bar{\lambda}_{i} \xi_{i} + \sum _{i = 1}^{k} \bar{\mu}_{j} \zeta_{j} \Biggr] + \sum_{j = 1}^{m} \bar{\mu}_{j} \beta_{j}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}< 0. $$

The above inequality along with (9) gives \(\sum_{j = 1}^{m}\bar{\mu}_{j} \beta_{j}\|\psi(x, \bar{x})\|^{\sigma}< 0\), which is not possible. Hence the result. □

Theorem 3.3

Let conditions (1)-(4) be satisfied. Suppose that \(f_{i}(\cdot) + (\cdot)^{T} w_{i}\), \(i = 1, 2, \ldots, k\), are strongly pseudo-invex type I of order σ and that \(g_{j}\), \(j \in I\) are strongly quasi-invex type II of order σ with respect to η and ψ. Then x̄ is a strict minimizer of order σ of with respect to ψ of (MP).

Proof

Condition (1) implies that there exist \(\bar{\xi}_{i} \in \partial f_{i}(\bar{x})\) and \(\bar{\zeta}_{i} \in\partial g_{j}(\bar{x})\) satisfying (9).

Now suppose that x̄ is not a strict minimizer of order σ with respect to ψ for (MP). Then, for \(c_{i} > 0\), \(i = 1, 2, \ldots, k\), there exists some \(x \in X\) such that

$$f_{i}(x) + s(x \vert D_{i}) < f_{i}(\bar{x}) + s(\bar{x} \vert D_{i}) + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma},\quad i \in K. $$

Since \(x^{T} w_{i} \leqq s(x \vert D_{i})\) and \(\bar{x}^{T} w_{i} = s(x \vert D_{i})\),

$$\begin{aligned} f_{i}(x) + x^{T} w_{i} \leqq& f_{i}(x) + s(x \vert D_{i}) \\ < & f_{i}(\bar{x}) + s(\bar{x} \vert D_{i}) + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma},\quad i \in K \\ =& f_{i}(\bar{x}) + \bar{x}^{T} w_{i} + c_{i} \bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}, \quad i\in K. \end{aligned}$$

As \(f_{i}(\cdot) + (\cdot)^{T} w_{i}\), \(i = 1, 2, \ldots, k\), are strongly pseudo-invex type I of order σ with respect to η and ψ,

$$\bigl\langle \xi_{i} + w_{i}, \eta(x, \bar{x})\bigr\rangle < 0,\quad i \in K. $$

For \(\bar{\lambda}_{i} \geqq0\) and \(\lambda^{T}e = 1\), we obtain

$$ \sum_{i = 1}^{k} \bar{\lambda}_{i} \bigl\langle \xi_{i} + w_{i}, \eta(x, \bar{x})\bigr\rangle < 0. $$
(12)

As \(x \in X\), we have

$$g_{j}(x) \leqq g_{j}(\bar{x}), \quad j \in I $$

or

$$g_{j}(x) \leqq g_{j}(\bar{x}) + \beta_{j}\bigl\Vert \psi(x, \bar{x})\bigr\Vert ^{\sigma}\quad \mbox{for } \beta_{j} > 0 , j \in M. $$

Since \(g_{j}\), \(j \in I\) is strongly quasi-invex type II of order σ with respect to η and ψ, therefore

$$\bigl\langle \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle \leqq 0. $$

The above inequality along with \(\bar{\mu}_{j} \geqq0\), \(j \in I\) yields

$$\sum_{j \in I} \bigl\langle \bar{\mu}_{j} \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle \leqq 0. $$

As \(\bar{\mu}_{j} = 0\) for \(j \in J\), we have

$$ \sum_{j = 1}^{m} \bigl\langle \bar{ \mu}_{j} \zeta_{j}, \eta(x, \bar{x}) \bigr\rangle \leqq 0. $$
(13)

On adding (12) and (13) we get

$$\sum_{i = 1}^{k} \Biggl\langle \bar{ \lambda}_{i} (\xi_{i} + w_{i}) +\sum _{j = 1}^{m} \bar{\mu}_{j} \zeta_{j} , \eta(x, \bar{x}) \Biggr\rangle < 0. $$

This contradicts (9). □

4 Mond-Weir type duality

For the primal problem (MP), we formulate the following Mond-Weir type dual problem:

$$\begin{aligned}& (\mathrm{MD})\quad \mbox{Maximize }f(u) = \bigl[f_{1}(u) + u^{T} w_{1}, f_{2}(u) + u^{T} w_{2}, \ldots, f_{k}(u) + u^{T}w_{k} \bigr] \\& \hphantom{(\mathrm{MD})\quad}\quad \mbox{subject to }0\in\sum_{i= 1}^{k} \lambda_{i} \bigl(\partial f_{i}(u) + w_{i} \bigr) + \sum_{j = 1}^{m} \mu_{j} \partial g_{j}(u), \end{aligned}$$
(14)
$$\begin{aligned}& \hphantom{(\mathrm{MD})\quad\quad \mbox{subject to }}\mu^{T}g(u) \geqq0, \end{aligned}$$
(15)
$$\begin{aligned}& \hphantom{(\mathrm{MD})\quad\quad \mbox{subject to }}\mu\geqq0,\qquad w_{i} \in D_{i},\quad i \in K, \end{aligned}$$
(16)
$$\begin{aligned}& \hphantom{(\mathrm{MD})\quad\quad \mbox{subject to }}\lambda\geqq0,\qquad \lambda^{T} e = 1,\quad e =(1, 1, \ldots, 1) \in R^{k}. \end{aligned}$$
(17)

Theorem 4.1

(Weak duality)

Let x and \(( u, \lambda, \mu, w_{1}, w_{2}, \ldots, w_{k})\) be feasible solutions for (MP) and (MD) respectively. Suppose \(\sum_{i =1}^{k}\lambda _{i}(f_{i}(\cdot) + (\cdot)^{T} w_{i})\), \(i \in K\) is strongly pseudo-invex type I and \(\sum_{j = 1}^{m} \mu_{j} g_{j}\) is strongly quasi-invex type I of order σ with respect to η and ψ, then there exists \(c \in \operatorname{int} R_{+}^{k}\) such that

$$f_{i}(x) + s(x\vert D_{i}) \nless f_{i}(u) + u^{T} w_{i} + c_{i}\bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}, \quad i \in K. $$

Proof

Since \((u, \lambda, \mu, w_{1}, w_{2},\ldots, w_{k})\) is a feasible solution for (MD), there exist \(\xi_{i} \in\partial f_{i}(u)\) and \(\zeta_{j} \in\partial g_{j}(u)\) such that

$$ 0 = \sum_{i = 1}^{k} \lambda_{i} (\xi_{i} +w_{i}) + \sum_{j = 1}^{m} \mu_{j} \zeta_{j}. $$
(18)

Since x is feasible for (MP) and \((u, \lambda, \mu, w_{1}, w_{2},\ldots, w_{k})\) is feasible for (MD), we have

$$\sum_{j =1}^{m} \mu_{j}g_{j}(x) \leqq \sum_{j =1}^{m} \mu_{j}g_{j}(u). $$

The strong quasi-invexity type I of \(\sum_{j =1}^{m} \mu_{j}g_{j}(\cdot)\) of order σ with respect to η and ψ at u implies that there exists a constant \(\beta> 0\) such that

$$\Biggl\langle \sum_{j = 1}^{m} \mu_{j} \zeta_{j}, \eta(x, u) \Biggr\rangle + \beta \bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}\leqq 0 , \quad \forall \zeta_{j} \in \partial g_{j}(u), j \in M. $$

Using (18), we have

$$\Biggl\langle \sum_{i = 1}^{k} \lambda_{i}(\xi_{i} + w_{i}), \eta(x, u) \Biggr\rangle - \beta\bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}\geqq 0 $$

or

$$\Biggl\langle \sum_{i = 1}^{k} \lambda_{i}(\xi_{i} + w_{i}), \eta(x, u) \Biggr\rangle \geqq0. $$

Now strong pseudo-invexity of type I of order σ of \(\sum_{i = 1}^{k} \lambda_{i}(f_{i}(\cdot) + (\cdot)^{T} w_{i})\) with respect to η and ψ at u implies that there exists a constant \(\gamma> 0\) such that

$$\sum_{i = 1}^{k} \lambda_{i} \bigl(f_{i}(x) + x^{T} w_{i}\bigr) \geqq\sum _{i = 1}^{k} \lambda_{i} \bigl(f_{i}(u) + u^{T} w_{i}\bigr) + \gamma\bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}$$

or

$$ \sum_{i = 1}^{k} \lambda_{i} \bigl(f_{i}(x) + x^{T} w_{i}\bigr) \geqq\sum _{i = 1}^{k} \lambda_{i} \bigl(f_{i}(u) + u^{T} w_{i} + c_{i} \bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}\bigr), $$
(19)

where \(c = \gamma e\) and \(\lambda^{T}e = 1\).

Suppose to the contrary that

$$f_{i}(x) + s(x\vert D_{i}) < f_{i}(u) + u^{T} w_{i} + c_{i}\bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}, \quad i \in K. $$

Since \(x^{T}w_{i} \leqq s(x\vert D_{i})\), \(i \in K\), we have

$$f_{i}(x) + x^{T}w < f_{i}(u) + u^{T} w_{i} + c_{i}\bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}, \quad i \in K. $$

Using \(\lambda\geqq0\) and \(\lambda^{T}e = 1\), we get

$$\sum_{i = 1}^{k} \lambda_{i} \bigl(f_{i}(x) + x^{T} w_{i}\bigr) < \sum _{i = 1}^{k} \lambda_{i} \bigl(f_{i}(u) + u^{T} w_{i} + c_{i} \bigl\Vert \psi(x, u)\bigr\Vert ^{\sigma}\bigr). $$

This contradicts (19). Hence the result. □

The following definition is needed in the proof of the strong duality theorem.

Definition 4.1

[14]

A point \(\bar{x} \in X\) is a strict maximizer of order σ for (MP) with respect to a nonlinear function \(\psi: X \times X \rightarrow R^{n}\) if there exists a constant \(c \in \operatorname{int} R_{+}^{k}\) such that

$$f(\bar{x}) + \bar{x}^{T} w + c \bigl\Vert \psi(x, \bar{x}) \bigr\Vert ^{\sigma}\nless f( x) \quad \mbox{for all }x \in X. $$

Theorem 4.2

(Strong duality)

Let x̄ be a strict minimizer of order σ with respect to ψ of (MP), and let the basic regularity hold at x̄. Then there exist \(\bar{\lambda}_{i} \geqq0\), \(\bar{w}_{i} \in D_{i}\), \(i \in K\) and \(\bar{\mu}_{j} \geqq0\), \(j \in M\) such that \((\bar{x}, \bar{\lambda}, \bar{\mu}, \bar{w}_{1}, \bar{w}_{2}, \ldots, \bar{w}_{k} )\) is a feasible solution of (MD) and \(\bar{x}^{T}w_{i} = s(\bar{x} \vert D_{i})\), \(i \in K\). Moreover, if the hypothesis of Theorem  4.1 is satisfied, then \((\bar{x}, \bar{\lambda}, \bar{\mu}, \bar{w}_{1}, \bar{w}_{2}, \ldots, \bar{w}_{k} )\) is a strict minimizer of order m with respect to ψ of (MD).

Proof

Since x̄ is a strict minimizer of order σ with respect to ψ for (NMP), by Theorem 3.1 there exist \({\bar{\lambda}_{i}} \geqq0\), \(i \in K\), \({\bar{\mu}_{j}} \geqq0\), \(j \in M\) and \(\bar{w}_{i} \in D_{i}\), \(i \in K\),

$$\begin{aligned}& 0 \in\sum_{i = 1}^{k} {\bar{ \lambda}_{i}}\bigl({\partial f_{i}}(\bar{x}) + \bar{w}_{i}\bigr)+ \sum_{j = 1}^{m}{ \bar{\mu}_{j}} {\partial g_{j}}(\bar{x}), \\& \bar{x}^{T} \bar{w}_{i} = s(\bar{x} \vert D_{i}), \quad i \in K, \\& \bar{\mu}_{j} g_{j}(\bar{x}) = 0,\quad j \in M, \\& \bar{\lambda}^{T} e = 1,\quad e =(1, 1, \ldots, 1) \in R^{k}. \end{aligned}$$

Therefore \((\bar{x}, \bar{\lambda}, \bar{\mu}, \bar{w}_{1}, \bar{w}_{2}, \ldots, \bar{w}_{k})\) is feasible for (MD). Now a strict minimizer of order σ with respect to ψ at \((\bar{x}, \bar{\lambda}, \bar{\mu}, \bar{w}_{1}, \bar{w}_{2}, \ldots, \bar{w}_{k})\) for (MD) follows from the weak duality theorem. □

5 Conclusion

In this paper, we have presented several Khun-Tucker type sufficient optimality conditions and Mond-Weir type duality results for a nondifferentiable multiobjective problem involving a support function of a compact convex set. The present results can be further generalized for the following fractional analogue of (MP):

$$\begin{aligned}& (\mathrm{FP})\quad \mbox{Minimize } \biggl( \frac{f_{1}(x) + s(x \vert D_{1})}{h_{1}(x) - s(x\vert E_{1})}, \frac{f_{2}(x) + s(x\vert D_{2})}{h_{2}(x) - s(x\vert E_{2})}, \ldots, \frac{f_{k}(x) + s(x\vert D_{k})}{h_{k}(x) - s(x\vert E_{k})} \biggr) \\& \hphantom{(\mathrm{FP})\quad}\quad \mbox{subject to }- g(x)\in C^{*},\quad x\in C, \end{aligned}$$

where \(f_{i} : X\rightarrow R\), \(h_{i} : X\rightarrow R\), \(i \in K\), \(g : X\rightarrow R^{m}\); \(D_{i}\) and \(E_{i}\), \(i\in K\) are compact sets in \(R^{n}\). C is a closed cone with nonempty interior in \(R^{m}\).