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<h1>Vector optimization problems and approximated vector optimization problems</h1>
<p class="authors">
<span class="author">Eugenia Duca\(^\ast \) Dorel I. Duca\(^\S \)</span>
</p>
<p class="date">August 18, 2010.</p>
</div>
<p>\(^\ast \)Technical University, Department of Mathematics, Bariţiu Street, no. 25–28, 400027 Cluj-Napoca, Romania, e-mail: <span class="tt">jeniduca@yahoo.com, educa@math.utcluj.ro</span>. </p>
<p>\(^\S \) “Babeş-Bolyai" University, Faculty of Mathematics and Computer Science, 1 M. Kogălniceanu Street, 400084 Cluj-Napoca, Romania, e-mail: <span class="tt">dorelduca@yahoo.com, dduca@math.ubbcluj.ro</span>. </p>

<div class="abstract"><p> In this paper, a so-called approximated vector optimization problem associated to a vector optimization problem is considered. The equivalence between the efficient solutions of the approximated vector optimization problem and efficient solutions of the original optimization problem is established. </p>
<p><b class="bf">MSC.</b> Primary: 90C29; Secondary: 90C30, 90C46. </p>
<p><b class="bf">Keywords.</b> Efficient solution, invex function, pseudoinvex function, approximation. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>We consider the vector optimization problem</p>
<div class="equation" id="a0000000003">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} C-\min &  f\left( x\right) \\ \text{s.t.} &  x\in X \\ &  g\left( x\right) \in -K,\end{array} \tag {$VOP$} \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p>where \(X\) is a subset of \(\mathbb {R}^{n},\) \(C\) is a convex cone in \(\mathbb {R}^{p},\) \(K\) is a convex cone in \(\mathbb {R}^{m},\) and \(f:X\rightarrow \mathbb {R}^{p}\), \(g:X\rightarrow \mathbb {R}^{m}\) are functions\(.\) </p>
<p>Let</p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \mathfrak {F}\left( VOP\right) :=\{ x\in X:\text{ }g\left( x\right) \in -K\} , \end{equation*}
</div>
<p>denote the set of all feasible solutions of Problem \(\left( VOP\right) .\) </p>
<p>Let \(L:X\times K^{\ast }\rightarrow \mathbb {R}^{n}\) be the <b class="bfseries">lagrangian</b> of Problem <span class="rm">(VOP)</span>, i.e. the function defined by</p>
<div class="displaymath" id="a0000000005">
  \begin{equation*}  L\left( x,v\right) :=f\left( x\right) +\left\langle g\left( x\right) ,v\right\rangle e,\text{ for all }\left( x,v\right) \in X\times K^{\ast }, \end{equation*}
</div>
<p>where </p>
<div class="displaymath" id="a0000000006">
  \begin{equation*}  K^{\ast }:=\{ u\in \mathbb {R}^{m}:\text{ }\left\langle u,v\right\rangle \geqq 0,\text{ for all }v\in K\}  \end{equation*}
</div>
<p>is the polar of the convex cone \(K,\) and \(e=\left( 1,...,1\right) \in \mathbb {R}^{n}.\) </p>
<p><div class="definition_thmwrapper " id="a0000000007">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1.1</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(x^{0}\) be a point of \(\  \mathfrak {F}\left( VOP\right) .\) We say that \(x^{0}\) is an <b class="bfseries">efficient</b> <b class="bfseries">solution</b> for Problem <span class="rm">(VOP)</span> if there exists no point \(x\in \mathfrak {F}\left( VOP\right) \) such that</p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  f\left( x^{0}\right) -f\left( x\right) \in C\backslash \{ 0\} . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000009">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1.2</span>
  </div>
  <div class="remark_thmcontent">
  <p>The point \(x^{0}\) is an efficient solution for Problem <span class="rm">(VOP)</span> if and only if</p>
<div class="displaymath" id="a0000000010">
  \begin{equation*}  f\left( x^{0}\right) -f\left( x\right) \notin C\backslash \{ 0\} ,\text{ for all }x\in \mathfrak {F}\left( VOP\right).\hfil \qed \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="a0000000011">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">1.3</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(x^{0}\) be a point of \(\mathfrak {F}\left( VOP\right) .\) We say that \(x^{0}\) is a <b class="bfseries">weak efficient</b> <b class="bfseries">solution</b> for Problem <span class="rm">(VOP)</span> if there exists no point \(x\in \mathfrak {F}\left( VOP\right) \) such that</p>
<div class="displaymath" id="a0000000012">
  \begin{equation*}  f\left( x^{0}\right) -f\left( x\right) \in \mathrm{int}C. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000013">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">1.4</span>
  </div>
  <div class="remark_thmcontent">
  <p>The point \(x^{0}\) is a weak efficient solution for Problem <span class="rm">(VOP)</span> if and only if</p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  f\left( x^{0}\right) -f\left( x\right) \notin {\rm int}C,\text{ for all }x\in \mathfrak {F}\left( VOP\right).\hfil \qed \end{equation*}
</div>

  </div>
</div> </p>
<p>If \(C\) is the closed convex cone \(\mathbb {R}_{+}^{p}\) and \(K\) is the closed convex cone \(\mathbb {R}_{+}^{m},\) then problem <span class="rm">(VOP)</span>, becomes the multicriteria optimization problem</p>
<div class="equation" id="a0000000015">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} v\text{-}\min &  f\left( x\right) \\ \text{s.t.} &  x\in X \\ &  g\left( x\right) \leqq 0.\end{array} \tag {MOP} \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p>For solving vector optimization problem <span class="rm">(VOP)</span>, there are various manners to approach. One of these manners is that for Problem <span class="rm">(VOP)</span> one attaches another optimization problem, problem whose solutions gives us the \((\)information about\()\) solutions of the initial problem <span class="rm">(VOP)</span>. </p>
<p>If \(x^{0}\) is a feasible solution for <span class="rm">(MOP)</span> and \(f\) is differentiable at \(x^{0},\) C.R. Bector, S. Chandra and C. Singh <span class="cite">
	[
	<a href="#BectorChandraSingh" >3</a>
	]
</span>, attached to Problem <span class="rm">(MOP)</span>, the problem</p>
<div class="equation" id="a0000000016">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} v\text{-}\min &  \left[ \nabla f\left( x^{0}\right) \right] \left( x\right) \\ \text{s.t.} &  x\in X \\ &  g\left( x\right) \leqq 0,\end{array} \tag {LMOP} \end{equation}
  </div>
  <span class="equation_label">1.3</span>
</p>
</div>
<p>and obtained connections of efficient solutions of the original problem <span class="rm">(MOP)</span> to the efficient solutions of the linearized multicriteria optimization problem <span class="rm">(LMOP)</span>. </p>
<p>If \(x^{0}\in \mathfrak {F}\left( MOP\right) \) is an interior point of \(X,\) \(f\) is differentiable at \(x^{0}\), Antczak <span class="cite">
	[
	<a href="#Antczak03" >2</a>
	]
</span>, proposed the following approximated multicriteria optimization problem</p>
<div class="equation" id="a0000000017">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} v\text{-}\min &  \left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \\ \text{s.t.} &  x\in X \\ &  g\left( x\right) \leqq 0,\end{array} \tag {$\eta MOP$} \end{equation}
  </div>
  <span class="equation_label">1.4</span>
</p>
</div>
<p>where \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) is a function, and obtained results to connect <span class="rm">(MOP)</span> and <span class="rm">(\(\eta \)MOP)</span>. </p>
<p>In this paper, assuming that \(x^{0}\in \mathfrak {F}\left( VOP\right) \) is an interior point of \(X,\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) is a function, we attach to Problem <span class="rm">(VOP)</span> the problems: </p>
<p>\(a)\) \(\  \)assuming that \(f\) is differentiable at \(x^{0}\)</p>
<div class="equation" id="a0000000018">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} C\text{-}\min &  f\left( x^{0}\right) +\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \\ \text{s.t.} &  x\in X \\ &  g\left( x\right) \in -K,\end{array} \tag {FAVOP} \end{equation}
  </div>
  <span class="equation_label">1.5</span>
</p>
</div>
<p>and \(b)\) \(\  \)assuming that \(g\) is differentiable at \(x^{0}\)</p>
<div class="equation" id="a0000000019">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} C-\min &  f\left( x\right) \\ \text{s.t.} &  x\in X \\ &  g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \in -K.\end{array} \tag {CAVOP} \end{equation}
  </div>
  <span class="equation_label">1.6</span>
</p>
</div>
<h1 id="a0000000020">2 Notions and Preliminary Results</h1>
<p>In the last few years, attempts have been made to weaken the convexity hypotheses and thus to explore the existence of optimality conditions applicability. Various classes of generalized convex functions have been suggested for the purpose of weakening the convexity limitation of the results. Among these, the concept of an invex function proposed by Hanson <span class="cite">
	[
	<a href="#Hanson81" >11</a>
	]
</span> has received more attention. The name of invex \((\)invariant convex\()\) function was given by Craven <span class="cite">
	[
	<a href="#Craven81" >5</a>
	]
</span>. </p>
<p><div class="definition_thmwrapper " id="Dinvex">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.1</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(X\) be a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0}\) be an interior point of \(X,\) \(C\) be a closed convex cone in \(\mathbb {R}^{p}\), \(f:X\rightarrow \mathbb {R}^{p}\) be a differentiable function at \(x^{0},\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function. </p>
<ul class="itemize">
  <li><p>We say that the function \(f\) is \(C\)-<b class="bfseries">invex</b> at<b class="bfseries"> </b>\(x^{0}\) with respect to \((\)w.r.t.\()\) \(\eta \) if</p>
<div class="equation" id="f1JD">
<p>
  <div class="equation_content">
    \begin{equation}  f\left( x\right) -f\left( x^{0}\right) -\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \in C,\text{ for all }x\in X. \label{f1JD} \end{equation}
  </div>
  <span class="equation_label">2.7</span>
</p>
</div>
</li>
  <li><p>We say that the function \(f\) is \(C\)-<b class="bfseries">incave</b> at<b class="bfseries"> </b>\(x^{0}\) with respect to \((\)w.r.t.\()\) \(\eta \) if</p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  f\left( x\right) -f\left( x^{0}\right) -\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \in -C,\text{ for all }x\in X. \end{equation*}
</div>
</li>
</ul>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000022">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.2</span>
  </div>
  <div class="remark_thmcontent">
  <p>The function \(f\) is \(C\)-incave at<b class="bfseries"> </b>\(x^{0}\) w.r.t. \(\eta \) if anf only if the function \(f\) is \(\left( -C\right) \)-invex at<b class="bfseries"> </b>\(x^{0}\) w.r.t. \(\eta .\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="Efctinv1">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">2.3</span>
  </div>
  <div class="example_thmcontent">
  <p>Let \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) the function defined by</p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  f\left( x\right) :=\left( x_{1}^{2}+\sin \tfrac {\pi x_{2}}{2},x_{2}^{2}+\sin \tfrac {\pi x_{1}}{3}\right) ,\text{ for all }x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2}. \end{equation*}
</div>
<p>\(a)\) The function \(f\) is \(\mathbb {R}_{+}^{2}\)-invex at \(x^{0}=\left( 0,0\right) \) w.r.t. \(\eta :\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) defined by</p>
<div class="displaymath" id="a0000000024">
  \begin{equation*}  \eta \left( x,u\right) :=\left( \tfrac {3}{\pi }\sin \tfrac {\pi x_{1}}{3},\tfrac {2}{\pi }\sin \tfrac {\pi x_{2}}{2}\right) , \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) </p>
<p>\(b)\) Also, the function \(f\) is \(\mathbb {R}_{+}^{2}\)-invex at \(x^{0}=\left( 0,0\right) \) w.r.t. \(\mu :\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) defined by</p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  \mu \left( x,u\right) :=\left( \tfrac {3}{\pi }\sin \tfrac {\pi x_{1}}{3}-4,\tfrac {2}{\pi }\sin \tfrac {\pi x_{2}}{2}-7\right) , \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) </p>
<p>Let’s remark that \(\mu \left( x,u\right) \neq \left( 0,0\right) ,\) for all \(\left( x,u\right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) </p>
<p>\(c)\) Also, the function \(f\) is \(\mathbb {R}_{+}^{2}\)-invex at \(x^{0}=\left( 0,0\right) \) w.r.t. \(\zeta :\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) defined by</p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  \zeta \left( x,u\right) :=\left( \tfrac {3}{\pi }\sin \tfrac {\pi x_{1}}{3}-x_{1}^{2},\tfrac {2}{\pi }\sin \tfrac {\pi x_{2}}{2}-x_{2}^{2}\right) , \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>After the works of Hanson and Craven, other types of differentiable functions have appeared with the intent of generalizing invex function from different points of view. </p>
<p>Ben-Israel and Mond <span class="cite">
	[
	<a href="#Ben-Israel-Mond86" >4</a>
	]
</span> defined the so-called pseudoinvex functions, generalizing pseudoconvex functions in the same way that invex functions generalize convex functions. Here we give the following notion of pseudoinvexity: </p>
<p><div class="definition_thmwrapper " id="pinv">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.4</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(X\) be a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) and \(L\) be two convex cones in \(\mathbb {R}^{p}\), \(f:X\rightarrow \mathbb {R}^{p}\) be a differentiable function at \(x^{0},\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function. We say that \(f\) is \(\left( K,L\right) \)-<b class="bfseries">pseudoinvex</b> at \(x^{0}\) with respect to \((\)w.r.t.\()\) \(\eta \) if, for each \(x\in X\backslash \{ x^{0}\} \) with the property that</p>
<div class="displaymath" id="a0000000027">
  \begin{equation*}  \left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \in K, \end{equation*}
</div>
<p>we have</p>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  f\left( x\right) -f\left( x^{0}\right) \in L. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000029">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.5</span>
  </div>
  <div class="remark_thmcontent">
  <p>The notion of \(K\)-pseudoinvexity is equivalent with the notion of \(\left( K,K\right) \)-pseudoinvexity.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="qinv">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.6</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(X\) be a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) and \(L\) be two convex cones in \(\mathbb {R}^{p}\), \(f:X\rightarrow \mathbb {R}\) be a differentiable function at \(x^{0},\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function. We say that \(f\) is \(\left( K,L\right) \)-<b class="bfseries">quasiinvex</b> at \(x^{0}\) with respect to \((\)w.r.t.\()\) \(\eta \) if, for each \(x\in X\backslash \{ x^{0}\} \) with the property that</p>
<div class="displaymath" id="a0000000030">
  \begin{equation*}  f\left( x^{0}\right) -f\left( x\right) \in K, \end{equation*}
</div>
<p>we have</p>
<div class="displaymath" id="a0000000031">
  \begin{equation*}  \left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \in L. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000032">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.7</span>
  </div>
  <div class="remark_thmcontent">
  <p>The notion of \(K\)-quasiinvexity is equivalent with the notion of \(\left( K,-K\right) \)-quasiinvexity.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000033">3 The modified criteria function of Vector Optimization Problems</h1>
<p>In this section, \(X\) is a subset of \(\mathbb {R}^{n},\) \(x^{0}\) is an interior point of \(X,\) \(f:X\rightarrow \mathbb {R}^{p}\) is a differentiable function at \(x^{0},\) \(C\) is a convex cone in \(\mathbb {R}^{p}\), \(K\) is a convex cone in \(\mathbb {R}^{m}\), and \(g:X\rightarrow \mathbb {R}^{m}\) is a function\(.\) </p>
<p>For \(\eta :X\times X\rightarrow \mathbb {R}^{n},\) we attach to Problem \(\left( VOP\right) \) the following optimization problem:</p>
<div class="equation" id="a0000000034">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} C\text{-}\min &  f\left( x^{0}\right) +\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \\ \text{s.t.} &  x\in X \\ &  g\left( x\right) \in -K.\end{array} \tag {FAVOP} \end{equation}
  </div>
  <span class="equation_label">3.8</span>
</p>
</div>
<p>Let</p>
<div class="displaymath" id="a0000000035">
  \begin{equation*}  \mathfrak {F}\left( FAVOP\right) :=\{ x\in X:\text{ }g\left( x\right) \in -K\} , \end{equation*}
</div>
<p>denote the set of all feasible solutions of Problem <span class="rm">(FAVOP)</span>. </p>
<p>Obviously</p>
<div class="displaymath" id="a0000000036">
  \begin{equation*}  \mathfrak {F}\left( FAVOP\right) =\mathfrak {F}\left( VOP\right) . \end{equation*}
</div>
<p>Let \(F:X\rightarrow \mathbb {R}^{p}\) the function defined by</p>
<div class="displaymath" id="a0000000037">
  \begin{equation*}  F\left( x\right) =f\left( x^{0}\right) +\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) ,\text{ for all }x\in X, \end{equation*}
</div>
<p>i.e. the criteria function of Problem <span class="rm">(FAVOP)</span>. </p>
<p><div class="theorem_thmwrapper " id="T1JDV">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(X\) be a subset of \(\mathbb {R} ^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) be a closed convex cone in \(\mathbb {R}^{m},\) \(C\) be a closed convex cone in \(\mathbb {R}^{p},\) \(g:X\rightarrow \mathbb {R} ^{m}\) be a function, \(\eta :X\times X\rightarrow \mathbb {R} ^{n}\) such that \(\eta \left( x^{0},x^{0}\right) =0\) and \(f:X\rightarrow \mathbb {R} ^{p}\) be a differentiable function at \(x^{0}\) and \(\left( -C\backslash \{ 0\} ,-C\backslash \{ 0\} \right) \)-pseudoinvex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>If \(x^{0}\) is an efficient solution for <span class="rm">(VOP)</span>, then \(x^{0}\) is an efficient solution for <span class="rm">(FAVOP)</span> . </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000038">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Assume that \(x^{0}\) is not an efficient solution for <span class="rm">(FAVOP)</span>, then there exists a feasible solution \(x^{1}\in \mathfrak {F}\left( FAVOP\right) \) such that</p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  F\left( x^{0}\right) -F\left( x^{1}\right) \in C\backslash \{ 0\} . \end{equation*}
</div>
<p>Since \(\eta \left( x^{0},x^{0}\right) =0,\) we have</p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  F\left( x^{0}\right) -F\left( x^{1}\right) = \end{equation*}
</div>
<div class="displaymath" id="a0000000041">
  \begin{equation*}  f\left( x^{0}\right) +\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) -f\left( x^{0}\right) -\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x^{1},x^{0}\right) \right) = \end{equation*}
</div>
<div class="displaymath" id="a0000000042">
  \begin{equation*}  =-\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x^{1},x^{0}\right) \right) , \end{equation*}
</div>
<p>hence</p>
<div class="displaymath" id="a0000000043">
  \begin{equation*}  \left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x^{1},x^{0}\right) \right) \in -C\backslash \{ 0\} . \end{equation*}
</div>
<p>But \(f\) is \(\left( -C\backslash \{ 0\} ,-C\backslash \{ 0\} \right) \)-pseudoinvex at \(x^{0}\) w.r.t \(\eta \) and then</p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  f\left( x^{1}\right) -f\left( x^{0}\right) \in -C\backslash \{ 0\} , \end{equation*}
</div>
<p>i.e. \(x^{0}\) is not an efficient solution for <span class="rm">(VOP)</span>. The theorem is proved. <div class="proof_wrapper" id="a0000000045">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="R1JDV">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.2</span>
  </div>
  <div class="remark_thmcontent">
  <p>The hypothesis that \(f\) is \(\left( -C\backslash \{ 0\} ,-C\backslash \{ 0\} \right) \)-pseudoinvex at \(x^{0}\) w.r.t. \(\eta \) is essential, as seen in the following example.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="E1JDV">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">3.3</span>
  </div>
  <div class="example_thmcontent">
  <p>Let’s consider Problem <span class="rm">(VOP)</span> with \(X:=\mathbb {R}^{2},\) \(C:=\mathbb {R}_{+}^{2},\) \(K:=\mathbb {R}_{+}^{2},\) and \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2},\) \(g:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) and \(\eta :\) \(\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) the functions defined by</p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  f\left( x\right) =\left( x_{1}^{2}+\sin \tfrac {\pi x_{2}}{2},x_{2}^{2}+\sin \tfrac {\pi x_{1}}{3}\right) , \end{equation*}
</div>
<div class="displaymath" id="a0000000047">
  \begin{equation*}  g\left( x\right) =\left( x_{1}^{2}-x_{2},x_{2}^{2}-x_{1}\right) , \end{equation*}
</div>
<p>for all \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\) and</p>
<div class="displaymath" id="a0000000048">
  \begin{equation*}  \eta \left( x,u\right) =\left( \tfrac {3}{\pi }\sin \tfrac {\pi x_{1}}{3}-\tfrac {3}{\pi }x_{2}^{2},\text{ }\tfrac {2}{\pi }\sin \tfrac {\pi x_{2}}{2}-\tfrac {4}{\pi }x_{1}^{2}\right) , \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) </p>
<p>The point \(x^{0}=\left( 0,0\right) \in \mathfrak {F}\left( VOP\right) \) is an efficient solution for Problem <span class="rm">(VOP)</span>. On the other hand, </p>
<div class="displaymath" id="a0000000049">
  \begin{equation*}  F\left( x\right) =f\left( x^{0}\right) +\left[ \bigtriangledown f(x^{0})\right] (\eta (x,x^{0}))=\left( \sin \tfrac {\pi x_{2}}{2}-2x_{1}^{2},\sin \tfrac {\pi x_{1}}{3}-x_{2}^{2}\right) , \end{equation*}
</div>
<p>for all \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\) and for \(x^{1}=\left( 1,1\right) \in \mathfrak {F}\left( VOP\right) = \mathfrak {F}\left( FAVOP\right) ,\)</p>
<div class="displaymath" id="a0000000050">
  \begin{equation*}  F\left( x^{0}\right) -F\left( x^{1}\right) =\left( 1,\tfrac {1-\sqrt{3}}{2}\right) \in C\backslash \{ 0\} =\mathbb {R}_{+}^{2}\backslash \{ 0\} . \end{equation*}
</div>
<p>Consequently, \(x^{0}\) is not an efficient solution for <span class="rm">(FAVOP)</span>. </p>
<p>Let’s remark that \(f\) is not \(\left( -C\backslash \{ 0\} ,-C\backslash \{ 0\} \right) \)-pseudoinvex at \(x^{0}\) w.r.t. \(\eta ,\) because</p>
<div class="displaymath" id="a0000000051">
  \begin{equation*}  \left[ \bigtriangledown f(x^{0})\right] (\eta (x^{1},x^{0}))=\left( -1,\tfrac {-\sqrt{3}-1}{2}\right) \in -C\backslash \{ 0\} =-\mathbb {R}_{+}^{2}\backslash \{ 0\}  \end{equation*}
</div>
<p>and</p>
<div class="displaymath" id="a0000000052">
  \begin{equation*}  f\left( x^{1}\right) -f\left( x^{0}\right) =\left( 2,1+\tfrac {\sqrt{3}}{2}\right) \notin -C\backslash \{ 0\} =-\mathbb {R}_{+}^{2}\backslash \{ 0\} . \end{equation*}
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="R2JDV">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.4</span>
  </div>
  <div class="remark_thmcontent">
  <p>The hypothesis that \(\eta \left( x^{0},x^{0}\right) =0\) is essential, as seen in the following example.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="E2JDV">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">3.5</span>
  </div>
  <div class="example_thmcontent">
  <p>Let’s consider Problem <span class="rm">(VOP)</span> with \(X:=\mathbb {R}^{2},\) \(C:=\mathbb {R}_{+}^{2},\) \(K:=\mathbb {R}_{+}^{2},\) and \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2},\) \(g:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) and \(\eta :\) \(\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) the functions defined by</p>
<div class="displaymath" id="a0000000053">
  \begin{equation*}  f\left( x\right) =\left( x_{1},x_{2}\right) ,\text{ \  }g\left( x\right) =\left( x_{1}^{2}-x_{2},x_{2}^{2}-x_{1}\right) , \end{equation*}
</div>
<p>for all \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}_{+}^{2},\) and</p>
<div class="displaymath" id="a0000000054">
  \begin{equation*}  \eta \left( x,u\right) =\left( x_{1}+\left( x_{1}-1\right) ^{2},x_{2}+\left( x_{2}-1\right) ^{2}\right) , \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) The point \(x^{0}=\left( 0,0\right) \) is an efficient solution for Problem <span class="rm">(VOP)</span>. The function \(f\) is \(\left( -C\right) \)-invex at \(x^{0}\) w.r.t. \(\eta ,\) because, for all \(\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\) we have:</p>
<div class="displaymath" id="a0000000055">
  \begin{equation*}  f\left( x\right) -f\left( x^{0}\right) -\left[ \nabla f\left( x_{0}\right) \left( \eta \left( x,x^{0}\right) \right) \right] =-\left( \left( 1-x_{1}\right) ^{2},\left( 1-x_{2}\right) ^{2}\right) \in -C. \end{equation*}
</div>
<p>It follows that \(f\) is \(\left( -C\backslash \{ 0\} ,-C\backslash \{ 0\} \right) \)-pseudoinvex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>Since, for each \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\) </p>
<div class="displaymath" id="a0000000056">
  \begin{equation*}  F\left( x\right) =f\left( x^{0}\right) +\left[ \nabla f\left( x^{0}\right) \left( \eta \left( x,x^{0}\right) \right) \right] =\left( x_{1}+\left( x_{1}-1\right) ^{2},x_{2}+\left( x_{2}-1\right) ^{2}\right) , \end{equation*}
</div>
<p>we deduce that \(x^{0}\) is not an efficient solution for Problem <span class="rm">(FAVOP)</span>. Why? Because, for \(x^{1}=\left( \tfrac {1}{2},\tfrac {1}{2}\right) \), we have</p>
<div class="displaymath" id="a0000000057">
  \begin{align*}  F\left( x^{0}\right) -F\left( x^{1}\right) & =f\left( x^{0}\right) +\left[ \nabla f\left( x_{0}\right) \left( \eta \left( x^{0},x^{0}\right) \right) \right] -f\left( x^{0}\right) -\\ & \quad -\left[ \nabla f\left( x_{0}\right) \left( \eta \left( x^{1},x^{0}\right) \right) \right] =\\ & =\left( \tfrac {3}{4},\tfrac {3}{4}\right) \in C\backslash \{ 0\} . \end{align*}
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="T1JDVs">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.6</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(X\) be a subset of \(\mathbb {R} ^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) be a closed convex cone in \(\mathbb {R}^{m},\) \(C\) be a closed convex cone with nonempty interior in \(\mathbb {R}^{p},\) \(g:X\rightarrow \mathbb {R} ^{m}\) be a function, \(\eta :X\times X\rightarrow \mathbb {R} ^{n}\) such that \(\eta \left( x^{0},x^{0}\right) =0\) and \(f:X\rightarrow \mathbb {R} ^{p}\) be a differentiable function at \(x^{0}\) and \(\left( -\mathrm{int}C,-\mathrm{int}C\right) \)-pseudoinvex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>If \(x^{0}\) is a weak efficient solution for <span class="rm">(VOP)</span>, then \(x^{0} \) is a weak efficient solution for <span class="rm">(FAVOP)</span>. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000058">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The proof is similar to the proof of Theorem <a href="#T1JDV">3.1</a>. <div class="proof_wrapper" id="a0000000059">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="T2JDV">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(X\) be a subset of \(\mathbb {R} ^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) be a closed convex cone in \(\mathbb {R}^{m},\) \(C\) be a closed convex cone in \(\mathbb {R}^{p},\) \(g:X\rightarrow \mathbb {R} ^{m}\) be a function, \(\eta :X\times X\rightarrow \mathbb {R} ^{n}\) such that \(\eta \left( x^{0},x^{0}\right) =0\) and \(f:X\rightarrow \mathbb {R} ^{p}\) be a differentiable function at \(x^{0}\) and \(\left( C\backslash \{ 0\} ,C\backslash \{ 0\} \right) \)-quasiinvex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>If \(x^{0}\) is an efficient solution for <span class="rm">(FAVOP)</span>, then \(x^{0}\) is an efficient solution for <span class="rm">(VOP)</span>. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000060">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Assume that \(x^{0}\) is not an efficient solution for <span class="rm">(VOP)</span>, then there exists a feasible solution \(x^{1}\in \mathfrak {F}\left( VOP\right) \) such that</p>
<div class="displaymath" id="a0000000061">
  \begin{equation*}  f\left( x^{0}\right) -f\left( x^{1}\right) \in C\backslash \{ 0\} . \end{equation*}
</div>
<p>But \(f\) is \(\left( C\backslash \{ 0\} ,-C\backslash \{ 0\} \right) \)-quasiinvex at \(x^{0}\) w.r.t \(\eta \) and hence</p>
<div class="displaymath" id="a0000000062">
  \begin{equation*}  \left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x^{1},x^{0}\right) \right) \in -C\backslash \{ 0\} . \end{equation*}
</div>
<p>It follows that</p>
<div class="displaymath" id="a0000000063">
  \begin{align*} & F\left( x^{0}\right) -F\left( x^{1}\right) =\\ & =f\left( x^{0}\right) +\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) -f\left( x^{0}\right) -\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x^{1},x^{0}\right) \right) \in C\backslash \{ 0\} , \end{align*}
</div>
<p>because \(\eta \left( x^{0},x^{0}\right) =0.\) Consequently, \(x^{0}\) is not an efficient solution for <br /><span class="rm">(FAVOP)</span> which is a contradiction. The theorem is proved. <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> </p>
<p><div class="remark_thmwrapper " id="R3JDV">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.8</span>
  </div>
  <div class="remark_thmcontent">
  <p>In Theorem <a href="#T2JDV">3.7</a>, the hypothesis that \(f\) is \(C\)-invex at \(x^{0}\) w.r.t. \(\eta \) is essential, as seen in the following example.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="E3JDV">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">3.9</span>
  </div>
  <div class="example_thmcontent">
  <p>Let’s consider Problem <span class="rm">(VOP)</span> with \(X:=\mathbb {R}^{2},\) \(C:=\mathbb {R}_{+}^{2},\) \(K:=\mathbb {R}_{+}^{2},\) and \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2},\) \(g:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) and \(\eta :\) \(\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) the functions defined by</p>
<div class="displaymath" id="a0000000065">
  \begin{equation*}  f\left( x\right) =\left( x_{1},x_{2}\right) , \end{equation*}
</div>
<div class="displaymath" id="a0000000066">
  \begin{equation*}  g\left( x\right) =\left( x_{1}^{2}-x_{2},x_{2}^{2}-x_{1}\right) , \end{equation*}
</div>
<p>for all \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\) and</p>
<div class="displaymath" id="a0000000067">
  \begin{equation*}  \eta \left( x,u\right) =\left( x_{1}+\left( x_{1}-1\right) ^{2},\text{ }x_{2}+\left( x_{2}-1\right) ^{2}\right) , \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) For \(x^{0}=\left( 0,0\right) \in \mathfrak {F}\left( VOP\right) ,\) we have</p>
<div class="displaymath" id="a0000000068">
  \begin{equation*}  f\left( x\right) -f\left( x^{0}\right) -\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) = \end{equation*}
</div>
<div class="displaymath" id="a0000000069">
  \begin{equation*}  =-\left( \left( x_{1}-1\right) ^{2},\left( x_{2}-1\right) ^{2}\right) \in -C=-\mathbb {R}_{+}^{2}, \end{equation*}
</div>
<p>for all \(x\in \mathbb {R}^{2}.\) Consequently, the function \(f\) is not \(C\)-invex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>Since, for each \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\)</p>
<div class="displaymath" id="a0000000070">
  \begin{align*} & F\left( x\right) =f\left( x^{0}\right) +\left[ \bigtriangledown f(x^{0})\right] (\eta (x,x^{0}))=\left( x_{1}+\left( x_{1}-1\right) ^{2},x_{2}+\left( x_{2}-1\right) ^{2}\right) =\\ & =\left( \left( x_{1}-\tfrac {1}{2}\right) ^{2}+\tfrac {3}{4},\left( x_{2}-\tfrac {1}{2}\right) ^{2}+\tfrac {3}{4}\right) \end{align*}
</div>
<p>it follows that \(x^{1}=\left( \tfrac {1}{2},\tfrac {1}{2}\right) \in \mathfrak {F}\left( VOP\right) =\mathfrak {F}\left( FAVOP\right) \) is an efficient solution for Problem <span class="rm">(FAVOP)</span>. </p>
<p>On the other hand,</p>
<div class="displaymath" id="a0000000071">
  \begin{equation*}  f\left( x^{1}\right) -f\left( x^{0}\right) =\left( \tfrac {1}{2},\tfrac {1}{2}\right) \in C\backslash \{ 0\} . \end{equation*}
</div>
<p>Consequently, \(x^{1},\) which is an efficient solution for Problem <span class="rm">(FAVOP)</span>, is not an efficient solution for problem <span class="rm">(VOP)</span>; the function \(f\) is not \(C\)-invex at \(x^{0}\) w.r.t. \(\eta .\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="R4JDV">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.10</span>
  </div>
  <div class="remark_thmcontent">
  <p>In Theorem <a href="#T2JDV">3.7</a>, the hypothesis that \(\eta \left( x^{0},x^{0}\right) =0\) is essential, as seen in the following example.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="example_thmwrapper " id="E4JDV">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">3.11</span>
  </div>
  <div class="example_thmcontent">
  <p>Let’s consider Problem <span class="rm">(VOP)</span> with \(X:=\mathbb {R}^{2},\) \(C:=\mathbb {R}_{+}^{2},\) \(K:=\mathbb {R}_{+}^{2},\) and \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2},\) \(g:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) and \(\eta :\) \(\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) the functions defined by</p>
<div class="displaymath" id="a0000000072">
  \begin{equation*}  f\left( x\right) =\left( x_{1}^{2}+x_{1},x_{2}^{2}+x_{2}\right) , \end{equation*}
</div>
<div class="displaymath" id="a0000000073">
  \begin{equation*}  g\left( x\right) =\left( x_{1}^{2}-x_{2},x_{2}^{2}-x_{1}\right) , \end{equation*}
</div>
<p>for all \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\) and</p>
<div class="displaymath" id="a0000000074">
  \begin{equation*}  \eta \left( x,u\right) =\left( x_{1}-\left( x_{1}^{2}+1\right) ^{2},\text{ }x_{2}-\left( x_{2}^{2}+1\right) ^{2}\right) , \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) </p>
<p>For \(x^{0}=\left( 0,0\right) \in \mathfrak {F}\left( VOP\right) ,\) we have</p>
<div class="displaymath" id="a0000000075">
  \begin{align*} & f\left( x\right) -f\left( x^{0}\right) -\left[ \nabla f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) =\\ & =\left( x_{1}^{2}+\left( x_{1}^{2}+1\right) ,x_{2}^{2}+\left( x_{2}^{2}+1\right) \right) \in C, \end{align*}
</div>
<p>for all \(x\in \mathbb {R}^{2}.\) Consequently, the function \(f\) is \(C\)-invex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>Since, for each \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\)</p>
<div class="displaymath" id="a0000000076">
  \begin{align*} & F\left( x\right) =f\left( x^{0}\right) +\left[ \bigtriangledown f(x^{0})\right] (\eta (x,x^{0}))=\left( x_{1}-\left( x_{1}^{2}+1\right) ^{2},x_{2}-\left( x_{2}^{2}+1\right) ^{2}\right) = \\ & =\left( -x_{1}^{4}-4x_{1}^{3}-6x_{1}^{2}-3x_{1}-1,-x_{2}^{4}-4x_{2}^{3}-6x_{2}^{2}-3x_{2}-1\right) , \end{align*}
</div>
<p>it follows that \(x^{1}=\left( 1,1\right) \in \mathfrak {F}\left( VOP\right) =\mathfrak {F}\left( FAVOP\right) \) is an efficient solution for Problem <span class="rm">(FAVOP)</span>. </p>
<p>On the other hand,</p>
<div class="displaymath" id="a0000000077">
  \begin{equation*}  f\left( x^{0}\right) -f\left( x^{1}\right) =\left( 2,2\right) \in C\backslash \{ 0\} . \end{equation*}
</div>
<p>Consequently, \(x^{1},\) which is an efficient solution for Problem <span class="rm">(FAVOP)</span>, is not an efficient solution for problem <span class="rm">(VOP)</span>. </p>
<p>Let’s remark that \(\eta \left( x^{0},x^{0}\right) =\left( -1,-1\right) \neq \left( 0,0\right) .\) <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="T2JDVs">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.12</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(X\) be a subset of \(\mathbb {R} ^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) be a closed convex cone in \(\mathbb {R}^{m},\) \(C\) be a closed convex cone in \(\mathbb {R}^{p},\) \(g:X\rightarrow \mathbb {R} ^{m}\) be a function, \(\eta :X\times X\rightarrow \mathbb {R} ^{n}\) such that \(\eta \left( x^{0},x^{0}\right) =0\) and \(f:X\rightarrow \mathbb {R} ^{p}\) be a differentiable function at \(x^{0}\) and \(\left( \mathrm{int}C,\mathrm{int}C\right) \)-quasiinvex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>If \(x^{0}\) is a weak efficient solution for <span class="rm">(FAVOP)</span>, then \(x^{0}\) is a weak efficient solution for <span class="rm">(VOP)</span>. </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000078">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> The proof is similar to the proof of Theorem <a href="#T2JDV">3.7</a>. <div class="proof_wrapper" id="a0000000079">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<h1 id="a0000000080">4 The modified constraint function of Vector Optimization Problems</h1>
<p>In this section, \(X\) is a subset of \(\mathbb {R}^{n},\) \(x^{0}\) is an interior point of \(X,\) \(f:X\rightarrow \mathbb {R}^{p}\) is a function, \(C\) is a convex cone in \(\mathbb {R}^{p}\), \(K\) is a convex cone in \(\mathbb {R}^{m}\), and \(g:X\rightarrow \mathbb {R}^{m}\) is a differentiable function at \(x^{0}.\) </p>
<p>For \(\eta :X\times X\rightarrow \mathbb {R}^{n},\) we attach to Problem <span class="rm">(VOP)</span> the following vector optimization problem: </p>
<div class="equation" id="a0000000081">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} C\text{-}\min &  f\left( x\right) \\ \text{s.t.} &  x\in X \\ &  g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \in -K\end{array} \tag {CAVOP} \end{equation}
  </div>
  <span class="equation_label">4.9</span>
</p>
</div>
<p>Let</p>
<div class="displaymath" id="a0000000082">
  \begin{equation*}  \mathfrak {F}\left( CAVOP\right) :=\{ x\in X:\text{ }g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \in -K\} , \end{equation*}
</div>
<p>denote the set of all feasible solutions of Problem <span class="rm">(CAVOP)</span>. </p>
<p><div class="theorem_thmwrapper " id="T5JDV">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(X\) be a subset of \(\mathbb {R} ^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) be a closed convex cone in \(\mathbb {R}^{m},\) \(C\) be a closed convex cone in \(\mathbb {R}^{p},\) \(\eta :X\times X\rightarrow \mathbb {R} ^{n}\) and \(f:X\rightarrow \mathbb {R} \) be two functions and \(g:X\rightarrow \mathbb {R} ^{m}\) be a differentiable function at \(x^{0}.\) </p>
<p>If the function \(g\) is \(K\)-incave at \(x^{0}\) w.r.t. \(\eta \), then every feasible solution for Problem <span class="rm">(CAVOP)</span> is a feasible solution for Problem <span class="rm">(VOP)</span>, i. e.</p>
<div class="displaymath" id="a0000000083">
  \begin{equation*}  \mathcal{F} (CAVOP)\subseteq \mathcal{F} (VOP). \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000084">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(x^{1}\in \mathcal{F} (CAVOP),\) i.e\(.\) </p>
<div class="displaymath" id="a0000000085">
  \begin{equation*}  g(x^{0})+\left[ \bigtriangledown g(x^{0})\right] (\eta (x^{1},x^{0}))\in -K. \end{equation*}
</div>
<p>Since \(g\) is \(K\)-incave at \(x^{0}\) w.r.t. \(\eta \), we have </p>
<div class="displaymath" id="a0000000086">
  \begin{equation*}  g(x^{1})-g(x^{0})-\left[ \bigtriangledown g(x^{0})\right] (\eta (x^{1},x^{0}))\in -K \end{equation*}
</div>
<p>From this, it follows </p>
<div class="displaymath" id="a0000000087">
  \begin{equation*}  g(x^{1})\in -K+\{ g\left( x^{0}\right) +\left[ \bigtriangledown g(x^{0})\right] (\eta (x^{1},x^{0}))\} \subseteq -K+\left( -K\right) =-K, \end{equation*}
</div>
<p>hence \(x\in \mathcal{F} (VOP)\) <div class="proof_wrapper" id="a0000000088">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="example_thmwrapper " id="E5JDV">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">4.2</span>
  </div>
  <div class="example_thmcontent">
  <p>Let’s consider Problem <span class="rm">(VOP)</span> with \(X=\mathbb {R}^{2},\) \(C=K=\mathbb {R}_{+}^{2},\) and \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2},\) \(g:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) and \(\eta :\) \(\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) the functions defined by</p>
<div class="displaymath" id="a0000000089">
  \begin{equation*}  f\left( x\right) =\left( \sin \tfrac {\left( x_{1}+x_{2}\right) \pi }{4},\text{ }x_{1}^{2}\left( x_{2}-7\right) ^{2}\right) , \end{equation*}
</div>
<div class="displaymath" id="a0000000090">
  \begin{equation*}  g\left( x\right) =\left( x_{1}^{2}-x_{2},\text{ }x_{2}^{2}-x_{1}\right) , \end{equation*}
</div>
<p>for all \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2}\) and</p>
<div class="displaymath" id="a0000000091">
  \begin{equation*}  \eta \left( x,u\right) =\left( x_{1}-u_{1},\text{ }x_{2}-u_{2}\right) \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) </p>
<p>The function \(g\) is not \(\mathbb {R}_{+}^{2}\)-incave at \(x^{0}=\left( 0,0\right) \) w.r.t. \(\eta ,\) because</p>
<div class="displaymath" id="a0000000092">
  \begin{equation*}  g\left( 1,1\right) -g\left( x^{0}\right) -\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( \left( 1,1\right) ,x^{0}\right) \right) =\left( 1,1\right) \notin -\mathbb {R}_{+}^{2} \end{equation*}
</div>
<p>Since </p>
<div class="displaymath" id="a0000000093">
  \begin{equation*}  g\left( x^{0}\right) +\left[ \bigtriangledown g(x^{0})\right] (\eta (x,x^{0}))=\left( -x_{2},-x_{1}\right) ,\text{ for all }x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2}, \end{equation*}
</div>
<p>the set of feasible solutions for Problem <span class="rm">(CAVOP)</span> is \(\mathfrak {F}\left( CAVOP\right) =\mathbb {R}_{+}^{2}.\) </p>
<p>Consequently</p>
<div class="displaymath" id="a0000000094">
  \begin{equation*}  \mathfrak {F}\left( CAVOP\right) =\mathbb {R}_{+}^{2}\supseteq \mathfrak {F}\left( VOP\right) =\{ \left( x_{1},x_{2}\right) :\text{ }x_{1}^{2}-x_{2}\leqq 0,\text{ }x_{2}^{2}-x_{1}\leqq 0\} . \end{equation*}
</div>
<p>Obviously, the point </p>
<div class="displaymath" id="a0000000095">
  \begin{equation*}  x^{1}=\left( 0,7\right) \in \mathfrak {F}\left( CAVOP\right) \text{ }\backslash \text{ }\mathfrak {F}\left( VOP\right) . \end{equation*}
</div>
<p>The point \(x^{0}=\left( 0,0\right) \) is an efficient solution for Problem <span class="rm">(VOP)</span> and \(x^{0}\) is not an efficient solution for Problem <span class="rm">(CAVOP)</span> because</p>
<div class="displaymath" id="a0000000096">
  \begin{equation*}  f\left( x^{0}\right) -f\left( 0,7\right) =\left( 1,0\right) \in C\backslash \{ 0\} =\mathbb {R}_{+}^{2}\backslash \{ 0\} . \end{equation*}
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="T6JDV">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(X\) be a subset of \(\mathbb {R} ^{n},\) \(x^{0}\) be an interior point of \(X,\) \(K\) be a closed convex cone in \(\mathbb {R}^{m},\) \(C\) be a closed convex cone in \(\mathbb {R}^{p},\) \(\eta :X\times X\rightarrow \mathbb {R} ^{n}\) and \(f:X\rightarrow \mathbb {R} \) be two functions and \(g:X\rightarrow \mathbb {R} ^{m}\) be a differentiable function at \(x^{0}.\) </p>
<p>If the function \(g\) is \(K\)-invex at \(x^{0}\) w.r.t. \(\eta \), then every feasible solution for Problem <span class="rm">(VOP)</span>, is a feasible solution for Problem <span class="rm">(CAVOP)</span>, i. e.</p>
<div class="displaymath" id="a0000000097">
  \begin{equation*}  \mathcal{F} (VOP)\subseteq \mathcal{F} (CAVOP). \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000098">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Let \(x^{1}\in \mathcal{F} (VOP),\) i.e. \(g(x^{1})\in -K.\) Since \(g\) is \(K\)-invex at \(x^{0}\) w.r.t. \(\eta \) we have </p>
<div class="displaymath" id="a0000000099">
  \begin{equation*}  g(x^{1})-g(x^{0})-\left[ \bigtriangledown g(x^{0})\right] (\eta (x^{1},x^{0}))\in K \end{equation*}
</div>
<p>From this, it follows </p>
<div class="displaymath" id="a0000000100">
  \begin{equation*}  g(x^{0})+\left[ \bigtriangledown g(x^{0})\right] (\eta (x^{1},x^{0}))\in -K+\{ g\left( x^{1}\right) \} \subseteq -K+\left( -K\right) =-K, \end{equation*}
</div>
<p>hence \(x^{1}\in \mathcal{F} (CAVOP).\) <div class="proof_wrapper" id="a0000000101">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p><div class="example_thmwrapper " id="E6JDV">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">4.4</span>
  </div>
  <div class="example_thmcontent">
  <p>Let’s consider Problem <span class="rm">(VOP)</span> with \(X:=\mathbb {R}^{2},\) \(C:=\mathbb {R}_{+}^{2},\) \(K:=\mathbb {R}_{+}^{2},\) and \(f:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2},\) \(g:\mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) and \(\eta :\) \(\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) the functions defined by</p>
<div class="displaymath" id="a0000000102">
  \begin{equation*}  f\left( x\right) =\left( \sin \tfrac {\left( x_{1}+x_{2}\right) \pi }{4},\text{ }x_{1}^{2}\left( x_{2}-7\right) ^{2}\right) , \end{equation*}
</div>
<div class="displaymath" id="a0000000103">
  \begin{equation*}  g\left( x\right) =\left( x_{1}^{2}-x_{2},\text{ }x_{2}^{2}-x_{1}\right) , \end{equation*}
</div>
<p>for all \(x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2},\) and</p>
<div class="displaymath" id="a0000000104">
  \begin{equation*}  \eta \left( x,u\right) =\left( x_{1}-u_{1},\text{ }x_{2}-u_{2}\right) \end{equation*}
</div>
<p>for all \(\left( x,u\right) =\left( \left( x_{1},x_{2}\right) ,\left( u_{1},u_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}.\) </p>
<p>We have </p>
<div class="displaymath" id="a0000000105">
  \begin{equation*}  \mathfrak {F}\left( VOP\right) =\{ \left( x_{1},x_{2}\right) :\text{ }x_{1}^{2}-x_{2}\leqq 0,\text{ }x_{2}^{2}-x_{1}\leqq 0\} \subseteq \left[ 0,1\right] \times \left[ 0,1\right] ; \end{equation*}
</div>
<p>the point \(x^{0}=\left( 0,0\right) \) is an efficient solution for <span class="rm">(VOP)</span> and the function \(g\) is \(\mathbb {R}_{+}^{2}\)-invex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>Since </p>
<div class="displaymath" id="a0000000106">
  \begin{equation*}  g\left( x^{0}\right) +\left[ \bigtriangledown g(x^{0})\right] (\eta (x,x^{0}))=\left( -x_{2},\text{ }-x_{1}\right) ,\text{ for all }x=\left( x_{1},x_{2}\right) \in \mathbb {R}^{2}, \end{equation*}
</div>
<p>the set of feasible solutions for Problem <span class="rm">(CAVOP)</span> is </p>
<div class="displaymath" id="a0000000107">
  \begin{equation*}  \mathfrak {F}\left( CAVOP\right) =\mathbb {R}_{+}^{2}\supseteq \mathfrak {F}\left( VOP\right) . \end{equation*}
</div>
<p>Easy to remark that \(x^{0}\) is not an efficient solution for Problem <span class="rm">(CAVOP)</span> because</p>
<div class="displaymath" id="a0000000108">
  \begin{equation*}  f\left( x^{0}\right) -f\left( 0,7\right) =\left( 1,0\right) \in C\backslash \{ 0\} =\mathbb {R}_{+}^{2}\backslash \{ 0\} . \end{equation*}
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000109">5 Conclusions</h1>
<p>In this paper one shows how, under some hypotheses, in order to obtain a solution for a vector optimization problem it is sufficient to solve another vector optimization problem. </p>

<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</dd>
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</dd>
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</dd>
</dl>


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