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<h1>Optimization problems and second order approximated optimization problems</h1>
<p class="authors">
<span class="author">Eugenia Duca\(^\ast \) Dorel I. Duca\(^\S \)</span>
</p>
<p class="date">August 11, 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 second order approximated optimization problem associated to an optimization problem is considered. The equivalence between the saddle points of the lagrangian of the second order approximated optimization problem and optimal solutions of the original optimization problem is established. </p>
<p><b class="bf">MSC.</b> 90C26, 90C30, 90C46. </p>
<p><b class="bf">Keywords.</b> Saddle points, invex functions, pseudoinvex functions, \(\eta \)-approximation. </p>
</div>
<h1 id="a0000000002">1 Introduction</h1>
<p>&#8195;&#8195;We consider the optimization problem</p>
<div class="equation" id="a0000000003">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{l} \min \text{ }f\left( x\right) \\ \text{s.t. \  }x\in X \\ \text{ \  \  \  \  \  }g\left( x\right) \leqq 0,\end{array} \tag {$P$} \end{equation}
  </div>
  <span class="equation_label">1.1</span>
</p>
</div>
<p>where \(X\) is a subset of \(\mathbb {R}^{n}\) and \(f:X\rightarrow \mathbb {R}\) and \(g=(g_{1},...,g_{m}):X\rightarrow \mathbb {R}^{m}\) are two functions\(.\) </p>
<p>Let</p>
<div class="displaymath" id="a0000000004">
  \begin{equation*}  \mathfrak {F}\left( P\right) :=\{ x\in X:\text{ }g\left( x\right) \leqq 0\}  \end{equation*}
</div>
<p>denote the set of all feasible solutions of Problem \(\left( P\right) .\) </p>
<p>For solving optimization problem \(\left( P\right) ,\) there are various manners to approach. One of these manners is that for Problem \(\left( P\right) \) one attached to another optimization problem, problem whose solution gives us the \((\)information about\()\) optimal solution of the initial problem \(\left( P\right) \). </p>
<p>Assuming that \(X\) is open\(,\) and that \(f\) and \(g\) are differentiable on \(X,\) Mangasarian <span class="cite">
	[
	<a href="#Mangasarian75" >12</a>
	]
</span> attached to Problem \(\left( P\right) \) and the point \(x^{0}\in X\), the problem</p>
<div class="displaymath" id="a0000000005">
  \begin{equation*} \begin{array}{l} \min \text{ }f\left( x^{0}\right) +\left\langle u,\nabla f\left( x^{0}\right) \right\rangle \\ \text{s.t. \  }u\in \mathbb {R}^{n} \\ \text{ \  \  \  \  \  }g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( u\right) \leqq 0.\end{array}\end{equation*}
</div>
<p>He took the dual of this linear optimization problem, and then considered \(x^{0}\) to be a variable. This last problem is precisely the classical dual of the nonlinear optimization problem, introduced in a different way by Wolfe <span class="cite">
	[
	<a href="#Wolfe" >16</a>
	]
</span> and investigated extensively (see, for example <span class="cite">
	[
	<a href="#Mangasarian69" >11</a>
	]
</span>). Connections between optimal solutions of the dual and the primal are known \((\)see, for example <span class="cite">
	[
	<a href="#Mangasarian69" >11</a>
	]
</span>\()\). </p>
<p>The above process is repeated but taking nonlinear instead of linear approximation of \(f\) and \(g\) around some fixed \(x^{0}\in X\) and taking the dual of the resulting optimization problem. One takes the dual of this optimization problem and then one considers \(x^{0}\) to be a variable in \(X\). One obtains the so called higher-order dual problem of Problem \(\left( \text{P}\right) \). In <span class="cite">
	[
	<a href="#Mangasarian75" >12</a>
	]
</span>, there are given connections between the optimal solutions of higher-order dual and initial problem \(\left( \text{P}\right) .\) D.I. Duca <span class="cite">
	[
	<a href="#Duca85" >7</a>
	]
</span>, <span class="cite">
	[
	<a href="#Duca06" >8</a>
	]
</span> used this idea for optimization problems in complex space. </p>
<p>Another idea came from Antczak <span class="cite">
	[
	<a href="#Antczak04" >4</a>
	]
</span>, <span class="cite">
	[
	<a href="#Antczak05" >3</a>
	]
</span>, <span class="cite">
	[
	<a href="#Antczak07" >2</a>
	]
</span>, who attached to Problem \(\left( \text{P}\right) \) the following problem</p>
<div class="equation" id="a0000000006">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} \min &  f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle \\ \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) \leqq 0,\end{array} \tag {$AP1$} \end{equation}
  </div>
  <span class="equation_label">1.2</span>
</p>
</div>
<p>where \(x^{0}\in X\) is an interior point of \(X,\) \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) is a function, and \(f:X\rightarrow \mathbb {R}\) and \(g:X\rightarrow \mathbb {R}^{m}\) are differentiable at \(x^{0}.\) He studied the connections between the saddle points of Problem \(\left( AP1\right) \) and optimal solutions of Problem \(\left( P\right) .\) </p>
<p>In <span class="cite">
	[
	<a href="#Antczak07a" >1</a>
	]
</span>, <span class="cite">
	[
	<a href="#Mishra-Lai" >14</a>
	]
</span>, <span class="cite">
	[
	<a href="#Mishra" >15</a>
	]
</span>, <span class="cite">
	[
	<a href="#Zhang" >17</a>
	]
</span> the another problems are attached to Problem \(\left( P\right) .\) </p>
<p>In this paper, we attached to Problem \(\left( P\right) ,\) the problem</p>
<div class="displaymath" id="a0000000007">
  \begin{equation*} \begin{array}{ll} \min &  F\left( x\right) \\ \text{s.t.} &  x\in X \\ &  G\left( x\right) \leqq 0,\end{array}\end{equation*}
</div>
<p>where \(F:X\rightarrow \mathbb {R}\) and \(G:X\rightarrow \mathbb {R}^{m}\) are the functions defined by</p>
<div class="displaymath" id="a0000000008">
  \begin{equation*}  F\left( x\right) :=f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle , \end{equation*}
</div>
<div class="displaymath" id="a0000000009">
  \begin{equation*}  G\left( x\right) :=g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) , \end{equation*}
</div>
<p>for all \(x\in X,\) and we study the connections between saddle points of this Problem and the optimal solutions of Problem \(\left( \text{P}\right) .\) </p>
<h1 id="a0000000010">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 in this result. Among these, the concept of an invex function proposed by Hanson (<span class="cite">
	[
	<a href="#Hanson81" >10</a>
	]
</span>) has received more attention. The name of invex \((\)invariant convex\()\) function was given by Craven (<span class="cite">
	[
	<a href="#Craven81" >6</a>
	]
</span>) </p>
<p><div class="definition_thmwrapper " id="invex">
  <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 \(X,\) \(f:X\rightarrow \mathbb {R}\) be a differentiable function at \(x^{0},\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function. We say that the function \(f\) is invex at \(x^{0}\) with respect to \((\)w.r.t.\()\) \(\eta \) if</p>
<div class="equation" id="0a">
<p>
  <div class="equation_content">
    \begin{equation}  f\left( x\right) -f\left( x^{0}\right) \geqq \left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle ,\text{ for all }x\in X. \label{0a} \end{equation}
  </div>
  <span class="equation_label">2.3</span>
</p>
</div>

  </div>
</div> </p>
<p>Hanson defined invex functions which allow the use of the Kuhn-Tucker conditions as sufficient conditions for optimality in constrained optimization problems. Later, Martin (<span class="cite">
	[
	<a href="#Martin85" >13</a>
	]
</span>) proved that invexity hypotheses are not only sufficient but also necessary when using the Kuhn-Tucker optimality conditions for unconstrained optimization problems. </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" >5</a>
	]
</span> defined the so-called pseudoinvex functions, generalizing pseudoconvex functions in the same way that invex functions generalize convex functions. </p>
<p><div class="definition_thmwrapper " id="pinv">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.2</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(X\) be a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0}\) be an interior point of \(X,\) \(f:X\rightarrow \mathbb {R}\) be a differentiable function at \(x^{0},\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function. We say that \(f\) is pseudoinvex at \(x^{0}\) w.r.t. \(\eta \) if, for each \(x\in X\) with the property that</p>
<div class="displaymath" id="a0000000011">
  \begin{equation*}  \left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle \geqq 0, \end{equation*}
</div>
<p>we have</p>
<div class="displaymath" id="a0000000012">
  \begin{equation*}  f\left( x\right) \geqq f\left( x^{0}\right) . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="qinv">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.3</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(X\) be a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0}\) be an interior point of \(X,\) \(f:X\rightarrow \mathbb {R}\) be a differentiable function at \(x^{0},\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function. We say that \(f\) is quasiinvex at \(x^{0}\) w.r.t. \(\eta \) if, for each \(x\in X\) with the property that</p>
<div class="displaymath" id="a0000000013">
  \begin{equation*}  f\left( x\right) \leqq f\left( x^{0}\right) , \end{equation*}
</div>
<p>we have</p>
<div class="displaymath" id="a0000000014">
  \begin{equation*}  \left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle \leqq 0. \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000015">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.4</span>
  </div>
  <div class="remark_thmcontent">
  <p>Note that, in general, there exists no unique function \(\eta =\eta _{x^{0}}\) such that the function \(f\) is invex, respectively pseudoinvex and quasiinvex at the point \(x^{0}\in X.\) </p>
<p>Indeed, the function \(f:\mathbb {R}\rightarrow \mathbb {R}\) defined by</p>
<div class="displaymath" id="a0000000016">
  \begin{equation*}  f\left( x\right) =\exp x,\text{ for all }x\in \mathbb {R}\text{,} \end{equation*}
</div>
<p>is invex at \(x^{0}=0\) w.r.t. the function \(\eta :\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}\) defined by</p>
<div class="displaymath" id="a0000000017">
  \begin{equation*}  \eta \left( x,u\right) =x-u,\text{ for all }\left( x,u\right) \in \mathbb {R}\times \mathbb {R}\text{.} \end{equation*}
</div>
<p>Also, the function \(f\) is invex at \(x^{0}=0\) w.r.t. the function \(\eta :\mathbb {R}^{2}\rightarrow \mathbb {R}\) defined by</p>
<div class="displaymath" id="a0000000018">
  \begin{equation*}  \eta \left( x,u\right) =x+\tfrac {x^{2}}{2}+\tfrac {x^{3}}{6},\text{ for all }\left( x,u\right) \in \mathbb {R}^{2}\text{.} \end{equation*}
</div>
<p>And also, the function \(f\) is invex at \(x^{0}\) w.r.t. the function \(\eta :\mathbb {R}^{2}\rightarrow \mathbb {R}\) defined by</p>
<div class="displaymath" id="a0000000019">
  \begin{equation*}  \eta \left( x,u\right) =x-2\text{, for all }\left( x,u\right) \in \mathbb {R}^{2}\text{.} \end{equation*}
</div>
<p><span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="Second_Invex">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.5</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(X\) be a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0}\) be an interior point of \(X,\) \(f:X\rightarrow \mathbb {R}\) be a twice differentiable function at \(x^{0}\) and \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function. We say that the function \(f\) is second order invex at \(x^{0}\) w.r.t. \(\eta \) if</p>
<div class="displaymath" id="Second">
  \begin{align}  f\left( x\right) -f\left( x^{0}\right) \geqq & \left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle + \label{Second} \\ & +\left\langle \left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( y\right) ,\eta \left( x,x^{0}\right) \right\rangle -\tfrac {1}{2}\left\langle y,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( y\right) \right\rangle , \notag \end{align}
</div>
<p>for all \(x\in X\) and \(y\in \mathbb {R}^{n}.\) </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000020">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.6</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(f\) is a second order invexity at \(x^{0}\) w.r.t. \(\eta ,\) then \(\left( \text{\ref{Second}}\right) \) is also satisfied for \(y=\eta \left( x,x^{0}\right) .\) Then \(\left( \text{\ref{Second}}\right) \) gives</p>
<div class="displaymath" id="a0000000021">
  \begin{equation*}  f\left( x\right) -f\left( x^{0}\right) \geqq \left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) ,\eta \left( x,x^{0}\right) \right\rangle , \end{equation*}
</div>
<p>for all \(x\in X.\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="definition_thmwrapper " id="Second_Pinvex">
  <div class="definition_thmheading">
    <span class="definition_thmcaption">
    Definition
    </span>
    <span class="definition_thmlabel">2.7</span>
  </div>
  <div class="definition_thmcontent">
  <p>Let \(X\) be a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0} \) be an interior point of \(X,\) \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) be a function\(,\) and \(f:X\rightarrow \mathbb {R}\) be a twice differentiable function at \(x^{0}.\) We say that \(f\) is second order pseudoinvex at \(x^{0}\) with respect to \((\)w.r.t.\()\) \(\eta \) if, for each \(x\in X\) with the property that</p>
<div class="displaymath" id="a0000000022">
  \begin{equation*}  \left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) ,\eta \left( x,x^{0}\right) \right\rangle \geqq 0, \end{equation*}
</div>
<p>we have</p>
<div class="displaymath" id="a0000000023">
  \begin{equation*}  f\left( x\right) \geqq f\left( x^{0}\right) . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000024">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">2.8</span>
  </div>
  <div class="remark_thmcontent">
  <p>Obviously, if the function \(f\) is second order invex at \(x^{0}\) w.r.t. \(\eta ,\) then the function \(f\) is second order pseudoinvex at \(x^{0}\) w.r.t. \(\eta .\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>Now let us attach to Problem \(\left( P\right) \) its lagrangian \(L:X\times \mathbb {R}_{+}^{m}\rightarrow \mathbb {R}\) defined by</p>
<div class="displaymath" id="a0000000025">
  \begin{equation*}  L\left( x,v\right) :=f\left( x\right) +\left\langle v,g\left( x\right) \right\rangle ,\text{ for all }\left( x,v\right) \in X\times \mathbb {R}_{+}^{m} \end{equation*}
</div>
<p>Then we have the following theorem (see, for example <span class="cite">
	[
	<a href="#Mangasarian69" >11</a>
	]
</span>): </p>
<p><div class="theorem_thmwrapper " id="Tpctsa">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">2.9</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(\left( x^{0},v^{0}\right) \in X\times \mathbb {R}_{+}^{m}\) is a saddle point of the lagrangian \(L\) of Problem <span class="rm">\(\left( P\right),\)</span> i.e. we have</p>
<div class="displaymath" id="a0000000026">
  \begin{equation*}  L\left( x^{0},v\right) \leqq L\left( x^{0},v^{0}\right) \leqq L\left( x,v^{0}\right) ,\text{ for all }\left( x,v\right) \in X\times \mathbb {R}_{+}^{m}, \end{equation*}
</div>
<p>then \(x^{0}\) is an optimal solution of Problem <span class="rm">\(\left( P\right) .\)</span> </p>

  </div>
</div> </p>
<h1 id="a0000000027">3 \(\protect \eta \)-Approximated Optimization Problem</h1>
<p>In what follows, \(X\) is a nonempty subset of \(\mathbb {R}^{n},\) \(x^{0}\) is an interior point of \(X,\) and \(f:X\rightarrow \mathbb {R}\) and \(g:X\rightarrow \mathbb {R}^{m}\) are two differentiable functions at \(x^{0}.\) </p>
<p>For \(\eta :X\times X\rightarrow \mathbb {R}^{n},\) Antczak (<span class="cite">
	[
	<a href="#Antczak04" >4</a>
	]
</span>) attaches to Problem \(\left( P\right) \) the problem \(\left( P_{\eta }\left( x^{0}\right) \right) ,\) called \(\eta \)-approximated at \(x^{0} \) of Problem \(\left( P\right) .\) </p>
<p>In <span class="cite">
	[
	<a href="#Antczak04" >4</a>
	]
</span> and <span class="cite">
	[
	<a href="#Duca1" >9</a>
	]
</span> one establishes the equivalence between saddle points of \(\eta \)-approximated problem \(\left( P_{\eta }\left( x^{0}\right) \right) \) and of the original problem \(\left( P\right) . \) </p>
<p>If \(x^{0}\) is a feasible solution of Problem \(\left( P\right) ,\) then</p>
<div class="displaymath" id="a0000000028">
  \begin{equation*}  I\left( x^{0}\right) =\{ i\in \{ 1,...,m\} :\text{ }g_{i}\left( x^{0}\right) =0\}  \end{equation*}
</div>
<p> denote the indices of the active restrictions at \(x^{0}.\) </p>
<p>In Ref. <span class="cite">
	[
	<a href="#Duca1" >9</a>
	]
</span>, generalizing a result from <span class="cite">
	[
	<a href="#Antczak07" >2</a>
	]
</span>, one proves the following statement: </p>
<p><div class="theorem_thmwrapper " id="Th2ca">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) such that \(\eta \left( x^{0},x^{0}\right) =0,\) \(f:X\rightarrow \mathbb {R}\) be pseudoinvex at \(x^{0}\) w.r.t. \(\eta \) and \(g=\left( g_{1},...,g_{m}\right) :X\rightarrow \mathbb {R}^{m}\) such that \(g_{i},\) \(i\in I\left( x^{0}\right) \) are quasiinvex at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>If \(\left( x^{0},v^{0}\right) \in X\times \mathbb {R}_{+}^{m}\) is a saddle point of the lagrangian \(L_{\eta }\) of Problem \(\left( P_{\eta }\left( x^{0}\right) \right) ,\) then \(x^{0}\) is an optimal solution of the original problem \(\left( P\right) .\) </p>

  </div>
</div> </p>
<p>Also, in <span class="cite">
	[
	<a href="#Duca1" >9</a>
	]
</span>, generalizing another result from <span class="cite">
	[
	<a href="#Antczak07" >2</a>
	]
</span>, we showed that, if \(x^{0}\) is an optimal solution of the original problem \(\left( P\right) ,\) then under certain conditions, there exists a point \(v^{0}\in \mathbb {R}_{+}^{m}\) such that \(\left( x^{0},v^{0}\right) \) is a saddle point of the \(\eta \)-approximated problem \(\left( \text{P}_{\eta }\left( x^{0}\right) \right) .\) </p>
<p>More exactly, the following statement is true <div class="theorem_thmwrapper " id="Th3ca">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(x^{0}\) be an optimal solution of the original problem \(\left( P\right) \) and assume that a suitable constraint qualification is satisfied at \(x^{0}\) \((\)CQ in <span class="cite">
	[
	<a href="#Mangasarian69" >11</a>
	]
</span>). If </p>
<ul class="itemize">
  <li><p>\(\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle \leqq 0;\) </p>
</li>
  <li><p>\(g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \leqq 0\) \((\)i.e. \(x^{0}\in \mathfrak {F}\left( P_{\eta }\left( x^{0}\right) \right) ),\)</p>
</li>
</ul>
<p> then there exists a point \(v^{0}\in \mathbb {R}_{+}^{m}\) such that \(\left( x^{0},v^{0}\right) \) is a saddle point of the lagrangian \(L_{\eta }\) of the \(\eta \)-approximated problem \(\left( P_{\eta }\left( x^{0}\right) \right) .\) </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000029">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.3</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(\eta \left( x^{0},x^{0}\right) =0,\) then the hypotheses \(\left( i\right) \) and \(\left( ii\right) \) from Theorem <a href="#Th3ca">3.2</a> are satisfied.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000030">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">3.4</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(f\) and \(g\) are invex at \(x^{0}\) w.r.t. \(\eta \), then the hypotheses \(\left( i\right) \) and \(\left( ii\right) \) from Theorem <a href="#Th3ca">3.2</a> are satisfied.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<h1 id="a0000000031">4 \(\left( 2,1\right) \)-\(\protect \eta \)-Approximated Optimization Problem</h1>
<p>In this section, \(X\) is a subset of \(\mathbb {R}^{n},\) \(x^{0}\) is an interior point of \(X,\) \(f:X\rightarrow \mathbb {R}\) is a twice continuously differentiable function at \(x^{0},\) 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 \(\left( P\right) \) the following optimization problem </p>
<div class="equation" id="a0000000032">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} \min &  F\left( x\right) \\ \text{s.t.} &  x\in X \\ &  G\left( x\right) \leqq 0,\end{array} \tag {$AP2$} \end{equation}
  </div>
  <span class="equation_label">4.5</span>
</p>
</div>
<p>where \(F:X\rightarrow \mathbb {R}\) and \(G:X\rightarrow \mathbb {R}^{m}\) are the functions defined by</p>
<div class="displaymath" id="a0000000033">
  \begin{equation*}  F\left( x\right) :=f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle , \end{equation*}
</div>
<div class="displaymath" id="a0000000034">
  \begin{equation*}  G\left( x\right) :=g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) , \end{equation*}
</div>
<p>for all \(x\in X.\) </p>
<p>Problem \(\left( AP2\right) \) is called \(\left( 2,1\right)\)-\(\eta \)-approximated at \(x^{0}\) of Problem \(\left( P\right) .\) </p>
<p><div class="remark_thmwrapper " id="a0000000035">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4.1</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(X=\mathbb {R}^{n}\) and \(\eta \left( x,x^{0}\right) =x-x_{0},\) for all \(x\in X,\) then Problem \(\left( AP2\right) \) is quadratic.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>Let</p>
<div class="displaymath" id="a0000000036">
  \begin{align*}  \mathfrak {F}\left( AP2\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) \leqq 0\}  \\ & =\{ x\in X:G\left( x\right) \leqq 0\} , \end{align*}
</div>
<p>denote the set of all feasible solutions of Problem \(\left( AP2\right) \). </p>
<p><div class="theorem_thmwrapper " id="T1sp">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.2</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Assume that \(g:X\rightarrow \mathbb {R}^{m}\) is invex at \(x^{0}\) w.r.t. \(\eta .\) If \(x\) is a feasible solution of Problem \(\left( P\right) \), then \(x\) is a feasible solution of Problem \(\left( AP2\right) ,\) i.e.</p>
<div class="equation" id="A">
<p>
  <div class="equation_content">
    \begin{equation}  \mathfrak {F}\left( P\right) \subseteq \mathfrak {F}\left( AP2\right) . \label{A} \end{equation}
  </div>
  <span class="equation_label">4.6</span>
</p>
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000037">
  <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 \(x\in X\) be a feasible solution of Problem \(\left( P\right) ,\) i.e. \(g\left( x\right) \leqq 0.\) </p>
<p>Since \(g\) is invex at \(x^{0}\) w.r.t. \(\eta ,\) we have</p>
<div class="displaymath" id="a0000000038">
  \begin{equation*}  \left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \leqq g\left( x\right) -g\left( x^{0}\right) , \end{equation*}
</div>
<p>i.e.</p>
<div class="displaymath" id="a0000000039">
  \begin{equation*}  \left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) +g\left( x^{0}\right) \leqq g\left( x\right) . \end{equation*}
</div>
<p>But \(g\left( x\right) \leqq 0\) and then</p>
<div class="displaymath" id="a0000000040">
  \begin{equation*}  \left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) +g\left( x^{0}\right) \leqq 0, \end{equation*}
</div>
<p>hence \(x\) is a feasible solution of Problem \(\left( AP2\right) .\) <div class="proof_wrapper" id="a0000000041">
  <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="a0000000042">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">4.3</span>
  </div>
  <div class="example_thmcontent">
  <p>For Problem</p>
<div class="equation" id="a0000000043">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} \min &  f\left( x\right) =x^{2}\medskip \\ \text{s.t.} &  x\in X=\mathbb {R} \\ &  g\left( x\right) =x^{2}-x\leqq 0,\end{array} \tag {$\widetilde{P}$} \end{equation}
  </div>
  <span class="equation_label">4.7</span>
</p>
</div>
<p>we have \(\mathfrak {F}\left( \widetilde{P}\right) =\left[ 0,1\right] .\) </p>
<p>The function \(g\) is invex at \(x^{0}=0\) w.r.t. the function \(\eta :\mathbb {R}^{2}\rightarrow \mathbb {R}\) defined by</p>
<div class="displaymath" id="a0000000044">
  \begin{equation*}  \eta \left( x,u\right) =x-u,\text{ for all }\left( x,u\right) \in \mathbb {R}^{2}\text{.} \end{equation*}
</div>
<p>On the other hand, the \(\left( 2,1\right)\)-\(\eta \)-approximated optimization problem \(\left( \widetilde{P}\right) \) is</p>
<div class="equation" id="a0000000045">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} \min &  x^{2} \\ \text{s.t.} &  x\in X=\mathbb {R} \\ &  -x\leqq 0,\end{array} \tag {$A\widetilde{P}2$} \end{equation}
  </div>
  <span class="equation_label">4.8</span>
</p>
</div>
<p>which has \(\mathfrak {F}\left( A\widetilde{P}2\right) =[0,+\infty \lbrack .\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="theorem_thmwrapper " id="T2sp">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Assume that \(f:X\rightarrow \mathbb {R}\) is a second order invex at \(x^{0}\) w.r.t. \(\eta ,\) and \(g:X\rightarrow \mathbb {R}^{m}\) is invex at \(x^{0}\) w.r.t. \(\eta .\) If \(x^{0}\) is an optimal solution of Problem \(\left( P\right) \), then</p>
<div class="displaymath" id="a0000000046">
  \begin{equation*}  \min \left\{  f\left( x\right) :x\in \mathfrak {F}\left( P\right) \right\}  \geqq \inf \left\{  F\left( x\right) :x\in \mathfrak {F}\left( AP2\right) \right\}  . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000047">
  <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 function \(f\) is second order invexity at \(x^{0}\) w.r.t. \(\eta ,\) then</p>
<div class="displaymath" id="a0000000048">
  \begin{equation*}  f\left( x\right) \geqq f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle , \end{equation*}
</div>
<p>for all \(x\in X.\) It follows that</p>
<div class="displaymath" id="a0000000049">
  \begin{align*} & \min \left\{  f\left( x\right) :x\in \mathfrak {F}\left( P\right) \right\}  =f\left( x^{0}\right) \geqq \\ & \geqq f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \eta \left( x^{0},x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle \geqq \\ & \geqq \inf \left\{  f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\right.\\ & \quad +\tfrac {1}{2}\left. \left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle :x\in \mathfrak {F}\left( P\right) \right\}  \geqq \text{ }(\text{from }\left( \ref{A}\right) )\\ & \geqq \inf \left\{  f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\right.\\ & \quad +\tfrac {1}{2}\left. \left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle :x\in \mathfrak {F}\left( AP2\right) \right\}  =\\ & =\inf \left\{  F\left( x\right) :x\in \mathfrak {F}\left( AP2\right) \right\}  . \end{align*}
</div>
<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> </p>
<p><div class="theorem_thmwrapper " id="T3sp">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.5</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(\eta \left( x^{0},x^{0}\right) =0,\) \(g:X\rightarrow \mathbb {R}^{m}\) is invex at \(x^{0}\) w.r.t. \(\eta ,\) and \(x^{0}\) is an optimal solution of Problem \(\left( P\right) \), then</p>
<div class="displaymath" id="a0000000051">
  \begin{equation*}  \min \left\{  f\left( x\right) :x\in \mathfrak {F}\left( P\right) \right\}  \geqq \inf \left\{  F\left( x\right) :x\in \mathfrak {F}\left( AP2\right) \right\}  . \end{equation*}
</div>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000052">
  <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 have</p>
<div class="displaymath" id="a0000000053">
  \begin{align*} & \min \left\{  f\left( x\right) :x\in \mathfrak {F}\left( P\right) \right\}  =f\left( x^{0}\right) =\\ & =f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \eta \left( x^{0},x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle \geqq \\ & \geqq \inf \left\{  f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\right.\\ & \quad +\tfrac {1}{2}\left. \left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle :x\in \mathfrak {F}\left( P\right) \right\}  \geqq \text{ }(\text{from }\left( \ref{A}\right) )\\ & \geqq \inf \left\{  f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\right.\\ & \quad +\tfrac {1}{2}\left. \left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle :x\in \mathfrak {F}\left( AP2\right) \right\}  =\\ & =\inf \left\{  F\left( x\right) :x\in \mathfrak {F}\left( AP2\right) \right\}  . \end{align*}
</div>
<p> <div class="proof_wrapper" id="a0000000054">
  <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>The lagrangian of Problem \(\left( AP2\right) \) will be denoted by \(L_{\eta }^{\left( 2,1\right) },\) i.e. \(L_{\eta }^{\left( 2,1\right) }:X\times \mathbb {R}_{+}^{m}\rightarrow \mathbb {R}\) is defined by</p>
<div class="displaymath" id="a0000000055">
  \begin{align*}  L_{\eta }^{\left( 2,1\right) }\left( x,v\right) &  :=F\left( x\right) +\left\langle v,G\left( x\right) \right\rangle =\medskip \\ &  =f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\medskip \\ & \quad +\tfrac {1}{2}\left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle +\medskip \\ & \quad +\left\langle v,g\left( x^{0}\right) \right\rangle +\left\langle v,\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle , \end{align*}
</div>
<p>for all \(\left( x,v\right) \in X\times \mathbb {R}_{+}^{m}.\) </p>
<p><div class="example_thmwrapper " id="a0000000056">
  <div class="example_thmheading">
    <span class="example_thmcaption">
    Example
    </span>
    <span class="example_thmlabel">4.6</span>
  </div>
  <div class="example_thmcontent">
  <p>Let us consider the optimization problem</p>
<div class="equation" id="a0000000057">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} \min &  f\left( x\right) =\exp x\medskip \\ \text{s.t.} &  x\in X=\mathbb {R} \\ &  g\left( x\right) =x^{2}-x\leqq 0.\end{array} \tag {$\overline{P}$} \end{equation}
  </div>
  <span class="equation_label">4.9</span>
</p>
</div>
<p>We have that \(\mathfrak {F}\left( \overline{P}\right) =\left[ 0,1\right] \) and \(x^{0}=0\) is the unique optimal solution of Problem \(\left( \overline{P}\right) .\) </p>
<p>The functions \(f\) and \(g\) are invex at \(x^{0}=0\) w.r.t. the function \(\eta :\mathbb {R}^{2}\rightarrow \mathbb {R}\) defined by</p>
<div class="displaymath" id="a0000000058">
  \begin{equation*}  \eta \left( x,u\right) =x-u,\text{ for all }\left( x,u\right) \in \mathbb {R}^{2}\text{.} \end{equation*}
</div>
<p>Then the \(\left( 2,1\right)\)-\(\eta \)-approximated optimization problem is</p>
<div class="equation" id="a0000000059">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} \min &  \left( 1+x+\tfrac {1}{2}x^{2}\right) \\ \text{s.t.} &  x\in X=\mathbb {R} \\ &  -x\leqq 0,\end{array} \tag {$A\overline{P}2$} \end{equation}
  </div>
  <span class="equation_label">4.10</span>
</p>
</div>
<p>which has the optimal solution \(x^{0}=0.\) </p>
<p>On the other hand, the lagrangian \(\overline{L}_{\eta }^{\left( 2,1\right) }\) of Problem \(\left( A\overline{P}2\right) \) is defined by</p>
<div class="displaymath" id="a0000000060">
  \begin{equation*}  \overline{L}_{\eta }^{\left( 2,1\right) }\left( x,v\right) =1+x+\tfrac {1}{2}x^{2}-vx,\text{ for all }\left( x,v\right) \in \mathbb {R}\times \mathbb {R}_{+}. \end{equation*}
</div>
<p>Obviously, \(\left( x^{0},v^{0}\right) =\left( 0,1\right) \) is a saddle point of the lagrangian \(\overline{L}_{\eta }^{\left( 2,1\right) }\) of Problem \(\left( A\overline{P}2\right) .\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>In this section we show the equivalence between saddle points of the lagrangian \(L_{\eta }^{\left( 2,1\right) },\) of Problem \(\left( AP2\right) ,\) and optimal solutions of Problem <br />\(\left( P_{\eta }^{\left( 2,1\right) }\left( x^{0}\right) \right) .\) </p>
<p>By Theorem <a href="#Tpctsa">2.9</a>, the following saddle point theorem follows: </p>
<p><div class="theorem_thmwrapper " id="Th1App">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.7</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(\left( x^{0},v^{0}\right) \in X\times \mathbb {R}_{+}^{m}\) is a saddle point of the lagrangian \(L_{\eta }^{\left( 2,1\right) }\) of Problem \(\left( AP2\right) ,\) then \(x^{0}\) is an optimal solution of Problem \(\left( AP2\right) .\) </p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000061">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">4.8</span>
  </div>
  <div class="remark_thmcontent">
  <p>We established Theorem <a href="#Th1App">4.7</a>, without any assumption about the function involved in Problem \(\left( AP2\right) .\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>Now, we can state the converse theorem of Theorem <a href="#Th1App">4.7</a>. </p>
<p><div class="theorem_thmwrapper " id="Th2App">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">4.9</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(x^{0}\in X\) be an optimal solution of Problem \(\left( AP2\right) ,\) \(\mu :X\times X\rightarrow \mathbb {R}^{n}\) be a function. Assume that \(\eta \left( \cdot ,x^{0}\right) :X\rightarrow \mathbb {R}^{n}\) is differentiable at \(x^{0},\) the functions \(F,\) \(G=\left( G_{1},...,G_{m}\right) :X\rightarrow \mathbb {R}^{m}\) are invex at \(x^{0}\) w.r.t. \(\mu \) and a suitable constraint qualification \((\)CQ\(,\) <span class="cite">
	[
	<a href="#Mangasarian69" >11</a>
	]
</span>) is satisfied at \(x^{0}.\) Then there exists a point \(v^{0}\in \mathbb {R}_{+}^{m}\) such that \(\left( x^{0},v^{0}\right) \) is a saddle point of Problem \(\left( AP2\right) .\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000062">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> In view of Karush-Kuhn-Tucker theorem, there exists a point \(v^{0}\in \mathbb {R}_{+}^{m}\) such that</p>
<div class="equation" id="20a">
<p>
  <div class="equation_content">
    \begin{equation}  \nabla F\left( x^{0}\right) +\left[ \nabla G\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) =0, \label{20a} \end{equation}
  </div>
  <span class="equation_label">4.11</span>
</p>
</div>
<div class="equation" id="21a">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle v^{0},G\left( x^{0}\right) \right\rangle =0. \label{21a} \end{equation}
  </div>
  <span class="equation_label">4.12</span>
</p>
</div>
<p>The functions \(F\) and \(G\) are invex at \(x^{0}\) w.r.t. \(\mu ,\) then, for each \(x\in X,\) we have</p>
<div class="equation" id="22a">
<p>
  <div class="equation_content">
    \begin{equation}  F\left( x\right) -F\left( x^{0}\right) \geqq \left\langle \nabla F\left( x^{0}\right) ,\mu \left( x,x^{0}\right) \right\rangle , \label{22a} \end{equation}
  </div>
  <span class="equation_label">4.13</span>
</p>
</div>
<div class="equation" id="23a">
<p>
  <div class="equation_content">
    \begin{equation}  G\left( x\right) -G\left( x^{0}\right) \geqq \left[ \nabla G\left( x^{0}\right) \right] \left( \mu \left( x,x^{0}\right) \right) . \label{23a} \end{equation}
  </div>
  <span class="equation_label">4.14</span>
</p>
</div>
<p>Since \(v^{0}\in \mathbb {R}_{+}^{m},\) by \(\left( \ref{23a}\right) ,\) we obtain</p>
<div class="equation" id="24a">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle v^{0},G\left( x\right) -G\left( x^{0}\right) \right\rangle \geqq \left\langle v^{0},\left[ \nabla G\left( x^{0}\right) \right] \left( \mu \left( x,x^{0}\right) \right) \right\rangle = \label{24a} \end{equation}
  </div>
  <span class="equation_label">4.15</span>
</p>
</div>
<div class="displaymath" id="a0000000063">
  \begin{equation*}  =\left\langle \left[ \nabla G\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) ,\mu \left( x,x^{0}\right) \right\rangle ,\text{ for all }x\in X. \end{equation*}
</div>
<p>Then, for each \(x\in X\)</p>
<div class="displaymath" id="a0000000064">
  \begin{align*} & L_{\eta }^{\left( 2,1\right) }\left( x,v^{0}\right) -L_{\eta }^{\left( 2,1\right) }\left( x^{0},v^{0}\right) =\\ & =F\left( x\right) -F\left( x^{0}\right) +\left\langle v^{0},G\left( x\right) -G\left( x^{0}\right) \right\rangle \geqq \text{ }(\text{by }\left( \text{\ref{22a}}\right) ,\text{ and }\left( \text{\ref{24a}}\right) )\\ & \geqq \left\langle \nabla F\left( x^{0}\right) +\left[ \nabla G\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) ,\mu \left( x,x^{0}\right) \right\rangle =\text{ }(\text{by }\left( \text{\ref{20a}}\right) )=0. \end{align*}
</div>
<p>Consequently, the second inequality of the definition of saddle point is satisfied. </p>
<p>In order to prove the first inequality of the definition of saddle point, let \(v\in \mathbb {R}_{+}^{m}.\) Then</p>
<div class="displaymath" id="a0000000065">
  \begin{align*} & L_{\eta }^{\left( 2,1\right) }\left( x^{0},v^{0}\right) -L_{\eta }^{\left( 2,1\right) }\left( x^{0},v\right) =\\ & =\left\langle v^{0},G\left( x^{0}\right) \right\rangle -\left\langle v,G\left( x^{0}\right) \right\rangle =\text{ }(\text{by }\left( \text{\ref{21a}}\right) )\\ & =-\left\langle v,G\left( x^{0}\right) \right\rangle \geqq 0, \end{align*}
</div>
<p>because \(G\left( x^{0}\right) \leqq 0\) and \(v\in \mathbb {R}_{+}^{m}.\) <div class="proof_wrapper" id="a0000000066">
  <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="a0000000067">5 Equivalence between saddle points of \(\left( 2,1\right)\)-\(\protect \eta \)-approximated problem and of the original problem</h1>
<p>In this section, \(X\) is a subset of \(\mathbb {R}^{n},\) \(x^{0}\) is an interior point of \(X,\) \(f:X\rightarrow \mathbb {R}\) is a twice continuously differentiable function at \(x^{0},\) and \(g:X\rightarrow \mathbb {R}^{m}\) is a differentiable function at \(x^{0}.\) </p>
<p>We will prove the equivalence between the original optimization problem \(\left( P\right) \) and its associated \(\left( 2,1\right)\)-\(\eta \)-approximated optimization problem \(\left( AP2\right) .\) We establish the results where one assumes that the function \(\eta \) satisfies only the condition \(\eta \left( x^{0},x^{0}\right) =0.\) </p>
<p>The following statement is true </p>
<p><div class="theorem_thmwrapper " id="Th2c">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(\eta :X\times X\rightarrow \mathbb {R}^{n}\) such that \(\eta \left( x^{0},x^{0}\right) =0,\) \(f:X\rightarrow \mathbb {R}\) be second order pseudoinvex function at \(x^{0}\) w.r.t. \(\eta \) and \(g=\left( g_{1},...,g_{m}\right) :X\rightarrow \mathbb {R}^{m}\) such that \(g_{i},\) \(i\in I\left( x^{0}\right) \) are quasiinvex functions at \(x^{0}\) w.r.t. \(\eta .\) </p>
<p>If \(\left( x^{0},v^{0}\right) \in X\times \mathbb {R}_{+}^{m}\) is a saddle point of the lagrangian \(L_{\eta }^{\left( 2,1\right) }\) of Problem \(\left( AP2\right) ,\) then \(x^{0}\) is an optimal solution of the original problem \(\left( P\right) .\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000068">
  <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 point \(\left( x^{0},v^{0}\right) \in X\times \mathbb {R}_{+}^{m}\) is a saddle point of the lagrangian \(L_{\eta }^{\left( 2,1\right) }\) of Problem \(\left( AP2\right) ;\) then</p>
<div class="displaymath" id="a0000000069">
  \begin{equation*}  L_{\eta }^{\left( 2,1\right) }\left( x^{0},v\right) \leqq L_{\eta }^{\left( 2,1\right) }\left( x^{0},v^{0}\right) ,\text{ for all }v\in \mathbb {R}_{+}^{m}, \end{equation*}
</div>
<p>i.e.</p>
<div class="equation" id="1c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle v-v^{0},g\left( x^{0}\right) \right\rangle \leqq 0,\text{ for all }v\in \mathbb {R}_{+}^{m}, \label{1c} \end{equation}
  </div>
  <span class="equation_label">5.16</span>
</p>
</div>
<p>because \(\eta \left( x^{0},x^{0}\right) =0.\) </p>
<p>Let \(i\in \{ 1,...,m\} ,\) and \(e^{i}=\left( 0,...,1,...,0\right) \in \mathbb {R}^{m}\) be the \(i\)-th unit point of \(\mathbb {R}^{m}.\) Then, for \(v=e^{i}+v^{0}\in \mathbb {R}_{+}^{m},\) relation \(\left( \text{\ref{1c}}\right) \) becomes \(g_{i}\left( x^{0}\right) \leqq 0.\) Hence </p>
<div class="displaymath" id="a0000000070">
  \begin{equation*}  g_{i}\left( x^{0}\right) \leqq 0,\text{ for all }i\in \{ 1,...,m\} . \end{equation*}
</div>
<p>Consequently, </p>
<div class="displaymath" id="a0000000071">
  \begin{equation*}  x^{0}\in \mathfrak {F}\left( P\right) . \end{equation*}
</div>
<p>If follows that</p>
<div class="equation" id="2c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle v^{0},g\left( x^{0}\right) \right\rangle \leqq 0, \label{2c} \end{equation}
  </div>
  <span class="equation_label">5.17</span>
</p>
</div>
<p> because \(v^{0}\in \mathbb {R}_{+}^{m}.\) But, from \(\left( \text{\ref{1c}}\right) \) we deduce</p>
<div class="equation" id="3c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle v^{0},g\left( x^{0}\right) \right\rangle \geqq 0, \label{3c} \end{equation}
  </div>
  <span class="equation_label">5.18</span>
</p>
</div>
<p> because \(v=0\in \mathbb {R}_{+}^{m}.\) </p>
<p>Thus, by \(\left( \text{\ref{2c}}\right) \) and \(\left( \text{\ref{3c}}\right) \)</p>
<div class="equation" id="4c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle v^{0},g\left( x^{0}\right) \right\rangle =0. \label{4c} \end{equation}
  </div>
  <span class="equation_label">5.19</span>
</p>
</div>
<p>From \(\left( \text{\ref{4c}}\right) \) it follows that</p>
<div class="equation" id="5c">
<p>
  <div class="equation_content">
    \begin{equation}  v_{i}^{0}=0,\text{ for all }i\in \{ 1,...,m\} \backslash I\left( x^{0}\right) . \label{5c} \end{equation}
  </div>
  <span class="equation_label">5.20</span>
</p>
</div>
<p>On the other hand, from</p>
<div class="displaymath" id="a0000000072">
  \begin{equation*}  L_{\eta }^{\left( 2,1\right) }\left( x^{0},v^{0}\right) \leqq L_{\eta }^{\left( 2,1\right) }\left( x,v^{0}\right) ,\text{ for all }x\in X, \end{equation*}
</div>
<p>we deduce that</p>
<div class="equation" id="6c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle \nabla f\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] ^{T}\left( v^{0}\right) +\tfrac {1}{2}\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) ,\eta \left( x,x^{0}\right) \right\rangle \geqq 0, \label{6c} \end{equation}
  </div>
  <span class="equation_label">5.21</span>
</p>
</div>
<p>for all \(x\in X.\) </p>
<p>In order to prove that \(x^{0}\) is an optimal solution of Problem \(\left( P\right) ,\) let \(x\in \mathfrak {F}\left( P\right) .\) Then</p>
<div class="displaymath" id="a0000000073">
  \begin{equation*}  g_{i}\left( x\right) \leqq 0,\text{ for all }i\in \{ 1,...,m\} . \end{equation*}
</div>
<p>Let \(i\in I\left( x^{0}\right) .\) Since</p>
<div class="displaymath" id="a0000000074">
  \begin{equation*}  g_{i}\left( x\right) -g_{i}\left( x^{0}\right) =g_{i}\left( x\right) \leqq 0, \end{equation*}
</div>
<p>and \(g_{i}\) is quasiinvex at \(x^{0}\) w.r.t. \(\eta ,\) we have</p>
<div class="displaymath" id="a0000000075">
  \begin{equation*}  \left\langle \nabla g_{i}\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle \leqq 0, \end{equation*}
</div>
<p>hence</p>
<div class="displaymath" id="a0000000076">
  \begin{equation*}  \left\langle v_{i}^{0}\nabla g_{i}\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle \leqq 0, \end{equation*}
</div>
<p>because \(v_{i}^{0}\geqq 0.\) Then</p>
<div class="equation" id="7c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle \left[ \nabla g\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle \leqq 0, \label{7c} \end{equation}
  </div>
  <span class="equation_label">5.22</span>
</p>
</div>
<p>because \(v_{i}^{0}=0,\) for all \(i\in \{ 1,...,m\} \backslash I\left( x^{0}\right) .\) </p>
<p>From \(\left( \text{\ref{6c}}\right) \) and \(\left( \text{\ref{7c}}\right) \) it follows that</p>
<div class="equation" id="8c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle \nabla f\left( x^{0}\right) +\tfrac {1}{2}\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) ,\eta \left( x,x^{0}\right) \right\rangle \geqq 0. \label{8c} \end{equation}
  </div>
  <span class="equation_label">5.23</span>
</p>
</div>
<p>But, the function \(f\) is second order pseudoinvex at \(x^{0}\) w.r.t. \(\eta ,\) and then, by \(\left( \text{\ref{8c}}\right) ,\) we deduce that</p>
<div class="displaymath" id="a0000000077">
  \begin{equation*}  f\left( x\right) \geqq f\left( x^{0}\right) . \end{equation*}
</div>
<p>Consequently, \(x^{0}\) is an optimal solution of the original problem \(\left( P\right) .\) The theorem is proved. <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> </p>
<p><div class="remark_thmwrapper " id="a0000000079">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5.2</span>
  </div>
  <div class="remark_thmcontent">
  <p>If the function \(f\) is second order invex at \(x^{0}\) w.r.t. \(\eta ,\) and \(g_{1},...,g_{m}\) are invex at \(x^{0}\) with respect to \(\eta ,\) then the hypotheses that \(f\) is second order pseudoinvex at \(x^{0}\) w.r.t. \(\eta \) and \(g_{i},\) \(i\in I\left( x^{0}\right) \) are quasiinvex at \(x^{0}\) w.r.t. \(\eta \) are satisfied.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000080">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5.3</span>
  </div>
  <div class="remark_thmcontent">
  <p>The assumption that the function \(\eta \) satisfies the condition \(\eta \left( x^{0},x^{0}\right) =0\) is essential in order to have the equivalence between the saddle points of the lagrangian \(L_{\eta }^{\left( 2,1\right) }\) of Problem \(\left( AP2\right) ,\) and the optimal solutions of the original problem \(\left( P\right) .\) \((\)see Example \(3.4\) from <span class="cite">
	[
	<a href="#Antczak07" >2</a>
	]
</span>)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>Now, we show that, if \(x^{0}\) is an optimal solution of the original problem \(\left( P\right) ,\) then under certain conditions, there exists a point \(v^{0}\in \mathbb {R}_{+}^{m}\) such that \(\left( x^{0},v^{0}\right) \) is a saddle point of the \(\eta \)-approximated problem \(\left( AP2\right) .\) </p>
<p>More exactly, the following statement is true: </p>
<p><div class="theorem_thmwrapper " id="Th3c">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">5.4</span>
  </div>
  <div class="theorem_thmcontent">
  <p>Let \(x^{0}\in X\) be an optimal solution of the original problem \(\left( P\right) \) and assume that a suitable constraint qualification is satisfied at \(x^{0}\) \((\)CQ in Ref. <span class="cite">
	[
	<a href="#Mangasarian69" >11</a>
	]
</span>). If the function \(\eta :X\times X\rightarrow \mathbb {R}^{m}\) satisfies </p>
<ul class="itemize">
  <li><p>\(\  \left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle \leqq 0;\)</p>
</li>
  <li><p>\(\  g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \leqq 0\) \(\left( \text{i.e. }x^{0}\in \mathfrak {F}\left( AP2\right) \right) ,\)</p>
</li>
  <li><p>\(x^{0}\) is an optimal solution of the problem</p>
<div class="displaymath" id="a0000000081">
  \begin{equation*} \begin{array}{l} \min \text{ }\left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle \\ \text{s.t. \  }x\in X,\end{array}\end{equation*}
</div>
</li>
</ul>
<p> then there exists a point \(v^{0}\in \mathbb {R}_{+}^{m}\) such that \(\left( x^{0},v^{0}\right) \) is a saddle point of the lagrangian \(L_{\eta }^{\left( 2,1\right) }\) of the \(\left( 2,1\right)\)-\(\eta \)-approximated problem \(\left( AP2\right) .\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000082">
  <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 \(x^{0}\) is an optimal solution of Problem \(\left( P\right) ,\) and a suitable constraint qualification at \(x^{0}\) is satisfied\(,\) by Karush-Kuhn-Tucker’s Theorem, there exists a point \(v^{0}\in \mathbb {R}_{+}^{m}\) such that</p>
<div class="equation" id="10c">
<p>
  <div class="equation_content">
    \begin{equation}  \nabla f\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) =0, \label{10c} \end{equation}
  </div>
  <span class="equation_label">5.24</span>
</p>
</div>
<div class="equation" id="11c">
<p>
  <div class="equation_content">
    \begin{equation}  \left\langle v^{0},g\left( x^{0}\right) \right\rangle =0. \label{11c} \end{equation}
  </div>
  <span class="equation_label">5.25</span>
</p>
</div>
<p>Let \(x\in X.\) Then, from \(\left( \ref{10c}\right) \) and hypothesis \(\left( iii\right) ,\) we have</p>
<div class="displaymath" id="a0000000083">
  \begin{align*} & L_{\eta }^{\left( 2,1\right) }\left( x,v^{0}\right) -L_{\eta }^{\left( 2,1\right) }\left( x^{0},v^{0}\right) =\\ & =f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle +\\ & \quad +\left\langle v^{0},g\left( x^{0}\right) \right\rangle +\left\langle v^{0}, \left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle -\\ & \quad -f\left( x^{0}\right) -\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle -\tfrac {1}{2}\left\langle \eta \left( x^{0},x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle -\\ & \quad -\left\langle v^{0},g\left( x^{0}\right) \right\rangle -\left\langle v^{0}, \left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle = \end{align*}
</div>
<div class="displaymath" id="a0000000084">
  \begin{align*} & =\left\langle \nabla f\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) ,\eta \left( x,x^{0}\right) \right\rangle -\\ & \quad -\left\langle \nabla f\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle +\\ & \quad +\tfrac {1}{2}\left[ \left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle - \right.\\ &  \quad \left. -\left\langle \eta \left( x^{0},x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle \right] =\\ & =\tfrac {1}{2}\left[ \left\langle \eta \left( x,x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x,x^{0}\right) \right) \right\rangle - \right.\\ & \quad \left. -\left\langle \eta \left( x^{0},x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle \right] \geqq 0. \end{align*}
</div>
<p>Consequently, the second inequality from the saddle point definition is true. </p>
<p>In order to prove the first inequality from the saddle point definition, let \(v\in \mathbb {R}_{+}^{m}.\) Then</p>
<div class="displaymath" id="a0000000085">
  \begin{align*} & L_{\eta }^{\left( 2,1\right) }\left( x^{0},v^{0}\right) -L_{\eta }^{\left( 2,1\right) }\left( x^{0},v\right) =\\ & =f\left( x^{0}\right) +\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle +\tfrac {1}{2}\left\langle \eta \left( x^{0},x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle +\\ & \quad +\left\langle v^{0},g\left( x^{0}\right) \right\rangle +\left\langle v^{0}, \left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle -\\ & \quad -f\left( x^{0}\right) -\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle -\tfrac {1}{2}\left\langle \eta \left( x^{0},x^{0}\right) ,\left[ \nabla ^{2}f\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle -\\ & \quad -\left\langle v,g\left( x^{0}\right) \right\rangle -\left\langle v,\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) \right\rangle =\\ & =\left\langle \nabla f\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] ^{\text{T}}\left( v^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle -\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle -\\ & \quad -\left\langle g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] ^{\text{T}}\left( \eta \left( x^{0},x^{0}\right) \right) ,v\right\rangle =\text{ }(\text{by }\left( \text{\ref{10c}}\right) \text{ and }\left( \text{\ref{11c}}\right) )\\ & =-\left\langle \nabla f\left( x^{0}\right) ,\eta \left( x^{0},x^{0}\right) \right\rangle -\left\langle g\left( x^{0}\right) +\left[ \nabla g\left( x^{0}\right) \right] \left( \eta \left( x^{0},x^{0}\right) \right) ,v\right\rangle \geqq \\ &  \geqq \text{ }(\text{by }\left( i\right) \text{ and }\left( ii\right) ) \geqq 0. \end{align*}
</div>
<p>Consequently, \(\left( x^{0},v^{0}\right) \) is a saddle point of the lagrangian of Problem \(\left( AP2\right) .\) <div class="proof_wrapper" id="a0000000086">
  <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="a0000000087">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5.5</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(\eta \left( x^{0},x^{0}\right) =0,\) then the hypotheses \(\left( i\right) \) and \(\left( ii\right) \) from Theorem <a href="#Th3c">5.4</a> are satisfied.<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000088">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5.6</span>
  </div>
  <div class="remark_thmcontent">
  <p>If \(f\) and \(g=\left( g_{1},...,g_{m}\right) \) are invex at \(x^{0}\) w.r.t. \(\eta \), then the hypotheses \(\left( i\right) \) and \(\left( ii\right) \) from Theorem <a href="#Th3c">5.4</a> are satisfied. <span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p><div class="remark_thmwrapper " id="a0000000089">
  <div class="remark_thmheading">
    <span class="remark_thmcaption">
    Remark
    </span>
    <span class="remark_thmlabel">5.7</span>
  </div>
  <div class="remark_thmcontent">
  <p>The hypothesis that the original problem \(\left( P\right) \) satisfies a suitable constraint qualification at \(x^{0}\) is essential. Indeed, for the problem</p>
<div class="equation" id="a0000000090">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{l} \min \text{ }f\left( x\right) =x_{2} \\ \text{s.t. \  }x\in X=\mathbb {R}^{2} \\ \begin{array}{l} \text{ \  \  \  \  \  }g_{1}\left( x\right) =x_{1}+x_{2}^{2}\leqq 0, \\ \text{ \  \  \  \  \  }g_{2}\left( x\right) =-x_{1}+x_{2}^{2}\leqq 0,\end{array}\end{array} \tag {$\widehat{P}$} \end{equation}
  </div>
  <span class="equation_label">5.26</span>
</p>
</div>
<p>we have the set of all feasible solutions \(\mathfrak {F}\big( \widehat{P}\big) =\{ \left( 0,0\right) \} ,\) and hence \(x^{0}=\left( 0,0\right) \) is the unique optimal solution. Let us remark that Problem \(\big( \widehat{P}\big) \) is convex, and then the functions \(f,g_{1},g_{2}\) are invex w.r.t. \(\eta :\mathbb {R}^{2}\times \mathbb {R}^{2}\rightarrow \mathbb {R}^{2}\) defined by</p>
<div class="displaymath" id="a0000000091">
  \begin{equation*}  \eta \left( x,u\right) =x-u,\text{ for all }\left( x,u\right) \in \mathbb {R}^{2}\times \mathbb {R}^{2}. \end{equation*}
</div>
<p>In this case, the \(\left( 2,1\right)\)-\(\eta \)-approximated optimization problem is</p>
<div class="equation" id="a0000000092">
<p>
  <div class="equation_content">
    \begin{equation}  \begin{array}{ll} \min &  x_{2} \\ \text{s.t.} &  \left( x_{1},x_{2}\right) \in \mathbb {R}^{2} \\ &  -x_{1}\leqq 0 \\ &  x_{1}\leqq 0.\end{array} \tag {$A\widehat{P}2$} \end{equation}
  </div>
  <span class="equation_label">5.27</span>
</p>
</div>
<p>Thus, \(\widehat{L}_{\eta }^{\left( 2,1\right) }:\mathbb {R}^{2}\times \mathbb {R}_{+}^{2}\rightarrow \mathbb {R}\) is defined by</p>
<div class="displaymath" id="a0000000093">
  \begin{equation*}  \widehat{L}_{\eta }^{\left( 2,1\right) }\left( x,v\right) =x_{2}-v_{1}x_{1}+v_{2}x_{1},\text{ } \end{equation*}
</div>
<p>for all \(\left( x,v\right) =\left( \left( x_{1},x_{2}\right) ,\left( v_{1},v_{2}\right) \right) \in \mathbb {R}^{2}\times \mathbb {R}_{+}^{2}.\)<span class="qed">â–¡</span></p>

  </div>
</div> </p>
<p>Easy to show that, for each \(v^{0}=\left( v_{1}^{0},v_{2}^{0}\right) \in \mathbb {R}_{+}^{2}\), the point \(\left( x^{0},v^{0}\right) \) is not a saddle point of the lagrangian of Problem \(\big( A\widehat{P}2\big) .\) </p>
<h1 id="a0000000094">6 Conclusions</h1>
<p>This paper shows how, under some hypotheses, to solve an optimization problem is equivalent with finding the saddle points \(\left( x^{0},v^{0}\right) \) of the so called \(\left( 2,1\right)\)-\(\eta \)- approximated problem at \(x^{0}\) of the original problem. </p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
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</dd>
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