- Research
- Open Access
Optimality and mixed duality in multiobjective E-convex programming
- Guang-Ri Piao^{1},
- Liguo Jiao^{2} and
- Do Sang Kim^{2}Email author
https://doi.org/10.1186/s13660-015-0854-6
© Piao et al. 2015
- Received: 31 July 2015
- Accepted: 5 October 2015
- Published: 16 October 2015
Abstract
In this paper, we consider a class of multiobjective E-convex programming problems with inequality constraints, where the objective and constraint functions are E-convex functions which were firstly introduced by Youness (J. Optim. Theory Appl. 102:439-450, 1999). Fritz-John and Kuhn-Tucker necessary and sufficient optimality theorems for the multiobjective E-convex programming are established under the weakened assumption of the theorems in Megahed et al. (J. Inequal. Appl. 2013:246, 2013) and Youness (Chaos Solitons Fractals 12:1737-1745, 2001). A mixed duality for the primal problem is formulated and weak and strong duality theorems between primal and dual problems are explored. Illustrative examples are given to explain the obtained results.
Keywords
- E-convex function
- mixed duality
- multiobjective programming
- optimality condition
MSC
- 90C29
- 90C30
- 69K05
1 Introduction
The concepts of an E-convex set and an E-convex function were introduced first by Youness [1]. Subsequently, necessary and sufficient optimality criteria for a class of E-convex programming problems were discussed by Youness [2], and E-Fritz-John and E-Kuhn-Tucker problems, which modified the Fritz-John and Kuhn-Tucker problems, were also presented. In Megahed et al. [3], the concept of an E-differentiable convex function which transforms a non-differentiable convex function to a differentiable function under an operator E: \(\Bbb{R}^{n} \rightarrow\Bbb{R}^{n}\) was presented, then a solution of mathematical programming with a non-differentiable function could be found by applying the Fritz-John and Kuhn-Tucker conditions due to Mangasarian [4].
However, on the other hand, the results on E-convex programming in Youness [1] were not correct, and some counterexamples were given by Yang [5]. The results concerning the characterization of an E-convex function f in terms of its E-epigraph in Youness [1] were also not correct, and some characterizations of E-convex functions using a different notion of epigraph were given by Duca et al. [6].
Based on the correct results in Youness [1], a class of semi-E-convex functions was introduced by Chen [7], the concepts of E-quasiconvex functions and strictly E-quasiconvex functions were introduced by Syau and Stanley Lee [8], respectively.
In fact, after defining the E-convex function in 1999, Youness [1] pointed out that the E-convex function that he defined had more generalized results than a convex function. He dealt mainly with some properties of an E-convex set and an E-convex function, a programming problem without E in both objective functions or constrained functions, and the relation between solutions of objective and constrained functions with and without E. He then drew the conclusion that the E-convex set and E-convex function were more generalized than the convex set and function proposed by Hanson [9], Hanson and Mond [10], and Kaul and Kaur [11].
This paper also addresses a counterexample of Theorem 4.1 in Youness [1]. Characterization of efficient solutions based on the modification of Theorem 4.2 in Youness [1] is presented. A sufficient optimality theorem is given by using this characterization and E-convexity conditions. We obtain the scalarization method due to Chankong and Haimes [12] for multiobjective E-convex programming. By employing this scalarization method, Fritz-John and Kuhn-Tucker necessary theorems for the multiobjective case are established under the weakened assumption of the theorems in Megahed et al. [3] and Youness [2]. Moreover, a mixed type dual for the primal problem is given. Under the assumption of the E-convex conditions, weak and strong duality theorems between the primal and dual problems are established, and we also propose some examples to illustrate our results.
2 Preliminaries
We present some concepts of E-convex set and E-convex function; for convenience, we recall the definition of E-convex set first.
Definition 2.1
[1]
A set \(M\subset{\Bbb{R}^{n}}\) is said to be E-convex iff there is a map \(E:{\Bbb{R}^{n}} \to{\Bbb{R}^{n}}\) such that \((1-\lambda)E(x)+ \lambda E(y) \in M\), for each \(x,y \in M\), and \(\lambda\in[0, 1]\).
It is clear that if \(M\subset\Bbb{R}^{n}\) is convex, then M is E-convex by taking a map \(E:{\Bbb{R}^{n}} \to{\Bbb{R}^{n}}\) as the identity map, but the converse may not be true; see the following example.
Example 2.1
Consider the set \(S_{1}= \{(x,y)\in\Bbb{R}^{2}\mid y\leqq x, 0\leqq x\leqq1 \}\). Let \(E(x,y)=(\sqrt{x},y)\), it is clear that \(S_{1}\) is E-convex (since \(S_{1}\) is convex). It is easy to check that \(E(S_{1})\) is E-convex by taking the map \(E(x,y)=(\sqrt{x},y)\), while \(E(S_{1})\) is not convex, where \(E(S_{1})= \{(x,y)\in\Bbb{R}^{2}\mid y\leqq x^{2}, 0\leqq x\leqq1 \}\).
However, if \(E: \Bbb{R}^{n} \to\Bbb{R}^{n}\) is a surjective map, it is easy to check that the converse also holds. Note that E is said to be surjective if there exists \(x\in M\) such that \(E(x)=y\), \(\forall y \in E(M)\).
Definition 2.2
[1]
Remark 2.1
Let f, g be E-convex on M. Then \(f + g\), αf (\(\alpha\geqq 0\)) are E-convex on the set M.
It is easy to check that every convex function f on a convex set M is an E-convex function, where E is the identity map. But the converse may not hold, we recall the example from [1].
Example 2.2
Obviously, if f is a real-valued differentiable function on an E-convex set \(M \subset{\Bbb{R}^{n}}\), we can define a differentiable E-convex function in the following.
Definition 2.3
3 Optimality criteria
In this section, we suppose that \(E:M\to M\) (\(M\subset{\Bbb{R}^{n}}\)) is a surjective map. In addition, as we know if a set \(M\subset\Bbb{R}^{n}\) is E-convex with respect to a mapping \(E: \Bbb{R}^{n} \to\Bbb{R}^{n}\), then \(E(M)\subset M\) (see [1], Proposition 2.2). For an E-convex function f, we say that the function \((f\circ E): M\to\Bbb{R}\) defined by \((f\circ E)(x)=f(E(x))\) for all \(x\in M\) is well defined (see [8]).
It states that, for a surjective map E, if f is E-convex, then \(f\circ E\) is obviously convex.
Definition 3.1
Now we give a counterexample which is easier to understand than the one in [5], to show that Theorem 4.1 (In (MP), the set M is an E-convex set.) in Youness [1] is incorrect.
Example 3.1
In (MP), \(g_{j}\), \(j\in Q\) are E-convex, but M does not always need to be E-convex set.
Let \(g(x)=x\in\Bbb{R} \) and define the map E as \(E(x)=|x|\). Then \(g(x)\) is E-convex. Take \(x=-1\), \(y=-1/2\). Then \(g(-1)=-1\), \(g(-1/2)=-1/2\).
Also, Theorem 4.2 in Youness [1] is incorrect. The counterexample was given by Yang [5].
Now we would like to present the characterization of efficient solutions modifying Theorem 4.2 in Youness [1] by using only surjective assumption of the mapping E as follows.
Theorem 3.1
Let \(E: M\to M\) be a surjective map. Then x̄ is an efficient solution of (MP_{E}) if and only if \(E(\bar{x})\) is an efficient solution of (MP).
Proof
Suppose that \(E(\bar{x})\) is not an efficient solution of (MP). Then there exists \(\bar{z}\in M\) such that \(f(\bar{z}) \leq f(E(\bar{x}))\). Since E is surjective, we have \(E(M)=M\), then there exists \(\bar{y} \in M\) such that \(\bar{z}=E(\bar{y})\), that is, \((f\circ E)(\bar{y}) \leq(f\circ E)(\bar{x})\), which contradicts that x̄ is an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)).
Conversely, suppose that x̄ is not an efficient solution of \({(\mathrm{MP}_{\mathrm{E}})}\), then there exists \(y^{*}\in E(M)\) such that \((f\circ E)(y^{*}) \leq(f\circ E)(\bar{x})\). Since E is surjective, there exists \(z^{*}\in M\) such that \(E(y^{*})=z^{*}\). Hence \(f(z^{*}) \leq f(E(\bar{x}))\), which contradicts that \(E(\bar{x})\) is an efficient solution of (MP). □
With the help of Theorem 3.1 and the E-convexity assumption, we now give the sufficient optimality condition.
Theorem 3.2
(Sufficient optimality condition)
Then x̄ is an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)).
Proof
Remark 3.1
If we replace the E-convexity of \(f_{i}\) and \(\bar{\lambda}> 0\) by the strictly E-convexity of \(f_{i}\) and \(\bar{\lambda}\geq0\), respectively, then Theorem 3.2 also holds.
Now we present the following lemma due to Chankong and Haimes [12] to deal with the relationship between the scalar and multiobjective programming problems.
Lemma 3.1
Proof
Conversely, assume that x̄ is a solution of \((\mathrm{MP}_{\mathrm{E}})_{k}\) for every \(k \in P\), then for all \(x\in E(M)\) with \((f_{i}\circ E)(x)\leqq (f_{i}\circ E)(\bar{x})\), \(i\neq k\), we have \((f_{k}\circ E)(\bar{x}) \leqq(f_{k}\circ E)(x)\). Then there exists no other \(x\in E(M)\) such that \((f_{i}\circ E)(x) \leqq(f_{i}\circ E )(\bar{x})\), \(i\in P\), with strict inequality holding for at least one i. This implies that x̄ is efficient for (\(\mathrm{MP}_{\mathrm{E}}\)). □
Remark 3.2
In order to obtain the necessary optimality condition, we employ the following generalized linearization lemma due to Mangasarian [4].
Lemma 3.2
We establish the following Fritz-John necessary optimality criteria by using Lemma 3.2.
Theorem 3.3
(Fritz-John necessary condition)
Proof
Since \(x^{*}\in E(M)\), \((g\circ E)(x^{*}) \leqq0\).
The proof is complete. □
Theorem 3.4
(Kuhn-Tucker necessary condition)
Proof
We only have to show that \(\bar{\lambda}\geq0\), that is, \(\bar{\lambda}_{k} >0\) for at least one \(k\in P\).
Since \((\bar{\lambda},\bar{\mu})\geq0\), \((\bar{\lambda}, \bar{\mu}_{W}) \geq0\), we have \(\bar{\lambda}_{k} > 0\) for at least one \(k\in P\) if W is empty. Now, we show that \(\bar{\lambda}_{k} > 0\) for at least one \(k\in P\) if W is nonempty by contradiction.
Remark 3.3
If we replace our surjective assumption of E by bijection (or linearity) of E, then our Fritz-John and Kuhn-Tucker necessary optimality results reduce to the ones in Megahed et al. [3] (or Youness [2]).
Example 3.2
- (a)
It is easy to check that the feasible sets of \({\widehat {(\mathrm{MP})}}\) and \({\widehat{(\mathrm{MP}_{\mathrm{E}})}}\) are \(M=[0, 1]\) and \(E(M)=[1,2]\), respectively.
- (b)
By the definition of an efficient solution, we see that \(x^{*}=0 \in M\) is the efficient solution of \({\widehat{(\mathrm{MP})}}\) and \(\bar{x}=E(x^{*})=1\in E(M)\) is the efficient solution of \({\widehat {(\mathrm{MP}_{\mathrm{E}})}}\), hence Theorem 3.1 holds.
- (c)
We can easily check that \((\bar{x}, (\bar{\lambda}_{1}, \bar{\lambda}_{2}), (\bar{\mu}_{1}, \bar{\mu}_{2}))=(1, ({1\over 2}, 1), (0, {1\over 2}))\) satisfy the conditions in Theorem 3.2, and \(\bar{x}=1\) is the efficient solution of \({\widehat{(\mathrm{MP}_{\mathrm{E}})}}\), hence Theorem 3.2 holds.
- (d)Since the efficient solution \(\bar{x}=1\) for \({\widehat {(\mathrm{MP}_{\mathrm{E}})}}\), also solves both \({\widehat{(\mathrm{MP}_{\mathrm{E}})}_{1}}\) and \({\widehat{(\mathrm{MP}_{\mathrm{E}})}_{2}}\), Lemma 3.1 holds, whereand$$\begin{aligned} \textstyle\begin{array}{@{}l@{\quad}l@{\quad}l@{}} \widehat{(\mathbf{MP}_{\mathbf{E}})}_{\mathbf{1}} &\mbox{Mimimize}& (f_{1}\circ E) (x),\\ &\mbox{subject to}& (f_{2}\circ E) (x)\leqq(f_{2}\circ E) ( \bar{x}),\\ &&x\in E(M), \end{array}\displaystyle \end{aligned}$$$$\begin{aligned} \textstyle\begin{array}{@{}l@{\quad}l@{\quad}l@{}} \widehat{(\mathbf{MP}_{\mathbf{E}})}_{\mathbf{2}} &\mbox{Mimimize}& (f_{2}\circ E) (x),\\ &\mbox{subject to}& (f_{1}\circ E) (x)\leqq(f_{1}\circ E) (\bar{x}),\\ &&x\in E(M). \end{array}\displaystyle \end{aligned}$$
- (e)
As \(\bar{x}=1\) is the efficient solution of \(\widehat {(\mathrm{MP}_{\mathrm{E}})}\), then there exist \(\bar{\lambda}=(\frac{1}{2}, 1)\) and \(\bar{\mu}= (0, \frac{1}{2})\) satisfy the conditions in Theorem 3.3, hence Theorem 3.3 holds.
- (f)
\(\bar{x}= 1\) is the efficient solution of \(\widehat{(\mathrm{MP}_{\mathrm{E}})}\) and it is easy to check the problem \(\widehat{(\mathrm{MP}_{\mathrm{E}})}_{1}\) satisfies the Kuhn-Tucker constraint qualification [4], and there exist \(\bar{\lambda}=(\frac{1}{2}, 1)\) and \(\bar{\mu}= (0, \frac{1}{2})\) satisfying the conditions in Theorem 3.4, hence Theorem 3.4 holds.
4 Duality
- (1)
If \(J_{0}=Q\), then our mixed dual type (\(\mathrm{MD}_{\mathrm{E}}\)) (or (MD)) reduces to the Wolfe dual type.
- (2)
If \(J_{0}=\emptyset\), then our mixed dual type (\(\mathrm{MD}_{\mathrm{E}}\)) (or (MD)) reduces to the Mond-Weir dual type.
Theorem 4.1
Let \(E:M\to M\) be a surjective map. Then ū is an efficient solution of (\(\mathrm{MD}_{\mathrm{E}}\)) if and only if \(E(\bar{u})\) is an efficient solution of (MD).
Proof
By Lemma 3.1, we can obtain this theorem. □
Assume that f is an E-convex function and \(E:M\to M \) (\(M\subset\Bbb{R}^{n}\)) is a surjective map, by Lemma 3.1, we can study dual problem between (MP) and (MD). Here, we would like to study the dual problem between \({(\mathrm{MP}_{\mathrm{E}})}\) and \({(\mathrm{MD}_{\mathrm{E}})}\).
Theorem 4.2
(Weak duality)
- (a)
\(\lambda_{i}>0\) for all \(i=1,2,\ldots,p\), or
- (b)
\(\sum_{i=1}^{p}\lambda_{i}f_{i}(\cdot)+\sum_{j=1}^{q}\mu_{j}g_{j}(\cdot)\) is strictly E-convex at u,
Proof
Corollary 4.1
Assume that weak duality (Theorem 4.2) holds between (\(\mathrm{MP}_{\mathrm{E}}\)) and (\(\mathrm{MD}_{\mathrm{E}}\)). If \((\bar{u},\bar{\lambda},\bar{\mu})\) is feasible for (\(\mathrm{MD}_{\mathrm{E}}\)) with \(\bar{\mu}^{T}(g\circ E)(\bar{u})=0\) and if ū is feasible for (\(\mathrm{MP}_{\mathrm{E}}\)), then ū is efficient for (\(\mathrm{MP}_{\mathrm{E}}\)) and \((\bar{u},\bar{\lambda},\bar{\mu})\) is efficient for (\(\mathrm{MD}_{\mathrm{E}}\)).
Proof
Theorem 4.3
(Strong duality)
Let x̄ be an efficient solution for \({(\mathrm{MP}_{\mathrm{E}})}\) and assume that x̄ satisfies a constraint qualification [4] for \({(\mathrm{MP}_{\mathrm{E}})_{k}}\) for at least one \(k=1,2,\ldots,p\). Then there exist \(\bar{\lambda}\in\Bbb{R}^{p}\) and \(\bar{\mu}\in\Bbb{R}^{q}\) such that \((\bar{x},\bar{\lambda},\bar{\mu})\) is feasible for \({(\mathrm{MD}_{\mathrm{E}})}\). Moreover, if weak duality (Theorem 4.2) holds between \({(\mathrm{MP}_{\mathrm{E}})}\) and \({(\mathrm{MD}_{\mathrm{E}})}\), then \((\bar{x},\bar{\lambda},\bar{\mu})\) is efficient for \({(\mathrm{MD}_{\mathrm{E}})}\).
Proof
Since x̄ is efficient for (\(\mathrm{MP}_{\mathrm{E}}\)), by Lemma 3.1, x̄ solves \((\mathrm{MP}_{\mathrm{E}})_{k}\) for all \(k=1,2,\ldots,p\). By hypothesis, there exists a \(k\in P= \{ 1,2,\ldots,p \}\) for which x̄ satisfies a constraint qualification of \((\mathrm{MP}_{\mathrm{E}})_{k}\).
Example 4.1
As we know the feasible set of \(\widehat{(\mathrm{MP}_{\mathrm{E}})}\) is \(E(M)=[1, 2]\) and it is easy to check that the feasible set of \(\widehat {(\mathrm{MD}_{\mathrm{E}})}\) denoted by G is \(G=\{ (u, \lambda, \mu)\in\Bbb{R}\times\Bbb{R}^{2}\times\Bbb{R}^{2}\mid \lambda_{2}(2u-3)+1+\mu_{1}-\mu_{2}=0, \mu_{2}(-u+1)\geqq0, 0\leqq \lambda_{2} \leqq1, \mu_{1}\geqq0, \mu_{2}\geqq0\}\).
Finally we turn to strong duality (Theorem 4.3), as we know \(\bar{x}=1\) is an efficient solution of \(\widehat{(\mathrm{MP}_{\mathrm{E}})}\), and with the satisfy of Kuhn-Tucker constraint qualification [4], it is easy to check that there exist \(\bar{\lambda}=(1, 0)\) and \(\bar{\mu}=(0, 1)\) such that \((\bar{x}, \bar{\lambda}, \bar{\mu})=(1, (1,0), (0, 1))\) is a feasible solution of \(\widehat{(\mathrm{MD}_{\mathrm{E}})}\). Moreover, if weak duality (Theorem 4.2) holds, \((\bar{x}, \bar{\lambda}, \bar{\mu})=(1, (1,0), (0, 1))\) is efficient for \(\widehat{(\mathrm{MD}_{\mathrm{E}})}\), hence strong duality (Theorem 4.3) holds.
Declarations
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2A10008908).
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
References
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