Sufficiency and duality in nondifferentiable multiobjective programming involving higher order strong invexity
- Izhar Ahmad^{1}Email author and
- Suliman Al-Homidan^{1}
https://doi.org/10.1186/s13660-015-0819-9
© Ahmad and Al-Homidan 2015
Received: 5 May 2015
Accepted: 10 September 2015
Published: 1 October 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.
Keywords
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].
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]
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]
Let \(\sigma\geq1\) be an integer throughout the paper.
Definition 2.3
[9]
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
Definition 2.5
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
Definition 2.7
Definition 2.8
Definition 2.9
Definition 2.10
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
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 \(\bar{\mu}_{j} = 0\).
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
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).
4 Mond-Weir type duality
Theorem 4.1
(Weak duality)
Proof
The following definition is needed in the proof of the strong duality theorem.
Definition 4.1
[14]
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
5 Conclusion
Declarations
Acknowledgements
This research is financially supported by King Fahd University of Petroleum and Minerals, Saudi Arabia under the Internal Research Project No. IN131038.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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