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# Sufficiency and duality in nondifferentiable multiobjective programming involving higher order strong invexity

*Journal of Inequalities and Applications*
**volume 2015**, Article number: 309 (2015)

## Abstract

In the present paper, we consider a nondifferentiable multiobjective programming problem with support functions and locally Lipschitz functions. Several sufficient optimality conditions are discussed for a strict minimizer of a nondifferentiable multiobjective programming problem under strong invexity and its generalizations of order *σ*. Weak and strong duality theorems are established for a Mond-Weir type dual.

## 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:

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.

## 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

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

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

Equivalently,

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

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:

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

We now consider the following multiobjective problem:

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

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

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

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

### 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

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 \(x, \bar{x} \in S\),

### 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 \(x, \bar{x} \in S\),

Or equivalently

### 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 \(x, \bar{x} \in S\),

### 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 \(x, \bar{x} \in S\),

### 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 \(x, \bar{x} \in S\),

## 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*

*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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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*,

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

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

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

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

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

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

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

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

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*,

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

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

or

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

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

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

On adding (12) and (13) we get

This contradicts (9). □

## Mond-Weir type duality

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

### 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*

### 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

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

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

Using (18), we have

or

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

or

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

Suppose to the contrary that

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

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

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

### 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 \(\bar{w}_{i} \in D_{i}\), \(i \in K\),

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. □

## 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):

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 \(R^{m}\).

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## Acknowledgements

This research is financially supported by King Fahd University of Petroleum and Minerals, Saudi Arabia under the Internal Research Project No. IN131038.

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### Authors’ contributions

Both authors carried out the proof. Both authors conceived of the study and participated in its design and coordination. Both authors read and approved the final manuscript.

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Ahmad, I., Al-Homidan, S. Sufficiency and duality in nondifferentiable multiobjective programming involving higher order strong invexity.
*J Inequal Appl* **2015, **309 (2015). https://doi.org/10.1186/s13660-015-0819-9

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### Keywords

- multiobjective programming
- locally Lipschitz function
- strict minimizer
- optimality conditions
- duality