\(^{\ast }\)Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, Jharkhand, India, e-mail: anurag_jais123@yahoo.com.
The research of this author was partially supported by the Indian School of Mines, Dhanbad under FRS(17)/2010-2011/AM

\(^{\S }\)Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 13 Septembrie Street, No. 13, 050711 Bucharest, Romania,
e-mail: stancu_minasian@yahoo.com

\(^{\bullet }\)Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi-835215, Jharkhand, India

1 INTRODUCTION

We consider the following minmax fractional programming problem:

subject to

where \(Y\ \)is a compact subset of \({\mathbb R}^{l},\) \(f\left( {\cdot ,\cdot }\right) :{\mathbb R}^{n}\times {\mathbb R}^{l}\rightarrow {\mathbb R},\ h\left( \cdot ,\cdot \right) :{\mathbb R}^{n}\times {\mathbb R}^{l}\rightarrow {\mathbb R}\ \)are \(C^{2}\) mappings on \({\mathbb R}^{n}\times {\mathbb R}^{l}\ \)and \(g\left( \cdot \right) :{\mathbb R}^{n}\rightarrow {\mathbb R}^{m}\ \)is \(C^{2}\) mapping on \({\mathbb R}^{n}\). It is assumed that for each \(\left( x,y\right) \) in \({\mathbb R}^{n}\times {\mathbb R}^{l},\) \(f\left( x,y\right) \geq 0\ \)and\(\ h\left( x,y\right) {\gt}0\).

In recent years, optimality conditions and duality for generalized minmax fractional programming have received much attention by many authors (see, for example, [1, 3, 8, 10–12, 14–17]). In particular, Crouzeix et al. [5] showed that the minmax fractional programming reduces to solving a minmax nonlinear parametric programming. In [3], Bector et al. used a parametric approach to obtain duality for the generalized minmax fractional programming involving differentiable pseudoconvex and quasiconvex functions.

Mangasarian [13] first formulated the second order dual for a nonlinear programming problem. Hanson [7] established second order duality theorems for nonlinear mathematical programming problem under defined second order type-I functions.

Zhang and Mond [18] introduced the concept of second order \(( F,\! \rho )\)-convexity and obtained some duality results concerning with nonlinear multiobjective programming problems. Ahmad and Husain [1] extended \(\left( F,\alpha ,\rho ,d\right) \)-convex functions which were introduced by Liang et al. [9] to second order \(\left( F,\alpha ,\rho ,d\right) \)-convex functions. Hachimi and Aghezzaf [6] further extended it to second order \(\left( F,\alpha ,\rho ,d\right) \)-type I functions.

Husain et al. [8] established duality theorems for two types of second order dual models related to minmax fractional programming problem (P) under the assumptions of \(\eta \)-bonvexity/generalized \(\eta \)-bonvexity.

Motivated by the earlier works and importance of the second order generalized convexity, in this paper we establish the second order duality theorems for the dual problem related to minmax fractional programming problem (P) under the assumption of generalized second order \(\left( F,\alpha ,\rho ,d\right) \)-type I functions.

The paper is organized as follows. Some definitions and notation are given in Section 2. Under the assumptions of generalized second order \(\left( F,\alpha ,\rho ,d\right) \)-type I functions, second order weak, strong and strict converse duality theorems related to problem (P) are given in Section 3. Concluding remarks are presented in Section 4.

2 NOTATION AND PRELIMINARIES

Let \({\mathbb R}^{n}\) be the \(n\)-dimensional Euclidean space and \({\mathbb R}_{+}^{n}\) its non-negative orthant. Let \(X\) be a nonempty open subset of \({\mathbb R}^{n}\). For \(x,\, y\, \in {\mathbb R}^{n}\), we let \(x\leq \, y\, \Leftrightarrow \, y-x\in {\mathbb R}_{+}^{n}\, ;\) \(x{\lt}\, y\, \Leftrightarrow \, y-x\in {\mathbb R}_{+}^{n}\backslash \left\{ 0\right\} \).

Throughout this paper, we denote by \(S=\left\{ x\in X\, :g\left( x\right) {\, \leq 0}\right\} \) the set of all feasible solutions of problem (P). For each\(\ \left( x,y\right) \in {\mathbb R}^{n}\times {\mathbb R}^{l}\), we define

and \(K\left( x\right) =\left\{ \left( s,t,\bar{y}\right) {\in }{\mathbb N}\times {\mathbb R}_{+}^{s}\times {\mathbb R}^{ls}:\ 1\leq s\leq n+1,\, t=\left( {t_{1},t_{2},...,t_{s}}\right) \in {\mathbb R}_{+}^{s}\right. \) with

By (ii) it is clear that\(\ F\left( x,\bar{x};\, 0{\, }\right) =0\).

Now, we let \(F\) be a sublinear functional and \(d\left( {\cdot ,\cdot }\right) :X\times X\rightarrow {\mathbb R}\). Let \(\alpha =\left( {\alpha ^{1},\alpha ^{2}}\right) \), where\(\ \alpha ^{1},\, \alpha ^{2}:X\times X\rightarrow {\mathbb R}_{+}\backslash \left\{ 0\right\} \), \(\rho =\left( \rho ^{1},\rho ^{2}\right) \), where \(\rho ^{1}=\left( \rho _{1}^{1},\rho _{2}^{1},...,\rho _{s}^{1}\right) \in {\mathbb R}^{s}\) and \(\rho ^{2}=\left( \rho _{1}^{2},\rho _{2}^{2},...,\rho _{m}^{2}\right) \in {\mathbb R}^{m}\). Let \(f\left( {\cdot ,\cdot }\right) :X\times Y\left( x\right) \rightarrow {\mathbb R}\) and \(g\left( \cdot \right) :X\rightarrow {\mathbb R}^{m}\) be two twice differentiable functions.

If the first inequality in the above definition is satisfied under the form

then we say that for each \(\ j\in M, \left( f,g_{j}\right)\) is second-order strictly \(\left( F,\alpha ,\rho ,d\right)\)-type I at \( \bar{x}\).

If the first implication in the above definition is satisfied under the form

then we say that for each \(j\in M\), \(\left( f,g_{j}\right) \, \)is second-order strictly pseudoquasi
\(\ \left( F,\alpha ,\rho ,d\right) \)-type I at\(\ \bar{x}\).

The following result will be needed in the sequel.

3 DUALITY

In this section, we consider the following dual model [8] for (P).

where \(H_{1}\left( s,t,\bar{y}\right) \) denotes the set of all \(\left( z,\mu ,\lambda ,p\right) \in {\mathbb R}^{n}\times {\mathbb R}_{+}^{m}\times {\mathbb R}_{+}\times {\mathbb R}^{n}\) satisfying

If, for a triplet\(\ \left( s,t,\bar{y}\right) \in K\left( z\right) \), the set\(\ H_{1}\left( s,t,\bar{y}\right) \) is empty, we define the supremum over it to be \(-\infty \).

Suppose contrary to the result that

Therefore, we have

It follows from \(t_{i}\geq 0,\, \, i=1,2,...,s,\ \)that

with at least one strict inequality, since \(t=\left( t_{1},t_{2},...,t_{s}\right) \neq 0.\) Taking summation over \(i,\) we have

which together with (3.2) gives

That is,

Using (3.3), (3.4) and hypothesis (i), we obtain

and

Since \(\alpha ^{1}\left( x,z\right) {\gt}0\ \)and\(\ \alpha ^{2}\left( x,z\right) {\gt}0\), by using the sublinearity of\(\ F\), the above two inequalities imply

and

From (3.1), (3.5), (3.6) and the sublinearity of\(\ F\), we get

Thus, we have a contradiction. Hence, the proof is complete.

Following the lines of proof of Theorem 3.1, we have:

Using (3.3), (3.7) and hypothesis (i), we obtain

and

Since \(\alpha ^{1}\left( x,z\right) {\gt}0\ \)and\(\ \alpha ^{2}\left( x,z\right) {\gt}0\), and the sublinearity of\(\ F\ \)in the above inequalities, we summarize to get

Since\(\ \displaystyle \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x,z\right) }+\displaystyle \tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x,z\right) }\geq 0\), by inequality (3.8), we have

which contradicts (3.1), as\(\ F\left( x,z{;0\, }\right) =0\). This completes the proof.

Since \(x^{\ast }\) is an optimal solution to (P) and \(\nabla g_{j}\left( x{^{\ast }}\right) ,j\in J\left( x{^{\ast }}\right) \), are linearly independent, then by Theorem 2.1, there exist \(\left( s^{\ast },t^{\ast },\bar{y}^{\ast }\right) \in K\left( x{^{\ast }}\right) \) and \(\left( x^{\ast },\mu ^{\ast },\lambda ^{\ast },p{^{\ast }=0}\right) \in H_{1}\left( s^{\ast },t^{\ast },\bar{y}{^{\ast }}\right) \) such that \(\left( x^{\ast },\mu ^{\ast },\lambda ^{\ast },s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }=0\right) \) is a feasible solution to (MD) and the two objectives have the same values. Optimality of \(\left( x^{\ast },\mu ^{\ast },\lambda ^{^{\ast }},s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }=0\right) \) for (MD) thus follows from weak duality Theorems 3.1 or 3.2.

Suppose to contrary that \(z^{\ast }\neq x^{\ast }\) and exhibit a contradiction. Since \(x^{\ast }\) and \(\left( z^{\ast },\mu ^{\ast },\lambda ^{\ast },s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }\right) \) are optimal solutions to (P) and (MD), respectively, and \(\nabla g_{j}\left( x^{\ast }\right) ,j\in J\left( x^{\ast }\right) \), are linearly independent, by Theorem 3.3 we have

Therefore, we have

It follows from \(t_{i}^{\ast }\geq 0,\, \, i=1,2,...,s,\ \)that

with at least one strict inequality, since \(t=\left( t_{1},t_{2},...,t_{s}\right) \neq 0.\) Taking summation over \(i,\) we have

which together with (3.2) gives

That is,

Using (3.3), (3.8) and hypothesis (ii), we obtain

and

Since \(\alpha ^{1}\left( x,z\right) {\gt}0\ \)and\(\ \alpha ^{2}\left( x,z\right) {\gt}0\), by using the sublinearity of\(\ F\), the above two inequalities imply

and

From (3.1), (3.10), (3.11) and the sublinearity of\(\ F\), we get

Thus, we have a contradiction. Hence\(\ z^{\ast }=x^{\ast }\).

Theorem 3.5 (Strict converse duality) Let \(x^{\ast }\) be an optimal solution to (P) and \(\left( z^{\ast },\mu ^{\ast },\lambda ^{^{\ast }},s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }\right) \) be optimal solution to (MD). Assume that are satisfied the conditions:

(i)\(\ \nabla g_{j}\left( x^{\ast }\right) ,j\in J\left( x^{\ast }\right) \), are linearly independent,

(ii)\(\left( \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( {\cdot },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( \cdot ,\bar{y}_{i}^{\ast }\right) \right) ,\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( \cdot \right) \right) \) is second order strictly
pseudoquasi \(\left( F,\alpha ,\rho ,d\right) \)-type I at \(z^{\ast },\)

(iii) \(\displaystyle \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x^{\ast },z^{\ast }\right) }+\displaystyle \tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x^{\ast },z^{\ast }\right) }\geq 0\).

Then, \(z^{\ast }=x^{\ast }\); that is, \(z^{\ast }\)is an optimal solution to (P).

We proceed as in the proof of Theorem 3.4 and obtain

From (3.3), and the second part of the hypothesis on
\(\left( \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( \cdot ,\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( \cdot ,\bar{y}_{i}^{\ast }\right) \right) ,\sum \limits _{j=1}^{m}g_{j}\left( \cdot \right) \right) \ \)at \(z^{\ast }\), we have

As\(\ \alpha ^{2}\left( x^{\ast },z^{\ast }\right) {\gt}0\ \)and as \(F\ \)is sublinear, it follows that

From relation (3.1), (3.13) and the sublinearity of\(\ F\), we obtain

In view of\(\ \displaystyle \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x^{\ast },z^{\ast }\right) }+\displaystyle \tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x^{\ast },z^{\ast }\right) }\geq 0\), \(\alpha ^{1}\left( x^{\ast },z^{\ast }\right) {\gt}0\) and the sublinearity of\(\ F\), the above inequality becomes

Using the first part of the hypothesis on
\(\ \left( \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( {\cdot },\bar{y}_{i}\right)\! \! -\! \! \lambda ^{\ast }h\left( \cdot ,\bar{y}_{i}\right) \right),\! \sum \limits _{j=1}^{m}g_{j}\left( \cdot \right) \right)\) at \(\ z^{\ast }\), it follows that

which is a contradiction to (3.12). Hence,\(\ z^{\ast }=x^{\ast }\).

4 CONCLUSIONS

In this paper, we have discussed the second order duality for dual model of minmax fractional programming problems under the assumptions of generalized \(\left( F,\alpha ,\rho ,d\right) \)-type I convexity. It will be interesting to see whether or not the second order duality results developed in this paper still hold for the following nondifferentiable minmax fractional programming problems:

where \(Y\) is a compact subset of\(\ {\mathbb R}^{m}\),\(\phi \left( {.,.}\right) ,\, \, \psi \left( {.,.}\right) \, :\, {\mathbb R}^{n}\times {\mathbb R}^{m}\rightarrow {\mathbb R}\) and \(g\left( {.,.}\right) :\, {\mathbb R}^{n}\rightarrow {\mathbb R}\ \)are continuously differentiable functions, and \(B\) and \(D\) are two positive semidefinite \(n\times n\ \)symmetric matrices.

where \(\xi =\left( z,\bar{z}\right) ,\quad v=\left( w,\bar{w}\right) \) for \(z\in C^{n},\quad w\in C^{l}\), \(\phi \left( {\ \cdot ,\cdot }\right) :C^{2n}\times C^{2l}\rightarrow C\) and \(\psi \left( {\cdot ,\cdot }\right) :C^{2n}\times C^{2l}\rightarrow C\) are analytic with respect to \(\xi \), \(W\) is a specified compact subset in \(C^{2l},\, \, S^{0}\ \)is a polyhedral cone in \(C^{m}\) and \(g:C^{2n}\rightarrow C^{m}\ \)is analytic. Also \(B,D\in C^{n\times n}\) are positive semidefinite Hermitian matrices.

This would be task of some of our forthcoming works.