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Optimality and mixed duality in multiobjective Econvex programming
Journal of Inequalities and Applications volume 2015, Article number: 335 (2015)
Abstract
In this paper, we consider a class of multiobjective Econvex programming problems with inequality constraints, where the objective and constraint functions are Econvex functions which were firstly introduced by Youness (J. Optim. Theory Appl. 102:439450, 1999). FritzJohn and KuhnTucker necessary and sufficient optimality theorems for the multiobjective Econvex 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 12:17371745, 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.
Introduction
The concepts of an Econvex set and an Econvex function were introduced first by Youness [1]. Subsequently, necessary and sufficient optimality criteria for a class of Econvex programming problems were discussed by Youness [2], and EFritzJohn and EKuhnTucker problems, which modified the FritzJohn and KuhnTucker problems, were also presented. In Megahed et al. [3], the concept of an Edifferentiable convex function which transforms a nondifferentiable 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 nondifferentiable function could be found by applying the FritzJohn and KuhnTucker conditions due to Mangasarian [4].
However, on the other hand, the results on Econvex programming in Youness [1] were not correct, and some counterexamples were given by Yang [5]. The results concerning the characterization of an Econvex function f in terms of its Eepigraph in Youness [1] were also not correct, and some characterizations of Econvex 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 semiEconvex functions was introduced by Chen [7], the concepts of Equasiconvex functions and strictly Equasiconvex functions were introduced by Syau and Stanley Lee [8], respectively.
In fact, after defining the Econvex function in 1999, Youness [1] pointed out that the Econvex function that he defined had more generalized results than a convex function. He dealt mainly with some properties of an Econvex set and an Econvex 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 Econvex set and Econvex 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 Econvexity conditions. We obtain the scalarization method due to Chankong and Haimes [12] for multiobjective Econvex programming. By employing this scalarization method, FritzJohn and KuhnTucker 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 Econvex conditions, weak and strong duality theorems between the primal and dual problems are established, and we also propose some examples to illustrate our results.
Preliminaries
Let \(\Bbb{R}^{n}\) denote the ndimensional Euclidean space. The following conventions for a vector in \(\Bbb{R}^{n}\) will be used in this paper:
We present some concepts of Econvex set and Econvex function; for convenience, we recall the definition of Econvex set first.
Definition 2.1
[1]
A set \(M\subset{\Bbb{R}^{n}}\) is said to be Econvex 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 Econvex 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 Econvex (since \(S_{1}\) is convex). It is easy to check that \(E(S_{1})\) is Econvex 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]
A function \(f: {\Bbb{R}^{n}}\to{\Bbb{R}}\) is said to be Econvex on \(M \in{\Bbb{R}^{n}}\) iff there is a map \(E: {\Bbb{R}^{n}} \to{\Bbb{R}^{n}}\) such that M is an Econvex set and
for each \(x,y \in M\) and \(0 \leqq\lambda\leqq1\). Moreover, if
then f is called Econcave on M. If the inequality signs in the above two inequalities are strict, then f is called strictly Econvex and strictly Econcave, respectively.
Remark 2.1
Let f, g be Econvex on M. Then \(f + g\), αf (\(\alpha\geqq 0\)) are Econvex on the set M.
It is easy to check that every convex function f on a convex set M is an Econvex function, where E is the identity map. But the converse may not hold, we recall the example from [1].
Example 2.2
Define the function \(f: \Bbb{R}\to\Bbb{R}\) as
and let \(E:{\Bbb{R}} \to{\Bbb{R}}\) be defined as \(E(x)=x^{2}\). Then \(\Bbb{R}\) is an Econvex set and f is Econvex but not convex.
Obviously, if f is a realvalued differentiable function on an Econvex set \(M \subset{\Bbb{R}^{n}}\), we can define a differentiable Econvex function in the following.
Definition 2.3
f is Econvex on M if and only if for each \(x,y \in M\)
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 Econvex 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 Econvex 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]).
Consider the following multiobjective nonlinear program:
where \(f_{i}:{\Bbb{R}^{n}}\to{\Bbb{R}}\) , \(i\in P=\{1,2,\ldots,p\}\) and \(g_{j}:{\Bbb{R}^{n}}\to{\Bbb{R}}\) , \(j\in Q=\{1,2,\ldots,m\}\) are Econvex functions.
Then we give the following Econvex program related to (MP):
where \(f_{i}\circ E\) , \(i\in P\) and \(g_{j}\circ E\) , \(j\in Q\) are differentiable on M.
It states that, for a surjective map E, if f is Econvex, then \(f\circ E\) is obviously convex.
Definition 3.1
A point \(\bar{x} \in E(M)\) is said to be an efficient solution of \({(\mathrm{MP}_{\mathrm{E}})}\) if and only if there is no other \(x\in E(M)\) such that
and
where \(P= \{1,2,\ldots,p \}\), that is
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 Econvex set.) in Youness [1] is incorrect.
Example 3.1
In (MP), \(g_{j}\), \(j\in Q\) are Econvex, but M does not always need to be Econvex set.
Let \(g(x)=x\in\Bbb{R} \) and define the map E as \(E(x)=x\). Then \(g(x)\) is Econvex. Take \(x=1\), \(y=1/2\). Then \(g(1)=1\), \(g(1/2)=1/2\).
So, \(1,1/2 \in M= \{x\in\Bbb{R}\mid g(x) \leqq0 \}\). But, for all \(\lambda\in[0,1]\),
Hence, M is not Econvex set.
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 Econvexity assumption, we now give the sufficient optimality condition.
Theorem 3.2
(Sufficient optimality condition)
Assume that \((\bar{x},\bar{\lambda},\bar{\mu})\) satisfies the following conditions:
where \(\bar{\lambda}\in\Bbb{R}^{p}\), \(\bar{\mu}\in\Bbb{R}^{m}\).
Then x̄ is an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)).
Proof
Suppose that x̄ is not an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)), then there exists \(x^{*}\in E(M)\) such that
Since \(f_{i}\) and \(g_{j}\) are Econvex and \(f_{i}\circ E\) and \(g_{j}\circ E\) are differentiable on M, for any \(x\in E(M)\), we have
Since \(\bar{\lambda}>0\), \(\bar{\mu}\geqq0\), from (3.2) and (3.3), for each \(i\in P\) and \(j\in Q\), we have
Since \(\bar{\lambda}\nabla(f\circ E)(\bar{x})+\bar{\mu}\nabla(g\circ E)(\bar{x})=0\), \(\bar{\mu}(g\circ E)(\bar{x})=0\) and \((g\circ E)(\bar{x}) \leqq0\), we get
which contradicts (3.1). □
Remark 3.1
If we replace the Econvexity of \(f_{i}\) and \(\bar{\lambda}> 0\) by the strictly Econvexity 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
x̄ is an efficient solution for \({(\mathrm{MP}_{\mathrm{E}})}\) if and only if x̄ solves
for each \(k=1,2,\ldots,p\).
Proof
Suppose that x̄ is not a solution of \((\mathrm{MP}_{\mathrm{E}})_{k}\). Then there exists \(x\in E(M)\) such that
From (3.4) and (3.5), we conclude that x̄ is not efficient for \({(\mathrm{MP}_{\mathrm{E}})}\).
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
Without loss of generality, we assume that \(P\cap Q= \emptyset\). Set
and \(T=P^{k} \cup Q\). Then \((\mathrm{MP}_{\mathrm{E}})_{k}\) is equivalent to the following problem:
In order to obtain the necessary optimality condition, we employ the following generalized linearization lemma due to Mangasarian [4].
Lemma 3.2
Let x̄ be a local solution of \((\mathrm{MP}_{\mathrm{E}})_{k}\), let \(f_{k}\circ E\), for each \(k\in P\) and \(G_{t}\circ E\), \(t\in T\) be differentiable at x̄. Then the system
has no solution \(z\in\Bbb{R}^{n}\), for each \(k\in P\), where we denote
We establish the following FritzJohn necessary optimality criteria by using Lemma 3.2.
Theorem 3.3
(FritzJohn necessary condition)
Assume that \(\bar{x}\in E(M)\) is an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)), then there exist \(\bar{\lambda}\in\Bbb{R}^{p}\), \(\bar{\mu}\in \Bbb{R}^{m}\) such that
Proof
Since x̄ is an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)), then from Lemma 3.1, x̄ solves \((\mathrm{MP}_{\mathrm{E}})_{k}\) for each \(k\in P\). By Lemma 3.2 and Remark 3.2, we see that the system
has no solution \(z\in\Bbb{R}^{n}\). Hence by Motzin’s theorem [4], there exist \(\bar{\lambda}_{k}\), \(\bar{\mu}_{W}\), \(\bar{\mu}_{V}\) such that
Since \((G_{W}\circ E)(\bar{x})=0\) and \((G_{V}\circ E)(\bar{x})=0\), it follows that if we define \(\bar{\mu}_{J}=0\) and \(\bar{\mu}=(\bar{\mu}_{W}, \bar{\mu}_{V}, \bar{\mu}_{J})\), then
here, we can reduce \(\bar{\mu}(g\circ E)(\bar{x})=0\). Thus \(\bar{\lambda}_{k}\nabla(f_{k}\circ E)(\bar{x})+\bar{\mu}(g\circ E)(\bar{x})=0\) and \((\bar{\lambda}_{k}, \bar{\mu}) \geq0\). Then, for each \(k\in P\), we have
Since \(x^{*}\in E(M)\), \((g\circ E)(x^{*}) \leqq0\).
The proof is complete. □
Theorem 3.4
(KuhnTucker necessary condition)
If \(\bar{x}\in E(M)\) is an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)) and \(G_{t}\circ E\), \(t\in T\) satisfies a constraint qualification [4] for \((\mathrm{MP}_{\mathrm{E}})_{k}\) for at least one \(k\in P\). Then there exist \(\bar{\lambda}\in\Bbb{R}^{p}\) and \(\bar{\mu}\in\Bbb{R}^{m}\) such that
Proof
Since x̄ is an efficient solution of (\(\mathrm{MP}_{\mathrm{E}}\)), then by Theorem 3.3 there exist \(\bar{\lambda}\in\Bbb{R}^{p}\), \(\bar{\mu}\in\Bbb{R}^{m}\) such that \((\bar{x}, \bar{\lambda}, \bar{\mu})\) satisfies
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.
Suppose that \(\bar{\lambda}_{k} = 0\) for all \(k\in P\). Since \(\bar{\mu}_{J} =0\) as we define in the proof of Theorem 3.3, we have \(\bar{\mu}_{W}\nabla(G_{W}\circ E)(\bar{x}) + \bar{\mu}_{V}\nabla (G_{V}\circ E)(\bar{x})=0\), \(\bar{\mu}_{W} \geq0\), \(\bar{\mu}_{V} \geqq0\). Since \(G_{t}\circ E\) satisfies the ArrowHurwiczUzawa constraint qualification [4] at x̄ for \({(\mathrm{MP}_{\mathrm{E}})_{k}}\) for at least one \(k\in P\), there exists \(\bar{z}\in\Bbb{R}^{n}\) such that
Multiplying (3.6) and (3.7) by \(\bar{\mu}_{W}\) and \(\bar{\mu}_{V}\), respectively, then yields
which contradicts the fact that
Hence \(\bar{\lambda}_{k} > 0\) for at least one \(k\in P\). Then we obtain \(\bar{\lambda}\geq0\). □
Remark 3.3
If we replace our surjective assumption of E by bijection (or linearity) of E, then our FritzJohn and KuhnTucker necessary optimality results reduce to the ones in Megahed et al. [3] (or Youness [2]).
Example 3.2
Consider the following problem:
where \(f_{1}(x)=x\), \(f_{2}(x)=x^{2}\), \(g_{1}(x)=x1\), and \(g_{2}(x)=x\).
Let \(E:M\to E(M)\) defined by \(E(x)=x+1\) be the surjective map, then we get the following Econvex programming problem related to \(\widehat{(\mathrm{MP})}\):
where \((f_{1}\circ E)(x)=x1\), \((f_{2}\circ E)(x)=x^{2}2x+1\), \((g_{1}\circ E)(x)=x2\), and \((g_{2}\circ E)(x)=x+1\).

(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, where
$$\begin{aligned} \textstyle\begin{array}{@{}l@{\quad}l@{\quad}l@{}} \widehat{(\mathbf{MP}_{\mathbf{E}})}_{\mathbf{1}} &\mbox{Mimimize}& (f_{1}\circ E) (x),\\ &\mbox{subject to}& (f_{2}\circ E) (x)\leqq(f_{2}\circ E) ( \bar{x}),\\ &&x\in E(M), \end{array}\displaystyle \end{aligned}$$and
$$\begin{aligned} \textstyle\begin{array}{@{}l@{\quad}l@{\quad}l@{}} \widehat{(\mathbf{MP}_{\mathbf{E}})}_{\mathbf{2}} &\mbox{Mimimize}& (f_{2}\circ E) (x),\\ &\mbox{subject to}& (f_{1}\circ E) (x)\leqq(f_{1}\circ E) (\bar{x}),\\ &&x\in E(M). \end{array}\displaystyle \end{aligned}$$ 
(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 KuhnTucker 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.
Duality
Recently, several researchers found some results on mixed dual model under some generalized convexity; see [13–15], for example. In this section, first we establish the following mixed dual problem (MD) to (MP):
where \(J_{\alpha}\subset Q=\{1,2,\ldots,q\}\), \(\alpha=0,1,\ldots,r\) with \(\bigcup_{\alpha=0}^{r} J_{\alpha}=Q\) and \(J_{\alpha}\cap J_{\beta}=\emptyset\) if \(\alpha\neq\beta\). \(\Lambda^{+}=\{\lambda\in\Bbb{R}^{p} \mid\lambda\geqq0, \lambda^{T}e=1, e=(1,\ldots, 1)^{T}\in\Bbb{R}^{p}\}\).
Then we formulate the following mixed dual problem (\(\mathrm{MD}_{\mathrm{E}}\)) to (\(\mathrm{MP}_{\mathrm{E}}\)):
where \(J_{\alpha}\subset Q=\{1,2,\ldots,q\}\), \(\alpha=0,1,\ldots,r\) with \(\bigcup_{\alpha=0}^{r} J_{\alpha}=Q\) and \(J_{\alpha}\cap J_{\beta}=\emptyset\) if \(\alpha\neq\beta\); \(\Lambda^{+}=\{\lambda\in\Bbb{R}^{p} \mid\lambda\geqq0, \lambda^{T}e=1, e=(1,\ldots, 1)^{T}\in\Bbb{R}^{p}\}\).

(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 MondWeir 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 Econvex 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)
Assume that for all feasible x of \({(\mathrm{MP}_{\mathrm{E}})}\) and all feasible \((u,\lambda,\mu)\) of \({(\mathrm{MD}_{\mathrm{E}})}\), \(f_{i}\), \(g_{j}\) are Econvex functions. If also either

(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 Econvex at u,
then the following cannot hold:
Proof
Suppose to the contrary that (4.1) and (4.2) hold. Since x is feasible for \({(\mathrm{MP}_{\mathrm{E}})}\) and \(\mu\geqq0\), from (4.1) and (4.2), we imply
If hypothesis (a) holds, then with \(\sum_{i=1}^{p}\lambda_{i}=1\), one has
and since \(f_{i}\), \(g_{j}\) are Econvex and \(\lambda_{i}>0\), \(i=1,2,\ldots ,p\), \(\mu\geqq0\), it now follows from (4.5) that
which contradicts the fact that
On the other hand, since \(\lambda_{i}\geqq0\), \(i=1,2,\ldots,p\) and \(\sum_{i=1}^{p}\lambda_{i}=1\), (4.3) and (4.4) imply
Now (4.6) and hypothesis (b) imply (4.5), which also contradicts the fact that
□
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
Suppose that ū is not efficient for (\(\mathrm{MP}_{\mathrm{E}}\)). Then there exists a feasible x for (\(\mathrm{MP}_{\mathrm{E}}\)) such that
By hypothesis \(\bar{\mu}^{T}(g\circ E)(\bar{u})=0\), so (4.7) and (4.8) can be written as
Since \((\bar{u},\bar{\lambda}, \bar{\mu})\) is feasible for (\(\mathrm{MD}_{\mathrm{E}}\)) and x is feasible for (\(\mathrm{MP}_{\mathrm{E}}\)), these inequalities contradict weak duality (Theorem 4.2).
Also, suppose that \((\bar{u},\bar{\lambda},\bar{\mu})\) is not efficient for (\(\mathrm{MD}_{\mathrm{E}}\)), then there exists a feasible solution \((u,\lambda ,\mu)\) for (\(\mathrm{MD}_{\mathrm{E}}\)) such that
Since \(\bar{\mu}^{T}(g\circ E)(\bar{u})=0\), (4.9) and (4.10) reduce to
Since ū is feasible for (\(\mathrm{MP}_{\mathrm{E}}\)), these inequalities contradict weak duality (Theorem 4.2). Therefore ū and \((\bar{u},\bar{\lambda},\bar{\mu})\) are efficient for their respective problems. □
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 \((\mathrm{MP}_{\mathrm{E}})_{k}\).
From the KuhnTucker necessary conditions [4], there exist \(\lambda_{i}\geqq0\) such that, for all \(i\neq k\) and \(\mu\geqq0\), \(\mu\in\Bbb{R}^{m}\),
Now we divide all terms in (4.11) and (4.12) by \(1+\sum_{i\neq k} \lambda_{i}\) and set \(\bar{\lambda}_{k}={\frac{1}{1+\sum_{i\neq k} \lambda_{i}}} > 0\), \(\bar{\lambda}_{j}={\frac{\lambda_{i}}{1+\sum_{i\neq k} \lambda_{i}}} \geqq0\), \(\bar{\mu}={\frac{\mu}{1+\sum_{i\neq k} \lambda_{i}}} \geqq0\). Since weak duality (Theorem 4.2) holds, from Corollary 4.1, we conclude that \((\bar{x},\bar{\lambda},\bar{\mu})\) is feasible as well as efficient for (\(\mathrm{MD}_{\mathrm{E}}\)). □
Example 4.1
Recall the problem in Example 3.2, and we now give the mixed dual problem to \(\widehat{(\mathrm{MP}_{\mathrm{E}})}\).
where \(\lambda=(\lambda_{1}, \lambda_{2})\) and \(\mu=(\mu_{1}, \mu_{2})\).
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 \(G=\{ (u, \lambda, \mu)\in\Bbb{R}\times\Bbb{R}^{2}\times\Bbb{R}^{2}\mid \lambda_{2}(2u3)+1+\mu_{1}\mu_{2}=0, \mu_{2}(u+1)\geqq0, 0\leqq \lambda_{2} \leqq1, \mu_{1}\geqq0, \mu_{2}\geqq0\}\).
Now we check the validity of weak duality, say Theorem 4.2, that is, for any feasible point \(x\in E(M)\) and \((u, \lambda, \mu)\in G\) with positive \(\lambda_{1}\) and \(\lambda_{2}\),
cannot hold. In fact, by the positivity of \(\lambda_{2}\), we have \(G=\{ (u, \lambda, \mu)\in\Bbb{R}\times\Bbb{R}^{2}\times\Bbb{R}^{2}\vert\ 1\leqq u\leqq{3\over 2}{{1+\mu_{1}}\over {2\lambda_{2}}}, 0< \lambda _{2} < 1, \mu_{1}\geqq0\}\), and
which implies (4.13) cannot hold, and we conclude that weak duality (Theorem 4.2) holds.
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 KuhnTucker 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.
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Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2013R1A1A2A10008908).
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MSC
 90C29
 90C30
 69K05
Keywords
 Econvex function
 mixed duality
 multiobjective programming
 optimality condition