Abstract
In this paper, we establish the sufficient Karush-Kuhn-Tucker (KKT) optimality conditions for the set-valued fractional programming problem (FP) via contingent epiderivative under ρ-cone convexity. We also study the duality results of parametric (PD), Mond-Weir (MWD), Wolfe (WD) and mixed (MD) types for the problem (FP).
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1 Introduction
In the past few years, the set-valued optimization theory has attracted the attention of many researchers towards this expanding branch of optimization. Many optimization problems in mathematical economics, optimal control, differential inclusions, image processing, viability theory and many more are set-valued optimization problems (SVOP) that involve set-valued maps as objective functions and constraints. Various types of differentiability notions of set-valued maps have been introduced in (SVOP). The notion of contingent epiderivative of set-valued maps, introduced by Jahn and Rauh [8], has a significant role to establish optimality conditions of (SVOP). The notion of cone convexity of set-valued maps, introduced by Borwein [6], also plays a vital role in this case. An important class of (SVOP) is set-valued fractional programming problems. In 1997, Bhatia and Mehra [4] introduced the notion of cone preinvexity for set-valued maps and obtained the Lagrangian duality results for the set-valued fractional programming problems. Later, they [5] proved the duality results for Geoffrion efficient solutions of the set-valued fractional programming problems under cone convexity assumptions. In 2013, Gadhi and Jawhar [7] established the necessary optimality conditions of the set-valued fractional programming problems without any convex separation approach. Many authors like Kaul and Lyall [9], Bhatia and Garg [3], Suneja and Gupta [11], Suneja and Lalitha [12] and Lee and Ho [10] established the optimality conditions and proved the duality theorems for vector-valued fractional programming problems under generalized convexity assumptions.
This paper is organized as follows. In Section 2, we recall some definitions and preliminary concepts of the set-valued optimization theory. In Section 3, we establish the sufficient optimality conditions for weak efficiency of the set-valued fractional programming problems under generalized cone convexity assumptions. We also establish the duality results of various types.
2 Definitions and preliminaries
Let Y be a real normed space and K be a nonempty subset of Y. Then K is called a cone if λ y∈K, for all y∈K and λ≥0. Further, the cone K is called pointed if K∩(−K)={0 Y }, solid if int (K)≠∅, closed if \(\overline {K} = K\) and convex if
where int(K) and \(\overline {K}\) denote the interior and closure of K, respectively and 0 Y is the zero element of Y.
The positive orthant \({\mathbb {R}}_{+}^{m}\) of \(\mathbb {R}^{m}\), defined by
is a solid pointed closed convex cone of \(\mathbb {R}^{m}\).
Various types of minimal points can be defined with respect to a solid pointed convex cone in a normed space.
Definition 2.1
Let B be a nonempty subset of a normed space Y, K be a solid pointed convex cone in Y and y ′∈B. Then,
-
(i) y ′ is an ideal minimal point of B if y ′−y∈−K, for all y∈B.
-
(ii) y ′ is a minimal point of B if there is no y∈B∖{y ′} such that y−y ′∈K.
-
(iii) y ′ is a weakly minimal point of B if there is no y∈B such that y−y ′∈int(K).
The contingent epiderivative of set-valued map is defined with the help of contingent cone. Aubin [1, 2] introduced the contingent cone in normed spaces.
Definition 2.2
[1, 2] Let B be a nonempty subset of a normed space Y and \(y^{\prime }\in \overline {B}\). Then, the contingent cone to B at y ′, denoted by T(B,y ′), is defined as:
y∈T(B,y ′) if there exist sequences {λ n } in \(\mathbb {R}\), with λ n →0+ and {y n } in Y, with y n →y, such that
or, there exist sequences {t n } in \(\mathbb {R}\), with t n >0 and \(\{{y}_{n}^{\prime }\}\) in B, with \({y}_{n}^{\prime } \rightarrow y^{\prime }\), such that
Let X and Y be real normed spaces, 2Y be the set of all subsets of Y and K be a solid pointed convex cone in Y. Let F:X→2Y be a set-valued map from X to Y i.e., F(x)⊆Y, for all x∈X. The effective domain, graph and epigraph of the set-valued map F are defined by:
and
In 1997, Jahn and Rauh [8] introduced the notion of contingent epiderivative of set-valued maps.
Definition 2.3
[8] Let F:X→2Y be a set-valued map and (x ′,y ′)∈gr(F). Then the single-valued map D ↑ F(x ′,y ′):X→Y is called the contingent epiderivative of F at (x ′,y ′) if
Jahn and Rauh [8] also showed that when \(f: X\rightarrow \mathbb {R}\) is a real-valued map, being continuous at the point x ′∈X and f is convex, then
where f ′(x ′)(u) is the directional derivative of f at x ′ in the direction u.
Borwein [6] introduced cone convexity of set-valued maps. Later, Jahn and Rauh [8] characterized cone convex set-valued maps in terms of contingent epiderivative.
Definition 2.4
[6] Let A be a nonempty convex subset of X. A set-valued map F:X→2Y, with A⊆dom(F), is called K-convex on A if ∀x 1,x 2∈A and λ∈[0,1],
It is clear that if the set-valued map F : X → 2Y is K-convex on A, then epi (F) is a convex subset of X×Y.
Lemma 2.1
[8] Let F:X→2 Y be K-convex on a nonempty convex subset A of X. Let x ′ ∈A and y ′ ∈F(x ′ ). Assume that F is contingent epiderivable at (x ′ ,y ′ ). Then, for all x∈A,
Let X be a real normed space and A be a nonempty subset of X. Let \(F:X\rightarrow 2^{\mathbb {R}^{m}}\), \(G:X\rightarrow 2^{\mathbb {R}^{m}}\) and \(H:X\rightarrow 2^{\mathbb {R}^{k}}\) be set-valued maps, with
Throughout the paper, denote
and
Let F=(F 1,F 2,...,F m ),G=(G 1,G 2,...,G m ) and H=(H 1,H 2,...,H k ), where the set-valued maps \(F_{i}:X \rightarrow 2^{\mathbb {R}}\), \(G_{i}:X \rightarrow 2^{\mathbb {R}}\), i=1,2,...,m and \(H_{j}:X \rightarrow 2^{\mathbb {R}}\), j=1,2,...,k, are defined by:
and
Assume that \(F_{i}(x)\subseteq \mathbb {R}_{+}\) and \(G_{i}(x)\subseteq \text {int}(\mathbb {R}_{+}), \forall i=1,2,...,m\) and x∈A. Let \(\lambda ^{\prime }=(\lambda _{1}^{\prime },\lambda _{2}^{\prime },...,\lambda _{m}^{\prime })\in {\mathbb {R}}_{+}^{m}\). Define \(\frac {y}{z}\in \mathbb {R}^{m}\) and \(\lambda ^{\prime } z\in \mathbb {R}^{m}\) by:
and
For x∈A, define the subset \(\frac {F(x)}{G(x)}\) of \(\mathbb {R}^{m}\) by:
Consider the set-valued fractional programming problem:
The feasible set of the problem (FP) is
Definition 2.5
A point \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\in X\times \mathbb {R}^{m}\), with x ′∈S,y ′∈F(x ′) and z ′∈G(x ′), is called a minimizer of the problem (FP) if there exist no x∈S, y∈F(x) and z∈G(x) such that
Definition 2.6
A point \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\in X\times \mathbb {R}^{m}\), with x ′∈S, y ′∈F(x ′) and z ′∈G(x ′), is called a weak minimizer of the problem (FP) if there exist no x∈S, y∈F(x) and z∈G(x) such that
Consider the parametric problem (\(FP_{\lambda ^{\prime }}\)) associated with the set-valued fractional programming problem (FP):
Definition 2.7
A point \((x^{\prime }, y^{\prime }-\lambda ^{\prime }z^{\prime })\in X\times \mathbb {R}^{m}\), with x ′∈S, y ′∈F(x ′) and z ′∈G(x ′), is called a minimizer of the problem (\(FP_{\lambda ^{\prime }}\)), if there exist no x∈S, y∈F(x) and z∈G(x) such that
Definition 2.8
A point \((x^{\prime }, y^{\prime }-\lambda ^{\prime }z^{\prime })\in X\times \mathbb {R}^{m}\), with x ′∈S,y ′∈F(x ′) and z ′∈G(x ′), is called a weak minimizer of the problem (\(FP_{\lambda ^{\prime }}\)), if there exist no x∈S, y∈F(x) and z∈G(x) such that
Gadhi and Jawhar [7] proved the relation between the solutions of the problems (FP) and (\(FP_{\lambda ^{\prime }}\)) in the following theorem.
Lemma 2.2
[7] A point \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\in X\times \mathbb {R}^{m}\) is a weak minimizer of the problem ( FP ) if and only if \((x^{\prime }, \mathbf {0}_{\mathbb {R}^{m}})\) is a weak minimizer of the problem (\(FP_{\lambda ^{\prime }}\)), where \(\lambda ^{\prime }= \frac {y^{\prime }}{z^{\prime }}\).
Lemma 2.3
[13] Let \(x_{1}, x_{2}\in \mathbb {R}^{n}\) and λ∈[0,1]. Then,
3 Main results
We introduce the notion of ρ-cone convexity of set-valued maps. For ρ=0, we have the usual notion of cone convexity of set-valued maps.
Definition 3.1
Let X, Y be real normed spaces, A be a nonempty convex subset of X, K be a solid pointed convex cone in Y, e∈int(K) and F:X→2Y be a set-valued map, with A⊆dom(F). Then F is called ρ-K-convex with respect to e on A if there exists \(\rho \in \mathbb {R}\) such that
In the following theorem, we characterize ρ-cone convexity for contingent epiderivable set-valued maps.
Theorem 3.1
Let A be a nonempty convex subset of X, e∈int(K) and F:X→2 Y be ρ-K-convex with respect to e on A. Let x ′ ∈A and y ′ ∈F(x ′ ). Assume that F is contingent epiderivable at (x ′ ,y ′ ). Then,
Proof
Let x∈A and y∈F(x). As F is ρ-K-convex with respect to e on A,
Let {λ n } be a sequence in \(\mathbb {R}\) such that λ n ∈(0,1) and λ n →0, as n→∞. Consider two sequences {x n } in X and {y n } in Y, defined by:
and
Therefore,
It is clear that
and
Therefore,
Consequently,
which is true for all y∈F(x). Therefore,
□
We also establish a relation between the notions of cone convexity and ρ-cone convexity of set-valued maps, when \(X=\mathbb {R}^{n}\).
Theorem 3.2
Let A be a nonempty convex subset of \(\mathbb {R}^{n}\) , e∈int(K) and \(F:\mathbb {R}^{n} \rightarrow 2^{Y}\) be a set-valued map, with A⊆dom(F). Then \(F:\mathbb {R}^{n} \rightarrow 2^{Y}\) is ρ-K-convex with respect to e on A if and only if there exists a K-convex set-valued map \(\widetilde {F}:\mathbb {R}^{n} \rightarrow 2^{Y}\) on A, such that
Proof
Suppose that there exists a K-convex set-valued map \(\widetilde {F}:\mathbb {R}^{n} \rightarrow 2^{Y}\) on A such that Eq. 3.1 holds. We show that \(F:\mathbb {R}^{n} \rightarrow 2^{Y}\) is ρ-K-convex with respect to e on A, i.e., for all x 1,x 2∈A and λ∈[0,1],
Let y 1∈F(x 1) and y 2∈F(x 2). Then,
and
for some \(z_{1}\in \widetilde {F}(x_{1})\) and \(z_{2}\in \widetilde {F}(x_{2})\). As \(\widetilde {F}:\mathbb {R}^{n} \rightarrow 2^{Y}\) is K-convex on A, we have
Therefore,
Again, from Lemma 2.3, we have
Hence,
It follows that
Therefore, \(F:\mathbb {R}^{n} \rightarrow 2^{Y}\) is ρ-K-convex with respect to e on A.
Conversely, let \(F:\mathbb {R}^{n} \rightarrow 2^{Y}\) be ρ-K-convex with respect to e on A. Consider the set-valued map \(\widetilde {F}:\mathbb {R}^{n} \rightarrow 2^{Y}\) defined by
To show that \(\widetilde {F}\) is K-convex on A, let \({x}_{1}^{\prime }, {x}_{2}^{\prime }\in A\), \(z_{1}^{\prime }\in \widetilde {F}({x}_{1}^{\prime })\) and \(z_{2}^{\prime }\in \widetilde {F}\left ({x}_{2}^{\prime }\right )\). Then,
and
for some \({y}_{1}^{\prime }\in F\left ({x}_{1}^{\prime }\right )\) and \({y}_{2}^{\prime }\in F\left ({x}_{2}^{\prime }\right )\). Therefore,
Hence,
Consequently, \(\widetilde {F}\) is a K-convex set-valued map on A. □
When ρ=0, ρ-cone convex set-valued map becomes cone convex. We construct an example of ρ-cone convex set-valued map, which is not cone convex.
Example 3.1
Consider the set-valued map \(F:[-1, 1]\subseteq \mathbb {R}\rightarrow 2^{{\mathbb {R}}^{2}}\) defined by:
Let t 1=−1, t 2=1 and \(\lambda =\frac {1}{2}\). So, λ t 1+(1−λ)t 2=0. Therefore,
It is clear that
So,
Hence, F is not \(\mathbb {R}^{2}_{+}\)-convex on [−1,1]. Assume that ρ=−2. Then the set-valued map \(\widetilde {F}:[-1, 1]\subseteq \mathbb {R}\rightarrow 2^{{\mathbb {R}}^{2}}\), defined by Eq. 3.2, is given by
We can easily prove that \(\widetilde {F}\) is \(\mathbb {R}^{2}_{+}\)-convex on [−1,1]. Hence, F is (−2)-\(\mathbb {R}^{2}_{+}\)-convex with respect to e=(1,1) on [−1,1].
3.1 Optimality conditions
We establish the sufficient optimality conditions for the problem (FP), assuming that the objective and constraint set-valued maps are ρ-cone convex as well as contingent epiderivable.
Theorem 3.3
(Sufficient optimality conditions) Let A be a nonempty convex subset of X, x ′ be an element of the feasible set S of the problem ( FP ), y ′ ∈F(x ′ ), z ′ ∈G(x ′), \(\lambda ^{\prime }= \frac {y^{\prime }}{z^{\prime }}\) and w ′ ∈H(x ′ )∩(−L). Assume that F is ρ 1 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , −λ ′ G is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\) , on A. Let F be contingent epiderivable at (x ′ ,y ′ ), −λ ′ G be contingent epiderivable at (x ′ ,−λ ′ z ′ ) and H be contingent epiderivable at (x ′ ,w ′ ). Suppose that there exists \((y^{\ast }, z^{\ast })\in {\mathbb {R}}_{+}^{m} \times \mathbb {R}^{k}_{+}\) , with \(y^{\ast }\neq \mathbf {0}_{\mathbb {R}^{m}}\) , and
such that
and
Then \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\) is a weak minimizer of the problem ( FP ).
Proof
We prove the theorem by the method of contradiction. Let \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\) be not a weak minimzer of the problem (FP). Then there exist x∈S, y∈F(x) and z∈G(x) such that
As y ′−λ ′ z ′=0, we have
So,
Hence,
Again, as y ′−λ ′ z ′=0, we have
Since x∈S, there exists an element \(w\in H(x)\cap \left (-\mathbb {R}^{k}_{+}\right )\). Therefore,
So,
Hence,
As F is ρ 1-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}, -\lambda ^{\prime }G\) is ρ 2-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3-\(\mathbb {R}^{k}_{+}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, we have
and
Hence,
and
Hence, from Eqs. 3.3 and 3.4, we have
which contradicts Eq. 3.7. Consequently, (x ′,y ′) is a weak minimizer of the problem (FP). □
We can also prove the following theorem by the above approach.
Theorem 3.4
(Sufficient optimality conditions) Let A be a nonempty convex subset of X, x ′ be an element of the feasible set S of the problem ( FP), y ′ ∈F(x ′ ), z ′ ∈G(x ′ ) and w ′ ∈H(x ′ )∩(−L). Assume that z ′ F is ρ 1-\({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}, -y^{\prime }G\) is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\) , on A. Suppose that there exists \((y^{\ast }, z^{\ast })\in {\mathbb {R}}_{+}^{m} \times \mathbb {R}^{k}_{+}\) , with \(y^{\ast }\neq \mathbf {0}_{\mathbb {R}^{m}}\) , and Eqs. 3.3 and 3.6 are satisfied, with
Then \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\) is a weak minimizer of the problem (FP).
Now, we formulate the duals of parametric (PD), Mond-Weir (MWD), Wolfe (WD) and mixed (MD) types for the problem (FP) and study the corresponding duality theorems. We give the proofs of the duality theorems of parametric (PD) and Mond-Weir (MWD) types. We state the duality theorems of Wolfe (WD) and mixed (MD) types whose proofs are very similar to the former ones, hence omitted.
3.2 Parametric type dual
We consider the parametric type dual (PD) associated the problem (FP).
A point (x ′,y ′,z ′,λ ′,w ′,y ∗,z ∗) satisfying all the constraints of the problem (PD) is called a feasible point of (PD).
Definition 3.2
A feasible point (x ′,y ′,z ′,λ ′,w ′,y ∗,z ∗) of the problem (PD) is called a weak maximizer of (PD) if there exists no feasible point \(\left (x, y, z, \lambda , w, y^{\ast }_{1}, z^{\ast }_{1}\right )\) of (PD) such that
Theorem 3.5
(Weak Duality) Let A be a nonempty convex subset of X, \(\overline {x}\) be an element of the feasible set S of the problem (FP) and (x ′ ,y ′ ,z ′ ,λ ′ ,y ∗ ,z ∗) be a feasible point of the problem (PD). Assume that F is ρ 1 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , −λ ′ G is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\) , on A, such that
Then,
Proof
We prove the theorem by the method of contradiction. Suppose that for some \(\overline {y}\in F(\overline {x})\) and \(\overline {z}\in G(\overline {x})\),
Therefore,
Hence,
So,
Therefore,
Again, from the constraints of (PD),
Therefore,
Again, since \(\overline {x}\in S\), we have
We choose \(\overline {w}\in H(\overline {x})\cap \left (-\mathbb {R}^{k}_{+}\right )\). So,
Again, from the constraints of (PD), we have
So,
Hence,
As F is ρ 1-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}, -\lambda ^{\prime }G\) is ρ 2-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3-\(\mathbb {R}^{k}_{+}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, we have
and
Hence,
and
Hence, from the constraints of (PD) and Eq. 3.9, we have
which contradicts Eq. 3.10. Therefore,
Since \(\overline {y}\in F(\overline {x})\) is arbitrary, we have
□
Theorem 3.6
(Strong Duality) Let \((x^{\prime }, \frac {y^{\prime }}{z^{\prime }})\) be a weak minimizer of the problem (FP) and \(w^{\prime }\in H(x^{\prime })\cap \left (-\mathbb {R}^{k}_{+}\right )\). Assume that for some \((y^{\ast }, z^{\ast })\in {\mathbb {R}}_{+}^{m} \times \mathbb {R}^{k}_{+} \) , with \(\langle y^{\ast }, \mathbf {1}_{\mathbb {R}^{m}}\rangle =1\) and \(\lambda ^{\prime } \in \mathbb {R}^{m}\), Eqs. 3.4, 3.5 and 3.6 are satisfied at (x ′ ,y ′ ,z ′ ,λ ′ ,y ∗ ,z ∗ ). Then (x ′ ,y ′ ,z ′ ,λ ′ ,y ∗ ,z ∗ ) is a feasible solution of the problem (PD). Further, if the weak duality Theorem 3.5 between (FP) and (PD) holds, then (x ′ ,y ′ ,z ′ ,λ ′ ,y ∗ ,z ∗) is a weak maximizer of (PD).
Proof
As the Eqs. 3.4, 3.5 and 3.6 are satisfied at (x ′,y ′,z ′,λ ′,y ∗,z ∗), we have
and
Hence (x ′,y ′,z ′,λ ′,y ∗,z ∗) is a feasible solution of (PD). Suppose that the weak duality Theorem 3.5 between (FP) and (PD) holds and (x ′,y ′,z ′,λ ′,y ∗,z ∗) is not a weak maximizer of (PD). Then there exits a feasible point \((x, y, z, \lambda , y^{\ast }_{1}, z^{\ast }_{1})\) of (PD) such that
As y ′−λ ′ z ′=0,
which contradicts the weak duality Theorem 3.5 between (FP) and (PD). Consequently, (x ′,y ′,z ′,λ ′,y ∗,z ∗) is a weak maximizer of (PD). □
Theorem 3.7
(Converse Duality) Let A be a nonempty convex subset of X and (x ′ ,y ′ ,z ′ ,λ ′ ,y ∗ ,z ∗) be a feasible point of the problem (PD), where \(\lambda ^{\prime }= \frac {y^{\prime }}{z^{\prime }}\) . Assume that F is ρ 1-\({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , −λ ′ G is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, satisfying Eq. 3.9. If x ′ is an element of the feasible set S of the problem (FP), then \(\left (x^{\prime },\frac {y^{\prime }}{z^{\prime }}\right )\) is a weak minimizer of the problem (FP).
Proof
We prove the theorem by the method of contradiction. Suppose \(\left (x^{\prime },\frac {y^{\prime }}{z^{\prime }}\right )\) is not a weak minimzer of the problem (FP). Therefore there exist x∈S,y∈F(x) and z∈G(x) such that
As \(\lambda ^{\prime }= \frac {y^{\prime }}{z^{\prime }}\), we have
So,
Hence,
Again, from the constraints of (PD),
Therefore,
Since x∈S, there exists an element
Therefore,
We have
Hence,
As F is ρ 1-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\), −λ ′ G is ρ 2-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3-\(\mathbb {R}^{k}_{+}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, we have
and
Hence,
and
Hence, from the constraints of (PD) and Eq. 3.9, we have
which contradicts Eq. 3.11. Consequently, \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\) is a weak minimizer of the problem (FP). □
3.3 Mond-Weir type dual
We consider the Mond-Weir type dual (MWD) associated the problem (FP).
A point (x ′,y ′,z ′,w ′,y ∗,z ∗) which satisfies all the constraints of the problem (MWD) is called a feasible point of (MWD).
Definition 3.3
A feasible point (x ′,y ′,z ′,w ′,y ∗,z ∗) of the problem (MWD) is called a weak maximizer of (MWD) if there exists no feasible point \((x, y, z, w, y^{\ast }_{1}, z^{\ast }_{1})\) of (MWD) such that
Theorem 3.8
(Weak Duality) Let A be a nonempty convex subset of X, \(\overline {x}\) be an element of the feasible set S of the problem (FP) and (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗) be a feasible point of the problem (MWD). Assume that z ′ F is ρ 1-\({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , −y ′ G is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, satisfying Eq. 3.9. Then,
Proof
We prove the theorem by the method of contradiction. Suppose that for some \(\overline {y}\in F(\overline {x})\) and \(\overline {z}\in G(\overline {x})\),
Therefore,
So,
Hence,
As \(\overline {x}\in S\), we have
We choose \(\overline {w}\in H(\overline {x})\cap \left (-\mathbb {R}^{k}_{+}\right )\). So,
Again, from the constraints of (MWD), we have
So,
Hence,
As z ′ F is ρ 1-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}, -y^{\prime }G\) is ρ 2-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3-\(\mathbb {R}^{k}_{+}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, we have
and
Hence,
and
Hence, from the constraints of (MWD) and Eq. 3.9, we have
which contradicts Eq. 3.12. Therefore,
Since \(\overline {y}\in F(\overline {x})\) is arbitrary, we have
□
Theorem 3.9
(Strong Duality) Let \(\left (x^{\prime },\frac {y^{\prime }}{z^{\prime }}\right )\) be a weak minimizer of the problem (FP) and \(w^{\prime }\in H(x^{\prime })\cap \left (-\mathbb {R}^{k}_{+}\right )\) . Assume that for some \((y^{\ast }, z^{\ast })\in {\mathbb {R}}_{+}^{m} \times \mathbb {R}^{k}_{+} \) , with \(\langle y^{\ast }, \mathbf {1}_{\mathbb {R}^{m}}\rangle =1\) , Eqs. 3.6 and 3.8 are satisfied at (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ). Then (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗) is a feasible solution of the problem (MWD). Further, if the weak duality Theorem 3.8 between (FP) and (MWD) holds, then (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗) is a weak maximizer of (MWD).
Proof
As Eqs. 3.6 and 3.8 are satisfied at (x ′,y ′,z ′,w ′,y ∗,z ∗), we have
and
So, (x ′,y ′,z ′,w ′,y ∗,z ∗) is a feasible solution of (MWD). Suppose that the weak duality Theorem 3.8 between (FP) and (MWD) holds and (x ′,y ′,z ′,w ′,y ∗,z ∗) is not a weak maximizer of (MWD). Then there exists a feasible point \((x, y, z, w, y^{\ast }_{1}, z^{\ast }_{1})\) of (MWD) such that
which contradicts the weak duality Theorem 3.8 between (FP) and (MWD). Consequently, (x ′,y ′,z ′,λ ′,y ∗,z ∗) is a weak maximizer of (MWD). □
Theorem 3.10
(Converse Duality) Let A be a nonempty convex subset of X and (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ) be a feasible point of the problem (MWD). Assume that z ′ F is ρ 1 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , −y ′ G is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, satisfying Eq. 3.9. If x ′ is an element of the feasible set S of the problem (FP), then \(\left (x^{\prime },\frac {y^{\prime }}{z^{\prime }}\right )\) is a weak minimizer of the problem (FP).
Proof
We prove the theorem by the method of contradiction. Suppose \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\) is not a weak minimzer of the problem (FP). Therefore there exist x∈S,y∈F(x) and z∈G(x) such that
So,
Therefore,
Again, since x∈S, we have
We choose \(w\in H(x)\cap \left (-\mathbb {R}^{k}_{+}\right )\). So,
From the constraints of (WD), we have
So,
Hence,
As z ′ F is ρ 1-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}, -y^{\prime }G\) is ρ 2-\({\mathbb {R}}_{+}^{m}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3-\(\mathbb {R}^{k}_{+}\)-convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, we have
and
Hence,
and
So, from the constraints of (WD) and Eq. 3.9, we have
which contradicts Eq. 3.13. Therefore \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\) is a weak minimizer of (FP). □
3.4 Wolfe type dual
We consider the Wolfe type dual (WD) associated the problem (FP).
A point (x ′,y ′,z ′,w ′,y ∗,z ∗) which satisfies all the constraints of the problem (WD) is called a feasible point of (WD).
Definition 3.4
A feasible point (x ′,y ′,z ′,w ′,y ∗,z ∗) of the problem (WD) is called a weak maximizer of (WD) if there exists no feasible point \((x, y, z, w, y^{\ast }_{1}, z^{\ast }_{1})\) of (WD) such that
Theorem 3.11
(Weak Duality) Let A be a nonempty convex subset of X, \(\overline {x}\) be an element of the feasible set S of the problem (FP) and (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗) be a feasible point of the problem (WD). Assume that z ′ F is ρ 1-\({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}, -y^{\prime }G\) is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, satisfying Eq. 3.9. Then,
Theorem 3.12
(Strong Duality) Let \((x^{\prime }, \frac {y^{\prime }}{z^{\prime }})\) be a weak minimizer of the problem (FP) and \(w^{\prime }\in H(x^{\prime })\cap \left (-\mathbb {R}^{k}_{+}\right )\). Assume that for some \((y^{\ast }, z^{\ast })\in {\mathbb {R}}_{+}^{m} \times \mathbb {R}^{k}_{+} \), with \(\langle y^{\ast }, \mathbf {1}_{\mathbb {R}^{m}}\rangle =1\), Eqs. 3.6 and 3.8 are satisfied at (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ). Then (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗) is a feasible solution of the problem (WD). Further, if the weak duality Theorem 3.11 between (FP) and (WD) holds, then (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ) is a weak maximizer of (WD).
Theorem 3.13
(Converse Duality) Let A be a nonempty convex subset of X, (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗) be a feasible point of the problem (WD) and 〈z ∗ ,w ′〉≥0. Assume that z ′ F is ρ 1 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , −y ′ G is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, satisfying Eq. 3.9. If x ′ is an element of the feasible set S of the problem (FP), then \((x^{\prime }, \frac {y^{\prime }}{z^{\prime }})\) is a weak minimizer of the problem (FP).
3.5 Mixed type dual
We consider the mixed type dual (MD) associated the problem (FP).
A point (x ′,y ′,z ′,w ′,y ∗,z ∗) satisfying all the constraints of the problem (MD) is called a feasible point of (MD).
Definition 3.5
A feasible point (x ′,y ′,z ′,w ′,y ∗,z ∗) of the problem (MD) is called a weak maximizer of (MD) if there exists no feasible point \((x, y, z, w, y^{\ast }_{1}, z^{\ast }_{1})\) of (MD) such that
Theorem 3.14
(Weak Duality) Let A be a nonempty convex subset of X, \(\overline {x}\) be an element of the feasible set S of the problem (FP) and (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ) be a feasible point of the problem (MD). Assume that z ′ F is \(\rho _{1} - {\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}, -y^{\prime }G\) is \(\rho _{2} - {\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\) , on A, satisfying Eq. 3.9. Then,
Theorem 3.15
(Strong Duality) Let \(\left (x^{\prime }, \frac {y^{\prime }}{z^{\prime }}\right )\) be a weak minimizer of the problem (FP) and \(w^{\prime }\in H(x^{\prime })\cap \left (-\mathbb {R}^{k}_{+}\right )\) . Assume that for some \((y^{\ast }, z^{\ast })\in {\mathbb {R}}_{+}^{m} \times \mathbb {R}^{k}_{+} \) , with \(\langle y^{\ast }, \mathbf {1}_{\mathbb {R}^{m}}\rangle =1\) , Eqs. 3.6 and 3.8 are satisfied at (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ). Then (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗) is a feasible solution of the problem (MD). Further, if the weak duality Theorem 3.14 between (FP) and (MD) holds, then (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ) is a weak maximizer of (MD).
Theorem 3.16
(Converse Duality) Let A be a nonempty convex subset of X and (x ′ ,y ′ ,z ′ ,w ′ ,y ∗ ,z ∗ ) be a feasible point of the problem (MD). Assume that z ′ F is ρ 1 - \({\mathbb{R}_{+}^{m}}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) , −y ′ G is ρ 2 - \({\mathbb {R}}_{+}^{m}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{m}}\) and H is ρ 3 - \(\mathbb {R}^{k}_{+}\) -convex with respect to \(\mathbf {1}_{\mathbb {R}^{k}}\), on A, satisfying Eq. 3.9. If x ′ is an element of the feasible set S of the problem (FP), then \(\left (x^{\prime },\frac {y^{\prime }}{z^{\prime }}\right )\) is a weak minimizer of the problem (FP).
4 Concluding remarks
In this paper, we establish the sufficient KKT conditions for the set-valued fractional programming problem (FP) via the contingent epiderivative. We assume generalized cone convexity assumptions on the objective and constraint set-valued maps. We also introduce the duals of parametric (PD), Mond-Weir (MWD), Wolfe (WD) and mixed (MD) types and prove the corresponding weak, strong and converse duality theorems under cone convexity assumptions.
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The first author is thankful to Council of Scientific and Industrial Research (CSIR), India, for his financial support in executing the study.
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Das, K., Nahak, C. Set-valued fractional programming problems under generalized cone convexity. OPSEARCH 53, 157–177 (2016). https://doi.org/10.1007/s12597-015-0222-9
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DOI: https://doi.org/10.1007/s12597-015-0222-9