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Secondorder duality for a nondifferentiable minimax fractional programming under generalized αunivexity
Journal of Inequalities and Applications volume 2012, Article number: 187 (2012)
Abstract
In this paper, we concentrate our study to derive appropriate duality theorems for two types of secondorder dual models of a nondifferentiable minimax fractional programming problem involving secondorder αunivex functions. Examples to show the existence of αunivex functions have also been illustrated. Several known results including many recent works are obtained as special cases.
MSC:49J35, 90C32, 49N15.
1 Introduction
After Schmitendorf [1], who derived necessary and sufficient optimality conditions for static minimax problems, much attention has been paid to optimality conditions and duality theorems for minimax fractional programming problems [2–17]. For the theory, algorithms, and applications of some minimax problems, the reader is referred to [18].
In this paper, we consider the following nondifferentiable minimax fractional programming problem:
where Y is a compact subset of {R}^{l}, f(\cdot ,\cdot ):{R}^{n}\times {R}^{l}\to R, h(\cdot ,\cdot ):{R}^{n}\times {R}^{l}\to R are twice continuously differentiable on {R}^{n}\times {R}^{l} and g(\cdot ):{R}^{n}\to {R}^{m} is twice continuously differentiable on {R}^{n}, B, and D are a n\times n positive semidefinite matrix, f(x,y)+{({x}^{T}Bx)}^{1/2}\ge 0, and h(x,y){({x}^{T}Dx)}^{1/2}>0 for each (x,y)\in \mathfrak{J}\times Y, where \mathfrak{J}=\{x\in {R}^{n}:g(x)\le 0\}.
Motivated by [7, 14, 15], Yang and Hou [17] formulated a dual model for fractional minimax programming problem and proved duality theorems under generalized convex functions. Ahmad and Husain [5] extended this model to nondifferentiable and obtained duality relations involving (F,\alpha ,\rho ,d)pseudoconvex functions. Jayswal [11] studied duality theorems for another two duals of (P) under αunivex functions. Recently, Ahmad et al.[4] derived the sufficient optimality condition for (P) and established duality relations for its dual problem under B\text{}(p,r)invexity assumptions. The papers [2, 4–7, 11–15, 17] involved the study of firstorder duality for minimax fractional programming problems.
The concept of secondorder duality in nonlinear programming problems was first introduced by Mangasarian [19]. One significant practical application of secondorder dual over firstorder is that it may provide tighter bounds for the value of objective function because there are more parameters involved. Hanson [20] has shown the other advantage of secondorder duality by citing an example, that is, if a feasible point of the primal is given and firstorder duality conditions do not apply (infeasible), then we may use secondorder duality to provide a lower bound for the value of primal problem.
Recently, several researchers [3, 8–10, 16] considered secondorder dual for minimax fractional programming problems. Husain et al.[8] first formulated secondorder dual models for a minimax fractional programming problem and established duality relations involving ηbonvex functions. This work was later on generalized in [10] by introducing an additional vector r to the dual models, and in Sharma and Gulati [16] by proving the results under secondorder generalized αtype I univex functions. The work cited in [3, 8, 10, 16] involves differentiable minimax fractional programming problems. Recently, Hu et al.[9] proved appropriate duality theorems for a secondorder dual model of (P) under ηpseudobonvexity/ηquasibonvexity assumptions. In this paper, we formulate two types of secondorder dual models for (P) and then derive weak, strong, and strict converse duality theorems under generalized αunivexity assumptions. Further, examples have been illustrated to show the existence of secondorder αunivex functions. Our study extends some of the known results of the literature [5, 6, 11, 12, 14].
2 Notations and preliminaries
For each (x,y)\in {R}^{n}\times {R}^{l} and M=\{1,2,\dots ,m\}, we define
Definition 2.1 Let \zeta :X\to R (X\subseteq {R}^{n}) be a twice differentiable function. Then ζ is said to be secondorder αunivex at u\in X, if there exist \eta :X\times X\to {R}^{n}, {b}_{0}:X\times X\to {R}_{+}, {\varphi}_{0}:R\to R, and \alpha :X\times X\to {R}_{+}\mathrm{\setminus}\{0\} such that for all x\in X and p\in {R}^{n}, we have
Example 2.1 Let \zeta :X\to R be defined as \zeta (x)={e}^{x}+{sin}^{2}x+{x}^{2}, where X=(1,\mathrm{\infty}). Also, let {\varphi}_{0}(t)=t+18, {b}_{0}(x,u)=u+1, \alpha (x,u)=\frac{{u}^{2}+2}{x+1} and \eta (x,u)=x+u. The function ζ is secondorder αunivex at u=1, since
But every αunivex function need not be invex. To show this, consider the following example.
Example 2.2 Let \mathrm{\Omega}:X=(0,\mathrm{\infty})\to R be defined as \mathrm{\Omega}(x)={x}^{2}. Let {\varphi}_{0}(t)=t, {b}_{0}(x,u)=\frac{1}{u}, \alpha (x,u)=2u, and \eta (x,u)=\frac{1}{2u}. Then we have
Hence, the function Ω is secondorder αunivex but not invex, since for x=3, u=2, and p=1, we obtain
Lemma 2.1 (Generalized Schwartz inequality)
Let B be a positive semidefinite matrix of order n. Then, for allx,w\in {R}^{n},
The equality holds ifBx=\lambda Bwfor some\lambda \ge 0.
Following Theorem 2.1 ([13], Theorem 3.1) will be required to prove the strong duality theorem.
Theorem 2.1 (Necessary condition)
If{x}^{\ast}is an optimal solution of problem (P) satisfying{x}^{\ast T}B{x}^{\ast}>0, {x}^{\ast T}D{x}^{\ast}>0, and\mathrm{\nabla}{g}_{j}({x}^{\ast}), j\in J({x}^{\ast})are linearly independent, then there exist({s}^{\ast},{t}^{\ast},\tilde{y})\in K({x}^{\ast}), {k}_{0}\in {R}_{+}, w,v\in {R}^{n}and{\mu}^{\ast}\in {R}_{+}^{m}such that
In the above theorem, both matrices B and D are positive semidefinite at {x}^{\ast}. If either {x}^{\ast T}B{x}^{\ast} or {x}^{\ast T}D{x}^{\ast} is zero, then the functions involved in the objective of problem (P) are not differentiable. To derive necessary conditions under this situation, for ({s}^{\ast},{t}^{\ast},\tilde{y})\in K({x}^{\ast}), we define
If in addition, we insert the condition {Z}_{\tilde{y}}({x}^{\ast})=\varphi, then the result of Theorem 2.1 still holds.
For the sake of convenience, let
and
where
3 Model I
In this section, we consider the following secondorder dual problem for (P):
where F(z)={sup}_{y\in Y}\frac{f(z,y)+{({z}^{T}Bz)}^{1/2}}{h(z,y){({z}^{T}Dz)}^{1/2}} and {H}_{1}(s,t,\tilde{y}) denotes the set of all (z,\mu ,w,v,p)\in {R}^{n}\times {R}_{+}^{m}\times {R}^{n}\times {R}^{n}\times {R}^{n} satisfying
If the set {H}_{1}(s,t,\tilde{y})=\varphi, we define the supremum of F(z) over {H}_{1}(s,t,\tilde{y}) equal to −∞.
Remark 3.1 If p=0, then using (3.3), the above dual model reduces to the problems studied in [6, 11, 12]. Further, if B and D are zero matrices of order n, then (DM1) becomes the dual model considered in [14].
Next, we establish duality relations between primal (P) and dual (DM1).
Theorem 3.1 (Weak duality)
Let x and(z,\mu ,w,v,s,t,\tilde{y},p)are feasible solutions of (P) and (DM1), respectively. Assume that

(i)
{\psi}_{1}(\cdot ) is secondorder αunivex at z,

(ii)
{\varphi}_{0}(a)\ge 0\Rightarrow a\ge 0 and {b}_{0}(x,z)>0.
Then
Proof Assume on contrary to the result that
Since {\tilde{y}}_{i}\in Y(z), i=1,2,\dots ,s, we have
From (3.4) and (3.5), for i=1,2,\dots ,s, we get
This further from {t}_{i}\ge 0, i=1,2,\dots ,s, t\ne 0 and {\tilde{y}}_{i}\in Y(z), we obtain
Now,
Therefore,
By hypothesis (i), we have
This follows from (3.1) that
which using hypothesis (ii) yields
This further from (2.6), (3.2), and the feasibility of x implies
This contradicts (3.7), hence the result. □
Theorem 3.2 (Strong duality)
Let{x}^{\ast}be an optimal solution for (P) and let\mathrm{\nabla}{g}_{j}({x}^{\ast}), j\in J({x}^{\ast})be linearly independent. Then there exist({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast})\in K({x}^{\ast})and({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{p}^{\ast}=0)\in {H}_{1}({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast}), such that({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0)is feasible solution of (DM1) and the two objectives have same values. If, in addition, the assumptions of Theorem 3.1 hold for all feasible solutions(x,\mu ,w,v,s,t,\tilde{y},p)of (DM1), then({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0)is an optimal solution of (DM1).
Proof Since {x}^{\ast} is an optimal solution of (P) and \mathrm{\nabla}{g}_{j}({x}^{\ast}), j\in J({x}^{\ast}) are linearly independent, then by Theorem 2.1, there exist ({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast})\in K({x}^{\ast}) and ({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{p}^{\ast}=0)\in {H}_{1}({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast}) such that ({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0) is feasible solution of (DM1) and the two objectives have same values. Optimality of ({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0) for (DM1), thus follows from Theorem 3.1. □
Theorem 3.3 (Strict converse duality)
Let{x}^{\ast}be an optimal solution to (P) and({z}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast})be an optimal solution to (DM1). Assume that

(i)
{\psi}_{1}(\cdot ) is strictly secondorder αunivex at {z}^{\ast},

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

(iii)
{\varphi}_{0}(a)>0\Rightarrow a>0 and {b}_{0}({x}^{\ast},{z}^{\ast})>0.
Then{z}^{\ast}={x}^{\ast}.
Proof By the strict αunivexity of {\psi}_{1}(\cdot ) at {z}^{\ast}, we get
which in view of (3.1) and hypothesis (iii) give
Using (2.6), (3.2), and feasibility of {x}^{\ast} in above, we obtain
Now, we shall assume that {z}^{\ast}\ne {x}^{\ast} and reach a contradiction. Since {x}^{\ast} and ({z}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}) are optimal solutions to (P) and (DM1), respectively, and \{\mathrm{\nabla}{g}_{j}({x}^{\ast}),j\in J({x}^{\ast})\}, are linearly independent, by Theorem 3.2, we get
Since {\tilde{y}}_{i}^{\ast}\in Y({z}^{\ast}), i=1,2,\dots ,{s}^{\ast}, we have
By (3.9) and (3.10), we get
for all i=1,2,\dots ,{s}^{\ast} and {\tilde{y}}_{i}^{\ast}\in Y. From {\tilde{y}}_{i}^{\ast}\in Y({z}^{\ast})\subset Y and {t}^{\ast}\in {R}_{+}^{{s}^{\ast}}, with {\sum}_{i=1}^{{s}^{\ast}}{t}_{i}^{\ast}=1, we obtain
From Lemma 2.1, (3.3), and (3.11), we have
which contradicts (3.8), hence the result. □
4 Model II
In this section, we consider another dual problem to (P):
where {H}_{2}(s,t,\tilde{y}) denotes the set of all (z,\mu ,w,v,p)\in {R}^{n}\times {R}_{+}^{m}\times {R}^{n}\times {R}^{n}\times {R}^{n} satisfying
If the set {H}_{2}(s,t,\tilde{y}) is empty, we define the supremum in (DM2) over {H}_{2}(s,t,\tilde{y}) equal to −∞.
Remark 4.1 If p=0, then using (4.3), the above dual model becomes the dual model considered in [5, 11, 12]. In addition, if B and D are zero matrices of order n, then (DM2) reduces to the problem studied in [14].
Now, we obtain the following appropriate duality theorems between (P) and (DM2).
Theorem 4.1 (Weak duality)
Let x and(z,\mu ,w,v,s,t,\tilde{y},p)are feasible solutions of (P) and (DM2), respectively. Suppose that the following conditions are satisfied:

(i)
{\psi}_{2}(\cdot ) is secondorder αunivex at z,

(ii)
{\varphi}_{0}(a)\ge 0\Rightarrow a\ge 0 and {b}_{0}(x,z)>0.
Then
Proof Assume on contrary to the result that
or
Using {t}_{i}\ge 0, i=1,2,\dots ,s and (4.3) in above, we have
Now,
Hence,
Now, by the secondorder αunivexity of {\psi}_{2}(\cdot ) at z, we get
which using (4.1) and hypothesis (ii) give
This from (4.2) follows that
which contradicts (4.5). This proves the theorem. □
By a similar way, we can prove the following theorems between (P) and (DM2).
Theorem 4.2 (Strong duality)
Let{x}^{\ast}be an optimal solution for (P) and let\mathrm{\nabla}{g}_{j}({x}^{\ast}), j\in J({x}^{\ast})be linearly independent. Then there exist({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast})\in K({x}^{\ast})and({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{p}^{\ast}=0)\in {H}_{2}({s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast}), such that({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0)is feasible solution of (DM2) and the two objectives have same values. If, in addition, the assumptions of weak duality hold for all feasible solutions(x,\mu ,w,v,s,t,\tilde{y},p)of (DM2), then({x}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast}=0)is an optimal solution of (DM2).
Theorem 4.3 (Strict converse duality)
Let{x}^{\ast}and({z}^{\ast},{\mu}^{\ast},{w}^{\ast},{v}^{\ast},{s}^{\ast},{t}^{\ast},{\tilde{y}}^{\ast},{p}^{\ast})are optimal solutions of (P) and (DM2), respectively. Assume that

(i)
{\psi}_{2}(\cdot ) is strictly secondorder αunivex at z,

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

(iii)
{\varphi}_{0}(a)>0\Rightarrow a>0 and {b}_{0}({x}^{\ast},{z}^{\ast})>0.
Then{z}^{\ast}={x}^{\ast}.
5 Concluding remarks
In the present work, we have formulated two types of secondorder dual models for a nondifferentiable minimax fractional programming problems and proved appropriate duality relations involving secondorder αunivex functions. Further, examples have been illustrated to show the existence of such type of functions. Now, the question arises whether or not the results can be further extended to a higherorder nondifferentiable minimax fractional programming problem.
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Acknowledgements
The authors wish to thank anonymous reviewers for their constructive and valuable suggestions which have considerably improved the presentation of the paper. The second author is also thankful to the Ministry of Human Resource Development, New Delhi (India) for financial support.
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Gupta, S., Dangar, D. & Kumar, S. Secondorder duality for a nondifferentiable minimax fractional programming under generalized αunivexity. J Inequal Appl 2012, 187 (2012). https://doi.org/10.1186/1029242X2012187
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DOI: https://doi.org/10.1186/1029242X2012187