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Higherorder duality in nondifferentiable minimax fractional programming involving generalized convexity
Journal of Inequalities and Applications volume 2012, Article number: 306 (2012)
Abstract
A higherorder dual for a nondifferentiable minimax fractional programming problem is formulated. Using the generalized higherorder ηconvexity assumptions on the functions involved, weak, strong and strict converse duality theorems are established in order to relate the primal and dual problems. Results obtained in this paper naturally unify and extend some previously known results on nondifferentiable minimax fractional programming in the literature.
MSC:26A51, 90C32, 49N15.
1 Introduction
The problem to be considered in the present analysis is the following nondifferentiable minimax fractional problem:
where Y is a compact subset of {R}^{l}, f(\cdot ,\cdot ), g(\cdot ,\cdot ):{R}^{n}\times {R}^{l}\to R, and h(\cdot ):{R}^{n}\to {R}^{m} are continuously differentiable functions. B and C are n\times n positive semidefinite symmetric matrices. It is assumed that for each (x,y) in {R}^{n}\times {R}^{l}, f(x,y)+{({x}^{T}Bx)}^{\frac{1}{2}}\ge 0 and g(x,y){({x}^{T}Cx)}^{\frac{1}{2}}>0.
In an earlier work, Schmittendorf [1] established necessary and sufficient optimality conditions for the following minimax programming problem:
where Y is a compact subset of {R}^{l}, the functions f(\cdot ,\cdot ):{R}^{n}\times {R}^{l}\to R, and h(\cdot ):{R}^{n}\to {R}^{m} are in {C}^{1}.
Tanimoto [2] applied the necessary conditions in [1] to formulate a dual problem and discussed the duality results, which were extended to fractional analogue of (P_{1}) by several authors [3–12]. Liu [13] proposed the secondorder duality theorems for (P_{1}) under generalized secondorder Binvex functions. Mishra and Rueda [14] and Ahmad, Husain and Sarita [15] discussed the secondorder duality results for the following nondifferentiable minimax programming problems:
where Y is a compact subset of {R}^{l}, f(\cdot ,\cdot ):{R}^{n}\times {R}^{l}\to R, and h(\cdot ):{R}^{n}\to {R}^{m} are twice differentiable functions. B is an n\times n positive semidefinite symmetric matrix. Ahmad, Husain and Sharma [16] formulated a unified higherorder dual of (P_{2}) and established weak, strong and strict converse duality theorems under higherorder (F,\alpha ,\rho ,d)Type I assumptions. Recently, Jayswal and StancuMinasian [17] obtained higherorder duality results for (P_{2}).
In this paper, we formulate a higherorder dual of (P) and establish weak, strong and strict converse duality theorems under generalized higherorder ηconvexity assumptions. More precisely, this paper is an extension of secondorder duality results of Hu, Chen and Jian [18] to a class of higherorder duality and it also presents an answer to a question raised in [17].
2 Notation and preliminaries
Let \mathcal{X}=\{x\in {R}^{n}:h(x)\le 0\} denote the set of all feasible solutions of (NFP). Any point x\in \mathcal{X} is called a feasible point of (NFP). For each (x,y)\in {R}^{n}\times {R}^{l}, we define
such that for each (x,y)\in \mathcal{X}\times Y,
For each x\in \mathcal{X}, we define
where
Since f and g are continuously differentiable and Y is compact in {R}^{l}, it follows that for each {x}^{\ast}\in \mathcal{X}, Y({x}^{\ast})\ne \mathrm{\varnothing}, and for any {\overline{y}}_{i}\in Y({x}^{\ast}), we have a positive constant
Lemma 2.1 (Generalized Schwarz inequality)
Let A be a positivesemidefinite matrix of order n. Then, for all x,w\in {R}^{n},
The equality Ax=\xi Aw holds for some \xi \ge 0. Clearly, if {({w}^{T}Aw)}^{\frac{1}{2}}\le 1, we have
We will use the following definitions.
Let \varphi :{R}^{n}\to R and k:{R}^{n}\times {R}^{n}\to R be differentiable functions.
Definition 2.1 [19]
A function ϕ is said to be higherorder ηconvex if there exists a certain mapping \eta :{R}^{n}\times {R}^{n}\to {R}^{n} such that for all x,p\in {R}^{n}, we have
Definition 2.2 [19]
A function ϕ is said to be higherorder (strictly) ηpseudoconvex if there exists a certain mapping \eta :{R}^{n}\times {R}^{n}\to {R}^{n} such that for all x,p\in {R}^{n}, we have
Definition 2.3 [19]
A function ϕ is said to be higherorder (strictly) ηquasiconvex if there exists a certain mapping \eta :{R}^{n}\times {R}^{n}\to {R}^{n} such that for all x,p\in {R}^{n}, we have
3 Higherorder duality model
In this section, we formulate the higherorder dual for (NFP) and derive duality results.
where L(s,t,\tilde{y}) denotes the set of all (z,\mu ,\lambda ,v,w,p)\in {R}^{n}\times {R}_{+}^{m}\times {R}_{+}\times {R}^{n}\times {R}^{n}\times {R}^{n} subject to
where {J}_{\alpha}\subseteq M=\{1,2,\dots ,m\}, \alpha =0,1,2,\dots ,r with {\bigcup}_{\alpha =0}^{r}{J}_{\alpha}=M and {J}_{\alpha}\cap J\beta =\mathrm{\varnothing} if \alpha \ne \beta. If for a triplet (s,t,\tilde{y})\in S(z), the set L(s,t,\tilde{y})=\mathrm{\varnothing}, then we define the supremum over it to be ∞.
Remark 3.1 (i) Let {J}_{0}=\mathrm{\varnothing}, F(z,{\overline{y}}_{i},p)={p}^{T}\u25bdf(z,{\overline{y}}_{i})+\frac{1}{2}{p}^{T}{\u25bd}^{2}f(z,{\overline{y}}_{i})p, G(z,{\overline{y}}_{i},p)={p}^{T}\u25bdg(z,{\overline{y}}_{i})+\frac{1}{2}{p}^{T}{\u25bd}^{2}g(z,{\overline{y}}_{i})p, i=1,2,\dots ,s and {H}_{j}(z,p)={p}^{T}\u25bd{h}_{j}(z)+\frac{1}{2}{p}^{T}{\u25bd}^{2}{h}_{j}(z)p, j=1,2,\dots ,m. Then (NMD) reduces to the secondorder dual in [18, 20]. If, in addition, p=0, then we get the dual formulated by Ahmad, Gupta, Kailey and Agarwal [5].
(ii) If {J}_{0}=\mathrm{\varnothing}, then the above dual becomes the dual formulated in [21].
Theorem 3.1 (Weak duality)
Let x and (z,\mu ,\lambda ,s,t,v,w,\tilde{y},p) be feasible solutions of (NFP) and (NMD) respectively. Assume that

(i)
[{\sum}_{i=1}^{s}{t}_{i}\{f(\cdot ,{\overline{y}}_{i})+{(\cdot )}^{T}Bw\lambda (g(\cdot ,{\overline{y}}_{i}){(\cdot )}^{T}Cv)\}+{\sum}_{j\in {J}_{0}}{\mu}_{j}{h}_{j}(\cdot )] is higherorder ηpseudoconvex at z,

(ii)
{\sum}_{j\in {J}_{\alpha}}{\mu}_{j}{h}_{j}(\cdot ), \alpha =1,2,\dots ,r, is higherorder ηquasiconvex at z.
Then
Proof Suppose to the contrary that
Then we have
It follows from {t}_{i}\ge 0, i=1,2,\dots ,s, that
with at least one strict inequality since t=({t}_{1},{t}_{2},\dots ,{t}_{s})\ne 0. Taking summation over i and using {\sum}_{i=1}^{s}{t}_{i}=1, we have
It follows from the generalized Schwarz inequality and (3.4) that
By the feasibility of x for (NFP)and \mu \ge 0, we obtain
The above inequality with (3.5) gives
Now, the feasibility of x for (NFP), \mu \ge 0 and (3.3) yields
The inequality (3.8) and Hypothesis (ii) gives
From (3.1) and (3.9), we have
which by virtue of Hypothesis (i) yields
This contradicts (3.7). □
Theorem 3.2 (Strong duality)
Let {x}^{\ast} be an optimal solution of (NFP) and let \mathrm{\nabla}{h}_{j}({x}^{\ast}), j\in J({x}^{\ast}) be linearly independent. Assume that
Then there exist ({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast})\in S and ({x}^{\ast},{\mu}^{\ast},{\lambda}^{\ast},{v}^{\ast},{w}^{\ast},{p}^{\ast})\in L({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast}) such that ({x}^{\ast},{\mu}^{\ast},{\lambda}^{\ast},{v}^{\ast},{w}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0) is a feasible solution of (NMD) and the two objectives have the same values. Furthermore, if the assumptions of weak duality (Theorem 3.1) hold for all feasible solutions of (NFP) and (NMD), then ({x}^{\ast},{\mu}^{\ast},{\lambda}^{\ast},{v}^{\ast},{w}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0) is an optimal solution of (NMD).
Proof Since {x}^{\ast} is an optimal solution of (NFP) and \mathrm{\nabla}{h}_{j}({x}^{\ast}), j\in J({x}^{\ast}) are linearly independent, by the necessary conditions obtained in [13], there exist ({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast})\in S and ({x}^{\ast},{\mu}^{\ast},{\lambda}^{\ast},{v}^{\ast},{w}^{\ast},{p}^{\ast})\in L({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast}) such that ({x}^{\ast},{\mu}^{\ast},{\lambda}^{\ast},{v}^{\ast},{w}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0) is a feasible solution of (NMD) and the problems (NFP) and (NMD) have the same objectives values and
□
Theorem 3.3 (Strict converse duality)
Let {x}^{\ast} and ({z}^{\ast},{\mu}^{\ast},{\lambda}^{\ast},{s}^{\ast},{t}^{\ast},{v}^{\ast},{w}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}) be the optimal solutions of (NFP) and (NMD), respectively. Assume that

(i)
\mathrm{\nabla}{h}_{j}({x}^{\ast}), j\in J({x}^{\ast}) are linearly independent,

(ii)
[{\sum}_{i=1}^{s}{t}_{i}^{\ast}\{f(\cdot ,{\overline{y}}_{i}^{\ast})+{(\cdot )}^{T}B{w}^{\ast}{\lambda}^{\ast}(g(\cdot ,{\overline{y}}_{i}^{\ast}){(\cdot )}^{T}C{v}^{\ast})\}+{\sum}_{j\in {J}_{0}}{\mu}_{j}^{\ast}{h}_{j}(\cdot )] is higherorder strictly ηpseudoconvex at {z}^{\ast},

(iii)
{\sum}_{j\in {J}_{\alpha}}{\mu}_{j}^{\ast}{h}_{j}(\cdot ), \alpha =1,2,\dots ,r, is higherorder ηquasiconvex at {z}^{\ast}.
Then, {z}^{\ast}={x}^{\ast}; that is, {z}^{\ast} is an optimal solution of (NFP).
Proof We will assume that {z}^{\ast}\ne {x}^{\ast} and reach a contradiction. From the strong duality theorem (Theorem 3.2), it follows that
Now, proceeding as in Theorem 3.1, we get
and
By Hypothesis (ii), (3.2) and (3.13), we have
which contradicts (3.12). Hence the results. □
4 Concluding remarks
The notion of higherorder invexity is adopted, which includes many other generalized convexity concepts in mathematical programming as special cases. If we take {J}_{0}=\mathrm{\varnothing}, F(z,{\overline{y}}_{i},p)={p}^{T}\u25bdf(z,{\overline{y}}_{i})+\frac{1}{2}{p}^{T}{\u25bd}^{2}f(z,{\overline{y}}_{i})p, G(z,{\overline{y}}_{i},p)={p}^{T}\u25bdg(z,{\overline{y}}_{i})+\frac{1}{2}{p}^{T}{\u25bd}^{2}g(z,{\overline{y}}_{i})p, i=1,2,\dots ,s and {H}_{j}(z,p)={p}^{T}\u25bd{h}_{j}(z)+\frac{1}{2}{p}^{T}{\u25bd}^{2}{h}_{j}(z)p, j=1,2,\dots ,m in Theorems 3.13.3, then we get Theorems 3.13.3 in [18].
The presented results in this paper can be further extended to the following related class of nondifferentiable minimax fractional programming problems:
where \xi =(z,\overline{z}), \nu =(\omega ,\overline{\omega}) for z\in {C}^{n}, \omega \in {C}^{l}. \varphi (\cdot ,\cdot ):{C}^{2n}\times {C}^{2l}\to C and \psi (\cdot ,\cdot ):{C}^{2n}\times {C}^{2l}\to C are analytic with respect to ξ, W is a specified compact subset in {C}^{2l}, {S}^{\circ} is a polyhedral cone in {C}^{m} and g:{C}^{2n}\to {C}^{m} is analytic. Also B, D\in {C}^{n\times n} are positive semidefinite Hermitian matrices.
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The research of the author is supported by the Internal Project No. IN111015 of King Fahd University of Petroleum and Minerals, Dhahran31261, Saudi Arabia.
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Ahmad, I. Higherorder duality in nondifferentiable minimax fractional programming involving generalized convexity. J Inequal Appl 2012, 306 (2012). https://doi.org/10.1186/1029242X2012306
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DOI: https://doi.org/10.1186/1029242X2012306