<!DOCTYPE html>
<html lang="en">
<head>
<script>
  MathJax = { 
    tex: {
		    inlineMath: [['\\(','\\)']]
	} }
</script>
<script type="text/javascript" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js">
</script>
<meta name="generator" content="plasTeX" />
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<title>Minmax fractional programming problem involving generalized convex functions: Minmax fractional programming problem involving generalized convex functions</title>
<link rel="stylesheet" href="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/styles/theme-white.css" />
</head>

<body>

<div class="wrapper">

<div class="content">
<div class="content-wrapper">


<div class="main-text">

<div class="titlepage">
<h1>Minmax fractional programming problem involving generalized convex functions</h1>
<p class="authors">
<span class="author">Anurag Jayswal \(^{\ast }\), I.M. Stancu-Minasian\(^{\S }\) Dilip Kumar\(^{\bullet }\)</span>
</p>
<p class="date">March 23, 2012.</p>
</div>
<p>\(^{\ast }\)Department of Applied Mathematics, Indian School of Mines, Dhanbad-826004, Jharkhand, India, e-mail: <span class="tt">anurag_jais123@yahoo.com</span>.<br />The research of this author was partially supported by the Indian School of Mines, Dhanbad under FRS(17)/2010-2011/AM </p>
<p>\(^{\S }\)Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 13 Septembrie Street, No. 13, 050711 Bucharest, Romania,<br />e-mail: <span class="tt">stancu_minasian@yahoo.com</span> </p>
<p>\(^{\bullet }\)Department of Applied Mathematics, Birla Institute of Technology, Mesra, Ranchi-835215, Jharkhand, India </p>

<div class="abstract"><p> In the present study we focus our attention on a minmax fractional programming problem and its second order dual problem. Duality results are obtained for the considered dual problem under the assumptions of second order \(\left( {F,\alpha ,\rho ,d}\right) \)<small class="small">-type I functions.</small> </p>
<p><b class="bf">MSC.</b> 26A51; 49J35; 90C32. </p>
<p><b class="bf">Keywords.</b> Minmax fractional programming, (\(F,\alpha ,\rho ,d\))-type I functions; second order duality. </p>
</div>
<h1 id="a0000000002">1 INTRODUCTION</h1>
<p>We consider the following minmax fractional programming problem:</p>
<div class="displaymath" id="a0000000003">
  \begin{equation*}  \text{Minimize\  }\psi \left( x\right) =\underset {y\in Y}{\  \sup }\tfrac {f\left( x,y\right) }{h\left( x,y\right) }\leqno \mathrm{(P)} \end{equation*}
</div>
<p>subject to</p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  g\left( x\right) \leq \, 0,\, \, x\in {\mathbb R}^{n}, \end{equation*}
</div>
<p>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\(\  h\left( x,y\right) {\gt}0\). </p>
<p>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 <i class="itshape">et al</i>. [5] showed that the minmax fractional programming reduces to solving a minmax nonlinear parametric programming. In [3], Bector <i class="itshape">et al</i>. used a parametric approach to obtain duality for the generalized minmax fractional programming involving differentiable pseudoconvex and quasiconvex functions. </p>
<p>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. </p>
<p>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 <i class="itshape">et al</i>. [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. </p>
<p>Husain <i class="itshape">et al</i>. [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. </p>
<p>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. </p>
<p>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. </p>
<h1 id="a0000000005">2 NOTATION AND PRELIMINARIES</h1>
<p>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 \(x\leq \, y\, \Leftrightarrow \, y-x\in {\mathbb R}_{+}^{n}\, ;\) \(x{\lt}\, y\, \Leftrightarrow \, y-x\in {\mathbb R}_{+}^{n}\backslash \left\{  0\right\}  \). </p>
<p>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</p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  J\left( x\right) =\left\{  j\in M=\left\{  1,2,...,m\right\} :\  g_{j}\left( x\right) =0\right\}  ,\quad \end{equation*}
</div>
<div class="displaymath" id="a0000000007">
  \begin{equation*}  Y\left( x\right) =\left\{  y\in Y:\  f\left( x,y\right) /h\left( x,y\right) {=\sup _{z\in Y}}f\left( x{,}z\right) /h\left( x,z\right) \right\}  , \end{equation*}
</div>
<p>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</p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  \sum \limits _{i=1}^{s}t_{i}=1,\bar{y}=\left( \bar{y}_{1},\bar{y}_{2}...,\bar{y}_{s}\right) ,{\, }\text{{with}}{\, }\left. \bar{y}_{i}\in Y\left( x\right) {,\, }i=1,2,...,s\right\}  . \end{equation*}
</div>
<p><div class="definition_thmwrapper " id="def2.1">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.1</span>
  </div>
  <div class="definition_thmcontent">
  <p>A functional \(F:X\times X\times {\mathbb R}^{n}\rightarrow {\mathbb R}\) is said to be sublinear in its third argument, if for any \(x,\bar{x}\in X\), </p>
<ul class="itemize">
  <li><p>\(F\left( x,\bar{x};a_{1}+a_{2}\right) \, \leq \, F\left( x,\bar{x};a_{1}\right) +F\left( x,\bar{x};a_{2}\right) \, \  \forall \, a_{1},\, a_{2}\in {\mathbb R}^{n};\) </p>
</li>
  <li><p>\(\  F\left( x,\bar{x};\alpha \, a\right) =\alpha \, F\left( x,\bar{x};a\right) \, \, \, \forall \alpha \in {\mathbb R}_{+},\forall \, a\in {\mathbb R}^{n}\). </p>
</li>
</ul>

  </div>
</div> By (ii) it is clear that\(\  F\left( x,\bar{x};\, 0{\, }\right) =0\). </p>
<p>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. </p>
<p><div class="definition_thmwrapper " id="def2.2">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.2</span>
  </div>
  <div class="definition_thmcontent">
  <p><span class="rm">[2]</span> For each \(\  j\in M\), \(\left( f,g_{j}\right) \) is said to be second-order \(\left( F,\alpha ,\rho ,d\right) \)-type I at \(\bar{x}\in X\) if for all \(\  x\in S\  \) and \( y_{i}\in Y\left( x\right) \), we have</p>
<div class="displaymath" id="a0000000009">
  \begin{align*} & f\left( x,y_{i}\right) -f\left( \bar{x},y_{i}\right) +\tfrac {1}{2}p^{T}\nabla ^{2}f\left( \bar{x},y{_{i}}\right) p\geq \end{align*}
</div>
<div class="displaymath" id="a0000000010">
  \begin{align*} & \geq F\left( x,\bar{x};\alpha ^{1}\left( x,\bar{x}\right) \left[ {\nabla }f\left( \bar{x},y_{i}\right) +{\nabla ^{2}}f\left( \bar{x},y_{i}\right) p\right] {\, }\right) +\rho _{i}^{1}d^{2}\left( x,\bar{x}\right) ,\, i=1,2,...,s, \end{align*}
</div>
<div class="displaymath" id="a0000000011">
  \begin{align*} & -g_{j}\left( \bar{x}\right) +\tfrac {1}{2}p^{T}\nabla ^{2}g_{j}\left( \bar{x}\right) p\geq \end{align*}
</div>
<div class="displaymath" id="a0000000012">
  \begin{align*} & \geq F\left( x,\bar{x}{;}\alpha ^{2}\left( x,\bar{x}\right) \left[ \nabla g_{j}\left( \bar{x}\right) +\nabla ^{2}g_{j}\left( \bar{x}\right) p\right] {\, }\right) +\rho _{j}^{2}d^{2}\left( x,\bar{x}\right) ,\, \, \, j=1,2,...,m, \end{align*}
</div>
<p>where \(\  p\in {\mathbb R}^{n}\). </p>

  </div>
</div> </p>
<p>If the first inequality in the above definition is satisfied under the form</p>
<div class="displaymath" id="a0000000013">
  \begin{align*} & f\left( x,y_{i}\right) -f\left( \bar{x},y_{i}\right) +\tfrac {1}{2}p^{T}\nabla ^{2}f\left( \bar{x},y_{i}\right) p{\gt}\\ & {\gt}\, F\left( x,\bar{x};\alpha ^{1}\left( x,\bar{x}\right) \left[ \nabla f\left( \bar{x},y_{i}\right) +\nabla ^{2}f\left( \bar{x},y_{i}\right) p\right] {\, }\right) +\rho _{i}^{1}d^{2}\left( x,\bar{x}\right) ,\, \, \, \, i=1,2,...,s, \end{align*}
</div>
<p>then we say that for each \(\  j\in M, \left( f,g_{j}\right)\) is <i class="it">second-order strictly \(\left( F,\alpha ,\rho ,d\right)\)-type I at \( \bar{x}\).</i> </p>
<p><div class="definition_thmwrapper " id="def2.3">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.3</span>
  </div>
  <div class="definition_thmcontent">
  <p> <span class="rm">[2]</span> For each \(\  j\in M\), \(\left( f,g_{j}\right) \) is said to be second-order pseudoquasi \(\left( F,\alpha ,\rho ,d\right) \)-type I at \(\bar{x}\in X\  \)if for all \(x\in S\) and \(y_{i}\in Y\left( x\right) \), we have</p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  f\left( x,y_{i}\right) {\lt}f\left( \bar{x},y_{i}\right) -\tfrac {1}{2}p^{T}\nabla ^{2}f\left( \bar{x},y_{i}\right) p \end{equation*}
</div>
<div class="displaymath" id="a0000000015">
  \begin{equation*}  \Rightarrow F\left( x,\bar{x};\alpha ^{1}\left( x,\bar{x}\right) \left[ \nabla f\left( \bar{x},y_{i}\right)\! +\! \nabla ^{2}f\left( \bar{x},y_{i}\right) p\right] {\, }\right)\! {\lt}-\rho _{i}^{1}d^{2}\left( x,\bar{x}\right) ,\, \, \, i=1,2,...,s, \end{equation*}
</div>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  -g_{j}\left( \bar{x}\right) +\tfrac {1}{2}p^{T}\nabla ^{2}g_{j}\left( \bar{x}\right) p\, \leq \, 0 \end{equation*}
</div>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  \Rightarrow \, \, F\left( x,\bar{x}{;}\alpha ^{2}\left( x,\bar{x}\right) \left[ \nabla g_{j}\left( \bar{x}\right) +\nabla ^{2}g_{j}\left( \bar{x}\right) p\right] {\, }\right) \leq -\rho _{j}^{2}d^{2}\left( x,\bar{x}\right) ,\, \, \, j=1,2,...,m, \end{equation*}
</div>
<p>where \(\  p\in {\mathbb R}^{n}\). </p>

  </div>
</div> </p>
<p>If the first implication in the above definition is satisfied under the form</p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  F\left( x,\bar{x}{;}\alpha ^{1}\left( x,\bar{x}\right) \left[ {\nabla }f\left( \bar{x},y_{i}\right) {+}\nabla ^{2}f\left( \bar{x},y_{i}\right) p\right] {\, }\right) \geq -\rho _{i}^{1}d^{2}\left( x,\bar{x}\right) ,\,  \end{equation*}
</div>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  \Rightarrow f\left( x,y_{i}\right) {\gt}f\left( \bar{x},y_{i}\right) -\tfrac {1}{2}p^{T}\nabla ^{2}f\left( \bar{x},y{_{i}}\right) p,\, \, \, i=1,2,...,s, \end{equation*}
</div>
<p>then we say that for each \(j\in M\), \(\left( f,g_{j}\right) \, \)<i class="itshape">is second-order strictly pseudoquasi</i> <br />\(\  \left( F,\alpha ,\rho ,d\right) \)-<i class="itshape">type</i> I <i class="itshape">at</i>\(\  \bar{x}\). </p>
<p>The following result will be needed in the sequel. </p>
<p><div class="theorem_thmwrapper " id="th2.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> <span class="rm">[4]</span> <i class="itshape">Let </i>\(x^{\ast }\)<i class="itshape"> be a solution of problem</i> (P)<i class="itshape"> and let </i>\(\nabla g_{j}\left( {x^{\ast }}\right) ,j\in J\left( {x^{\ast }}\right) \)<i class="itshape">, be linearly independent. Then there exist </i>\(\left( s^{\ast },t^{\ast },\bar{y}^{\ast }\right) \in K\left( x^{\ast }\right) ,\, \lambda ^{\ast }\in {\mathbb R}_{+}\)<i class="itshape">, and </i>\(\mu ^{\ast }\in {\mathbb R}_{+}^{m}\  \)<i class="itshape">such that</i></p>
<div class="displaymath" id="a0000000020">
  \begin{equation*}  \nabla \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) {-\lambda }^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \right) {+}\nabla \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( x^{\ast }\right) ={0}, \end{equation*}
</div>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) =0,\, i=1,2,...,s^{\ast }, \end{equation*}
</div>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( x^{\ast }\right) =0, \end{equation*}
</div>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  t_{i}^{\ast }\geq 0,\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }=1,\quad \bar{y}_{i}^{\ast }\in Y\left( x^{\ast }\right) ,\, \, i=1,2,...,s^{\ast }. \end{equation*}
</div>

  </div>
</div> </p>
<h1 id="a0000000024">3 DUALITY</h1>
<p>In this section, we consider the following dual model [8] for (P). </p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  \max _{\left( {s,t,\bar{y}}\right) \in K\left( z\right) }\sup _{\left( z,\mu ,\lambda ,p\right) \in H_{1}\left( s,t,\bar{y}\right) }\, \, \, \lambda ,\leqno \mathrm{(MD)} \end{equation*}
</div>
<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</p>
<div class="displaymath" id="a0000000026">
  \begin{align} & \nabla \sum \limits _{i=1}^{s}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( x,\bar{y}_{i}\right) \right) +\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( x,\bar{y}_{i}\right) \right) p +\tag {3.1}\\ & \quad +\nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p=0,\nonumber \end{align}
</div>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  \sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -{\lambda }h\left( z,\bar{y}_{i}\right) \right) -\tfrac {1}{2}p^{T}\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p\geq 0,\leqno (3.2) \end{equation*}
</div>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  \sum \limits _{j=1}^{m}{\mu _{j}g_{j}\left( z\right) }-\tfrac {1}{2}p^{T}\nabla ^{2}\sum \limits _{j=1}^{m}{\mu _{j}g_{j}\left( z\right) p}\geq 0.\leqno (3.3) \end{equation*}
</div>
<p>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 \(-\infty \). </p>
<p><div class="remark_thmwrapper " id="r3.1">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.1</span>
  </div>
  <div class="remark_thmcontent">
  <p> If\(\  p=0\), then (MD) becomes the dual problem considered in [11].<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="th3.1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p> (Weak duality) <i class="itshape">Let </i>\(x\  \)<i class="itshape">and </i>\(\left( z,\mu ,\lambda ,s,t,\bar{y},p\right) \  \)<i class="itshape">be feasible solutions to </i><span class="rm">(P)</span><i class="itshape"> and </i><span class="rm">(MD)</span>, <i class="itshape">respectively. Assume that </i> </p>
<ul class="itemize">
  <li><p>\(\  \left( \sum \limits _{i=1}^{s}t_{i}\left( f\left( \cdot ,\bar{y}_{i}\right) \  {-}\lambda h\left( {\cdot ,}\bar{y}_{i}\right) \right) {,}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( \cdot \right) \right) \)<i class="itshape"> is second order </i>\(\left( F,\alpha ,\rho ,d\right) \)<i class="itshape">-type I at </i>\(z,\) </p>
</li>
  <li><p>\(\displaystyle \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x,z\right) }+\displaystyle \tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x,z\right) }\geq 0\). </p>
</li>
</ul>
<p> <i class="itshape">Then </i></p>
<div class="displaymath" id="a0000000029">
  \begin{equation*}  \sup _{y\in Y}\tfrac {f\left( x,y\right) }{h\left( x,y\right) }\geq \lambda . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000030">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Suppose contrary to the result that </p>
<div class="displaymath" id="a0000000031">
  \begin{equation*}  \sup _{y\in Y}\tfrac {f\left( x,y\right) }{h\left( x,y\right) }{\lt}\lambda . \end{equation*}
</div>
<p>Therefore, we have</p>
<div class="displaymath" id="a0000000032">
  \begin{equation*}  f\left( x,\bar{y}_{i}\right) -\lambda h\left( x,\bar{y}_{i}\right) {\lt}0\  \text{for\  all\  }\bar{y}_{i}\in Y\left( x\right) ,\, \, \, i=1,2,...,s. \end{equation*}
</div>
<p>It follows from \(t_{i}\geq 0,\, \, i=1,2,...,s,\  \)that</p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  t_{i}\left( f\left( x,\bar{y}_{i}\right) {-}\lambda h\left( x,\bar{y}_{i}\right) \right) \leq 0, \end{equation*}
</div>
<p>with at least one strict inequality, since \(t=\left( t_{1},t_{2},...,t_{s}\right) \neq 0.\) Taking summation over \(i,\) we have</p>
<div class="displaymath" id="a0000000034">
  \begin{equation*}  \sum \limits _{i=1}^{s}t_{i}\left( f\left( x,\bar{y}_{i}\right) {-}\lambda h\left( x,\bar{y}_{i}\right) \right) {\lt}0, \end{equation*}
</div>
<p>which together with (3.2) gives</p>
<div class="displaymath" id="a0000000035">
  \begin{align*} & \sum \limits _{i=1}^{s}t_{i}\left( f\left( x,\bar{y}_{i}\right) -\lambda h\left( x,\bar{y}_{i}\right) \right) {\lt}0\leq \\ & \leq \sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right)-\tfrac {1}{2}p^{T}\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p. \end{align*}
</div>
<p>That is,</p>
<div class="displaymath" id="a0000000036">
  \begin{align} & \sum \limits _{i=1}^{s}t_{i}\left( f\left( x,\bar{y}_{i}\right) -\lambda h\left( x,\bar{y}_{i}\right) \right) -\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right)+\tag {3.4}\\ & \quad +\tfrac {1}{2}p^{T}\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p{\lt}0.\nonumber \end{align}
</div>
<p>Using (3.3), (3.4) and hypothesis (i), we obtain</p>
<div class="displaymath" id="a0000000037">
  \begin{align*}  0& {\gt}\sum \limits _{i=1}^{s}t_{i}\left( f\left( x,\bar{y}_{i}\right) -\lambda h\left( x,\bar{y}_{i}\right) \right) -\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right)\\ & \quad +\tfrac {1}{2}p^{T}\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p\\ & \geq \! F\! \left(\! x,\! z;\! \alpha ^{1}\! \! \left(x,\! z\! \right)\! \! \left( \nabla \! \! \sum \limits _{i=1}^{s}\! t_{i}\! \left( f\left(z,\bar{y}_{i}\right)\!  \! -\! \! \lambda h\left( z,\bar{y}_{i}\right) \right)\! \! +\! \! \nabla ^{2}\! \! \sum \limits _{i=1}^{s}\! t_{i}\left( f\left(z,\bar{y}{_{i}}\right)\!  \! -\! \! \lambda h\left( z,\bar{y}_{i}\right) \right)\!  p\! \right)\!  \right) \\ & \quad +\rho _{1}^{1}d^{2}\left( x,z\right) , \end{align*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000038">
  \begin{align*}  0& \geq -\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) +\tfrac {1}{2}p^{T}\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\\ & \geq F\left( x,z;\alpha ^{2}\left( x,z\right) \left( \nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) {+\nabla ^{2}}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\right) \right) +\rho _{1}^{2}d^{2}\left( x,z\right) . \end{align*}
</div>
<p>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</p>
<div class="displaymath" id="a0000000039">
  \begin{align} & F\left(\!  x,z;\nabla \! \sum \limits _{i=1}^{s}\! t_{i}\left(\!  f\left( z,\bar{y}_{i}\right)\!  -\! \lambda h\left( z,\bar{y}_{i}\right) \right)\!  +\! \nabla ^{2}\! \sum \limits _{i=1}^{s}\! t_{i}\left( f\left( z,\bar{y}_{i}\right) \! -\! \lambda h\left( z,\bar{y}_{i}\right) \right) p\right)\tag {3.5}\\ & {\lt}-\tfrac {\rho _{1}^{1}d^{2}\left( x,z\right) }{\alpha ^{1}\left( x,z\right) }\nonumber \end{align}
</div>
<p>and</p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  F\left( x,z;\nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\right) \leq -\tfrac {\rho _{1}^{2}d^{2}\left( x,z\right) }{\alpha ^{2}\left( x,z\right) }.\leqno (3.6) \end{equation*}
</div>
<p>From (3.1), (3.5), (3.6) and the sublinearity of\(\  F\), we get</p>
<div class="displaymath" id="a0000000041">
  \begin{align*}  0& =F\left( x,z;\nabla \sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) +\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p\right.\\ & \quad \left. +\nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\right) \end{align*}
</div>
<div class="displaymath" id="a0000000042">
  \begin{align*} & \leq F\left( x,z;\nabla \sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) +\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p\right)\\ & \quad +F\left( x,z;\nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\right)\\ &  {\lt}-\left( \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x,z\right) }+\tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x,z\right) }\right) d^{2}\left( x,z\right){\lt}0.\   \text{(by(ii))} \end{align*}
</div>
<p>Thus, we have a contradiction. Hence, the proof is complete. </p>
<p><div class="theorem_thmwrapper " id="th3.2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>(Weak duality) <i class="itshape">Let </i>\(x\  \)<i class="itshape">and </i>\(\left( z,\mu ,\lambda ,s,t,\bar{y},p\right) \  \)<i class="itshape">be feasible solutions to </i>(P)<i class="itshape"> and </i>(MD), <i class="itshape">respectively, Assume that </i> </p>
<ul class="itemize">
  <li><p>\(\left( \sum \limits _{i=1}^{s}t_{i}\left( f\left( \cdot ,\bar{y}_{i}\right) -\lambda h\left( \cdot ,\bar{y}_{i}\right) \right) {,}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( \cdot \right) \right) \)<i class="itshape"> is second order pseudoquasi<br /></i>\(\left( F,\alpha ,\rho ,d\right) \)<i class="itshape">-type I at </i>\(z,\) </p>
</li>
  <li><p>\(\displaystyle \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x,z\right) }+\displaystyle \tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x,z\right) }\geq 0\). </p>
</li>
</ul>
<p> <i class="itshape">Then </i></p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  \sup _{y\in Y}\tfrac {f\left( x,y\right) }{h\left( x,y\right) }\geq \lambda . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000044">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>Following the lines of proof of Theorem 3.1, we have:</p>
<div class="displaymath" id="a0000000045">
  \begin{align} & \sum \limits _{i=1}^{s}t_{i}\left( f\left( x,\bar{y}_{i}\right) -\lambda h\left( x,\bar{y}_{i}\right) \right){\lt}\tag {3.7}\\ & {\lt}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right)-\tfrac {1}{2}p^{T}\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p.\nonumber \end{align}
</div>
<p>Using (3.3), (3.7) and hypothesis (i), we obtain</p>
<div class="displaymath" id="a0000000046">
  \begin{align*} & F\! \left( \! x,\! z;\! \alpha ^{1}\! \left(\!  x,\! z\right)\!  \left( \! \nabla \!  \sum \limits _{i=1}^{s}\! t_{i}\left( f\left( z,\! \bar{y}_{i}\right)\! \!  -\! \! \lambda h\left( z,\! \bar{y}_{i}\right) \right)\! \!  +\! \! \nabla ^{2}\sum \limits _{i=1}^{s}\! t_{i}\left( f\left( z,\bar{y}_{i}\right)\! \!  -\! \! \lambda h\left( z,\bar{y}_{i}\right) \right) p\right) \right)\\ & {\lt}-\rho _{1}^{1}d^{2}\left( x,z\right) \end{align*}
</div>
<p> and</p>
<div class="displaymath" id="a0000000047">
  \begin{equation*}  F\left( x,z;\alpha ^{2}\left( x,z\right) \left( \nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) {+}\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\right) \right) \leq -\rho _{1}^{2}d^{2}\left( x,z\right) . \end{equation*}
</div>
<p>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</p>
<div class="displaymath" id="a0000000048">
  \begin{align} & F\left(\!  x,z;\nabla \! \sum \limits _{i=1}^{s}\! t_{i}\left( f\left( z,\bar{y}_{i}\right)\!  -\! \lambda h\left( z,\bar{y}_{i}\right) \right) \! +\! \nabla ^{2}\! \sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right)\!  -\! \lambda h\left( z,\bar{y}_{i}\right) \right) p+\tag {3.8}\right.\\ & \quad \left.+\nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right)+\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\right)\nonumber \\ &  {\lt}-\left( \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x,z\right) }+\tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x,z\right) }\right) d^{2}\left( x,z\right) {\lt}0.\nonumber \end{align}
</div>
<p>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</p>
<div class="displaymath" id="a0000000049">
  \begin{align*} & F\left( x,z;\nabla \sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) +\nabla ^{2}\sum \limits _{i=1}^{s}t_{i}\left( f\left( z,\bar{y}_{i}\right) -\lambda h\left( z,\bar{y}_{i}\right) \right) p+\right.\\ & \quad \left.+\nabla \sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) {+} \nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}g_{j}\left( z\right) p\right) {\lt}0,\end{align*}
</div>
<p>which contradicts (3.1), as\(\  F\left( x,z{;0\, }\right) =0\). This completes the proof. </p>
<p><div class="theorem_thmwrapper " id="th3.3">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p> (Strong duality) <i class="itshape">Assume that </i>\(x^{\ast }\)<i class="itshape"> is an optimal solution to </i><span class="rm">(P)</span><i class="itshape"> and </i>\(\nabla g_{j}\left( x^{\ast }\right) ,j\in J\left( x^{\ast }\right) \)<i class="itshape">, are linearly independent. Then there exist </i>\(\left( s^{\ast },t^{\ast },\bar{y}^{\ast }\right) \in K\left( x^{\ast }\right) \)<i class="itshape"> and </i>\(\left( x^{\ast },\mu ^{\ast },\lambda ^{\ast },p^{\ast }=0\right) \in H_{1}\left( s^{\ast },t^{\ast },\bar{y}^{\ast }\right) \)<i class="itshape"> such that </i><br />\(\left( x^{\ast },\mu ^{\ast },\lambda ^{\ast },s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }=0\right) \)<i class="itshape"> is a feasible solution to </i><span class="rm">(MD)</span><i class="itshape"> and the two objectives have the same values. Further, if the hypotheses of weak duality Theorems </i><span class="rm">3.1</span> <i class="itshape">or </i><span class="rm">3.2</span><i class="itshape"> hold for all feasible solutions </i>\(\left( z,\mu ,\lambda ,s,t,\bar{y},p\right) \)<i class="itshape"> to </i><span class="rm">(MD)</span>,<i class="itshape"> then </i>\(\left( x^{\ast },\mu ^{\ast },\lambda ^{\ast },s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }=0\right) \)<i class="itshape"> is an optimal solution to </i><span class="rm">(MD)</span>. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000050">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>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. </p>
<p><div class="theorem_thmwrapper " id="th3.4">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>(Strict converse duality) Let \(x^{\ast }\) be an optimal solution to <span class="rm">(P)</span> and \(\left( z^{\ast },\mu ^{\ast },\lambda ^{^{\ast }},s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }\right)\) be optimal solution to <span class="rm">(MD)</span>. Assume that are satisfied the conditions: </p>
<ul class="itemize">
  <li><p>\(\  \nabla g_{j}\left( x^{\ast }\right) ,j\in J\left( x^{\ast }\right) \), <i class="itshape">are linearly independent</i>, </p>
</li>
  <li><p><i class="itshape"> </i>\(\left( \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( {\cdot },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( \cdot ,\bar{y}_{i}^{\ast }\right) \right) {,}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( \cdot \right) \right) \) <i class="itshape">is second order </i><br />\(\left( F,\alpha ,\rho ,d\right) \)<i class="itshape">-type I at </i>\(z^{\ast },\) </p>
</li>
  <li><p>\(\displaystyle \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x^{\ast },z^{\ast }\right) }+\displaystyle \tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x^{\ast },z^{\ast }\right) }{\gt}0\). </p>
</li>
</ul>
<p> <i class="itshape">Then </i>\(z^\ast = x^\ast \), <i class="itshape">that is, </i>\(z^\ast \)<i class="itshape">is an optimal solution to</i> <span class="rm">(P)</span>. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000051">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>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</p>
<div class="displaymath" id="a0000000052">
  \begin{equation*}  \sup _{y^{\ast }\in Y}\tfrac {f\left( x^{\ast },y^{\ast }\right) }{h\left( x^{\ast },y^{\ast }\right) }=\lambda ^{\ast }. \end{equation*}
</div>
<p>Therefore, we have </p>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \leq 0\  \text{for\  all\  }\bar{y}_{i}^{\ast }\in Y\left( x^{\ast }\right) ,\, \, \, i=1,2,...,s^{\ast }. \end{equation*}
</div>
<p>It follows from \(t_{i}^{\ast }\geq 0,\, \, i=1,2,...,s,\  \)that</p>
<div class="displaymath" id="a0000000054">
  \begin{equation*}  t_{i}^{\ast }\left( f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \right) \leq 0, \end{equation*}
</div>
<p>with at least one strict inequality, since \(t=\left( t_{1},t_{2},...,t_{s}\right) \neq 0.\) Taking summation over \(i,\) we have</p>
<div class="displaymath" id="a0000000055">
  \begin{equation*}  \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \right) {\lt}0, \end{equation*}
</div>
<p>which together with (3.2) gives</p>
<div class="displaymath" id="a0000000056">
  \begin{align*} & \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \right) {\lt}0\leq \\ & \leq \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right)\! -\! \tfrac {1}{2}p^{\ast T}\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }. \end{align*}
</div>
<p>That is,</p>
<div class="displaymath" id="a0000000057">
  \begin{align} & \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \right) -\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right)\tag {3.9}\\ & \quad +\tfrac {1}{2}p^{\ast T}\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }{\lt}0.\nonumber \end{align}
</div>
<p>Using (3.3), (3.8) and hypothesis (ii), we obtain</p>
<div class="displaymath" id="a0000000058">
  \begin{align*}  0& {\gt}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \right) -\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right)\\ & \quad +\tfrac {1}{2}p^{\ast T}\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }\\ & \geq \, F\left( x^{\ast },z^{\ast };\alpha ^{1}\left( x^{\ast },z^{\ast }\right) \, \left( \nabla \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) \right. \right.\\ & \quad \left. \left. +\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }\right) \right) +\rho _{1}^{1}d^{2}\left( x^{\ast },z^{\ast }\right) , \end{align*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000059">
  \begin{align*}  0& \geq -\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) +\tfrac {1}{2}p^{\ast T}\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) p^{\ast }\\ & \geq F\left( x^{\ast },z^{\ast };\alpha ^{2}\left( x^{\ast },z^{\ast }\right) \left( \nabla \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) +{\nabla ^{2}}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) p^{\ast }\right) \right)\\ & \quad +\rho _{1}^{2}d^{2}\left( x^{\ast },z^{\ast }\right) . \end{align*}
</div>
<p>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</p>
<div class="displaymath" id="a0000000060">
  \begin{align} & F\left( x^{\ast },z^{\ast };{\, }\left( \nabla \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) \right. \right.\tag {3.10}\\ & \quad \left. \left. +\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }\right) \right) {\lt}-\tfrac {\rho _{1}^{1}d^{2}\left( x^{\ast },z^{\ast }\right) }{\alpha ^{1}\left( x^{\ast },z^{\ast }\right) },\nonumber \end{align}
</div>
<p>and</p>
<div class="displaymath" id="a0000000061">
  \begin{equation*}  F\left( x^{\ast },z^{\ast };\, \left( \nabla \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) p^{\ast }\right) \right) \leq -\tfrac {\rho _{1}^{2}d^{2}\left( x^{\ast },z^{\ast }\right) }{\alpha ^{2}\left( x^{\ast },z^{\ast }\right) }.\leqno (3.11) \end{equation*}
</div>
<p>From (3.1), (3.10), (3.11) and the sublinearity of\(\  F\), we get</p>
<div class="displaymath" id="a0000000062">
  \begin{align*}  0& =F\left( x^{\ast },z^{\ast };\nabla \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) \right.\\ & \quad \left.+\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }\right. \end{align*}
</div>
<div class="displaymath" id="a0000000063">
  \begin{align*} & \quad \left.+ \nabla \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) p^{\ast }\right)\\ & \leq F\left( x^{\ast },z^{\ast };\nabla \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right)\right.\\ & \quad \left.+\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }\right)\\ & \quad +F\left( x,z;\nabla \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) p^{\ast }\right)\\ & {\lt}-\left( \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) }\right) d^{2}\left( x^{\ast },z^{\ast }\right) {\lt}0\  \text{(by (iii))}. \end{align*}
</div>
<p>Thus, we have a contradiction. Hence\(\  z^{\ast }=x^{\ast }\). </p>
<p><span class="scshape">Theorem 3.5 </span>(Strict converse duality) <i class="itshape">Let </i>\(x^{\ast }\)<i class="itshape"> be an optimal solution to </i>(P) <i class="itshape">and </i> \(\left( z^{\ast },\mu ^{\ast },\lambda ^{^{\ast }},s^{\ast },t^{\ast },\bar{y}^{\ast },p^{\ast }\right) \) <i class="itshape">be optimal solution to </i>(MD). <i class="itshape">Assume that are satisfied the conditions:</i> </p>
<p>(i)\(\  \nabla g_{j}\left( x^{\ast }\right) ,j\in J\left( x^{\ast }\right) \), <i class="itshape">are linearly independent</i>, </p>
<p>(ii)\(\left( \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( {\cdot },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( \cdot ,\bar{y}_{i}^{\ast }\right) \right) ,\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( \cdot \right) \right) \) <i class="itshape">is second order strictly <br />pseudoquasi </i>\(\left( F,\alpha ,\rho ,d\right) \)<i class="itshape">-type I at </i>\(z^{\ast },\) </p>
<p>(iii) \(\displaystyle \tfrac {\rho _{1}^{1}}{\alpha ^{1}\left( x^{\ast },z^{\ast }\right) }+\displaystyle \tfrac {\rho _{1}^{2}}{\alpha ^{2}\left( x^{\ast },z^{\ast }\right) }\geq 0\). </p>
<p><i class="itshape">Then, </i>\(z^{\ast }=x^{\ast }\); <i class="itshape">that is</i>, \(z^{\ast }\)<i class="itshape">is an optimal solution to </i>(P). </p>
<p><div class="proof_wrapper" id="a0000000064">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div>We proceed as in the proof of Theorem 3.4 and obtain</p>
<div class="displaymath" id="a0000000065">
  \begin{align}  \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) & {\lt}\, \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda {^{\ast }}h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right)\nonumber \\ & \quad -\tfrac {1}{2}p^{\ast T}\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }.\tag {3.12} \end{align}
</div>
<p>From (3.3), and the second part of the hypothesis on <br />\(\left( \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( \cdot ,\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( \cdot ,\bar{y}_{i}^{\ast }\right) \right) ,\sum \limits _{j=1}^{m}g_{j}\left( \cdot \right) \right) \  \)at \(z^{\ast }\), we have</p>
<div class="displaymath" id="a0000000066">
  \begin{equation*}  F\left( x^{\ast },z^{\ast };\alpha ^{1}\! \! \left( x^{\ast },z^{\ast }\right) \, \left( \nabla \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right)\! \!  +\! \! \nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) p^{\ast }\right) \right)\! \!  \leq \! \!  -\rho _{1}^{2}d^{2}\left( x^{\ast },z^{\ast }\right) . \end{equation*}
</div>
<p>As\(\  \alpha ^{2}\left( x^{\ast },z^{\ast }\right) {\gt}0\  \)and as \(F\  \)is sublinear, it follows that</p>
<div class="displaymath" id="a0000000067">
  \begin{equation*}  F\left( x^{\ast },z^{\ast };\, \left( \nabla \sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) +\nabla ^{2}\sum \limits _{j=1}^{m}\mu _{j}^{\ast }g_{j}\left( z^{\ast }\right) p^{\ast }\right) \right) \leq -\tfrac {\rho _{1}^{2}d^{2}\left( x^{\ast },z^{\ast }\right) }{\alpha ^{2}\left( x^{\ast },z^{\ast }\right) }.\leqno (3.13) \end{equation*}
</div>
<p>From relation (3.1), (3.13) and the sublinearity of\(\  F\), we obtain </p>
<div class="displaymath" id="a0000000068">
  \begin{align*} & F\left( x^{\ast },z^{\ast };{\, }\left( \nabla \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) \right. \right.\\ & \quad \left. \left. +\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }\right) \right) \geq \tfrac {\rho _{1}^{2}d^{2}\left( x^{\ast },z^{\ast }\right) }{\alpha ^{2}\left( x^{\ast },z^{\ast }\right) }. \end{align*}
</div>
<p>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 </p>
<div class="displaymath" id="a0000000069">
  \begin{align*} & F\left( x^{\ast },z^{\ast };\alpha ^{1}\left( x^{\ast },z^{\ast }\right) \, \left( \nabla \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },y_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },y_{i}^{\ast }\right) \right) \right. \right.\\ & \quad \left. {\  }+\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },y_{i}^{\ast }\right) -\lambda ^{\ast }h\left( z^{\ast },y_{i}^{\ast }\right) \right) p^{\ast }\right) \geq -\rho _{1}^{1}d^{2}\left( x^{\ast },z^{\ast }\right) . \end{align*}
</div>
<p>Using the first part of the hypothesis on <br />\(\  \left( \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( {\cdot },\bar{y}_{i}\right)\! \! -\! \! \lambda ^{\ast }h\left( \cdot ,\bar{y}_{i}\right) \right),\! \sum \limits _{j=1}^{m}g_{j}\left( \cdot \right) \right)\) at \(\  z^{\ast }\), it follows that </p>
<div class="displaymath" id="a0000000070">
  \begin{align*}  \sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( x^{\ast },\bar{y}_{i}^{\ast }\right) -\lambda ^{\ast }h\left( x^{\ast },\bar{y}_{i}^{\ast }\right) \right) & {\gt}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) -{\lambda }^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right)\\ & \quad -\tfrac {1}{2}p^{\ast T}\nabla ^{2}\sum \limits _{i=1}^{s^{\ast }}t_{i}^{\ast }\left( f\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \! \! -\! \! \lambda ^{\ast }h\left( z^{\ast },\bar{y}_{i}^{\ast }\right) \right) p^{\ast }. \end{align*}
</div>
<p> which is a contradiction to (3.12). Hence,\(\  z^{\ast }=x^{\ast }\). </p>
<h1 id="a0000000071">4 CONCLUSIONS</h1>
<p>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: </p>
<div class="displaymath" id="a0000000072">
  \begin{equation*}  \mathrm{Min}\sup _{y\in Y}\tfrac {\phi \left( x,y\right) +\left( x^{T}Bx\right) ^{1/2}}{\psi \left( x,y\right) -\left( x^{T}Dx\right) ^{1/2}}\  \  \text{subject\  to}\  g\left( x\right) \, \leq \, 0,\, \, \, \, x\in {\mathbb R}^{n},\leqno \mathbf{(P1)} \end{equation*}
</div>
<p>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. </p>
<div class="displaymath" id="a0000000073">
  \begin{equation*}  \mathrm{Min}\sup _{v\in W}\tfrac {\mathrm{Re}\left[ {\phi }\left( \xi ,v\right) +\left( z^{T}Bz\right) {^{1/2}}\right] }{\mathrm{Re}\left[ {\psi }\left( \xi ,v\right) -\left( z^{T}Dz\right) {^{1/2}}\right] },\  \  \text{subject\  to\  }-g\left( \xi \right) \, \in S^{0},\, \, \, \, \xi \in C^{2n},\leqno \mathbf{(P2)} \end{equation*}
</div>
<p>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. </p>
<p>This would be task of some of our forthcoming works. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
<dl class="bibliography">
  <dt><a name="AhmadHusain">1</a></dt>
  <dd><p><i class="sc">I. Ahmad</i> and <i class="sc">Z. Husain</i>, <i class="it">Duality in nondifferentiable minimax fractional programming with generalized convexity</i>, Appl. Math. Comp., <b class="bf">176</b>, pp.&#160; 545–551, 2006. </p>
</dd>
  <dt><a name="AhmadHusainSharma">2</a></dt>
  <dd><p><i class="sc">I. Ahmad, Z. Husain</i> and <i class="sc">S. Sharma</i>, <i class="it">Second-order duality in nondifferentiable minmax programming involving type</i>-I <i class="itshape">functions</i>, J. Comp. Appl. Math., <b class="bf">215</b>, pp.&#160;91–102, 2008. </p>
</dd>
  <dt><a name="BectorChandraHusain">3</a></dt>
  <dd><p><i class="sc">C.R. Bector, S. Chandra</i> and <i class="sc">I. Husain</i>, <i class="it">Second order duality for a minimax programming problem</i>, Opsearch, <b class="bf">28</b>, pp.&#160;249–263, 1991. </p>
</dd>
  <dt><a name="ChandraKumar">4</a></dt>
  <dd><p><i class="sc">S. Chandra</i> and <i class="sc">V. Kumar</i>, <i class="it">Duality in fractional minimax programming</i>, J. Austral. Math. Soc. Ser. A, <b class="bf">58</b>, pp.&#160;376–386, 1995. </p>
</dd>
  <dt><a name="CrouzeixFerlandSchaible">5</a></dt>
  <dd><p><i class="sc">J.P. Crouzeix, J.A. Ferland</i> and <i class="sc">S. Schaible</i>, <i class="it">Duality in generalized fractional programming</i>, Math. Programming, <b class="bf">27</b>, pp.&#160;342–354, 1983. </p>
</dd>
  <dt><a name="HachimiAghezzaf">6</a></dt>
  <dd><p><i class="sc">M. Hachimi</i> and <i class="sc">B. Aghezzaf</i>, <i class="it">Second order duality in multiobjective programming involving generalized type-I functions</i>, Numer. Funct. Anal. Optim., <b class="bf">25</b>, pp.&#160;725–736, 2004. </p>
</dd>
  <dt><a name="Hanson">7</a></dt>
  <dd><p><i class="sc">M.A. Hanson</i>, <i class="it">Second order invexity and duality in mathematical programming</i>, Opsearch <b class="bf">30</b>, pp.&#160; 313–320, 1993. </p>
</dd>
  <dt><a name="HusainAhmadSharma">8</a></dt>
  <dd><p><i class="sc">Z. Husain, I. Ahmad</i> and <i class="sc">S. Sharma</i> <i class="it">Second order duality for minmax fractional programming</i>, Optim. Lett., <b class="bf">3</b>, pp.&#160;277–286, 2009. </p>
</dd>
  <dt><a name="LiangHuangPardalos">9</a></dt>
  <dd><p>Z.A. Liang, H.X. Huang and <i class="sc">P. Pardalos, P.M.</i>, <i class="it">Optimality conditions and duality for a class of nonlinear fractional programming problems</i>, J. Optim. Theory Appl., <b class="bf">110</b>, pp.&#160;611–619, 2001. </p>
</dd>
  <dt><a name="LiangShi">10</a></dt>
  <dd><p>Z. Liang and <i class="sc">Z. Shi</i>, <i class="it">Optimality conditions and duality for a minimax fractional programming with generalized convexity</i>, J. Math. Anal. Appl., <b class="bf">277</b>, pp.&#160; 474–488, 2003. </p>
</dd>
  <dt><a name="LiuWu">11</a></dt>
  <dd><p><i class="sc">J.C. Liu</i> and <i class="sc">C.S. Wu</i>, <i class="it">On minimax fractional optimality conditions with</i> \((F,\rho )\)-<i class="itshape">convexity</i>, J. Math. Anal. Appl, <b class="bf">219</b>, pp.&#160;36–51 1998. </p>
</dd>
  <dt><a name="LongQuan">12</a></dt>
  <dd><p><i class="sc">X.J. Long</i> and <i class="sc">J. Quan</i>, <i class="it">Optimality conditions and duality for minimax fractional programming involving nonsmooth generalized univexity</i>, Numer. Algebra Control Optim. <b class="bf">1</b>, pp.&#160;361–370, 2011. </p>
</dd>
  <dt><a name="Mangasarian">13</a></dt>
  <dd><p><i class="sc">O.L. Mangasarian</i>, <i class="it">Second and higher order duality in nonlinear programming</i>, J. Math. Anal. Appl. <b class="bf">51</b>, pp.&#160;607–620, 1975. </p>
</dd>
  <dt><a name="HuYangJian">14</a></dt>
  <dd><p><i class="sc">Q. Hu, G. Yang</i> and <i class="sc">J. Jian</i>, <i class="it">On second order duality for minimax fractional programming</i>, Nonlinear Anal. Real World Appl., <b class="bf">12</b>, pp.&#160;3509–3514, 2011. </p>
</dd>
  <dt><a name="HuChenJian">15</a></dt>
  <dd><p><i class="sc">Q. Hu, Y. Chen</i> and <i class="sc">J. Jian</i>, <i class="it">Second-order duality for nondifferentiable minimax fractional programming</i>, Int. J. Comput. Math., <b class="bf">89</b>, pp.&#160;11–16, 2012. </p>
</dd>
  <dt><a name="YadavMukherjee">16</a></dt>
  <dd><p><i class="sc">S.R. Yadav</i> and <i class="sc">R.N. Mukherjee</i>, <i class="it">Duality for fractional minimax programming problems</i>, J. Austral. Math. Soc. Ser. B, <b class="bf">31</b>, pp.&#160;482–492, 1990. </p>
</dd>
  <dt><a name="YangHou">17</a></dt>
  <dd><p><i class="sc">X.M. Yang</i> and <i class="sc">S.H. Hou</i>, <i class="it">On minimax fractional optimality and duality with generalized convexity</i>, J. Global Optim., <b class="bf">31</b>, pp.&#160;235–252, 2005. </p>
</dd>
  <dt><a name="ZhangMond">18</a></dt>
  <dd><p><i class="sc">J. Zhang</i> and <i class="sc">B. Mond</i>, <i class="it">Second order duality for multiobjective nonlinear programming involving generalized convexity</i>, in: B.M. Glover, B.D. Craven, D. Ralph (eds.), Proceeding of Optimization Miniconference III, University of Ballarat, pp.&#160;79–95, 1997. </p>
</dd>
</dl>


</div>
</div> <!--main-text -->
</div> <!-- content-wrapper -->
</div> <!-- content -->
</div> <!-- wrapper -->

<nav class="prev_up_next">
</nav>

<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/jquery.min.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/plastex.js"></script>
<script type="text/javascript" src="/var/www/clients/client1/web1/web/files/jnaat-files/journals/1/articles/js/svgxuse.js"></script>
</body>
</html>