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'''Part 5.''' (20 pts)
 
'''Part 5.''' (20 pts)
  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ Consider the following optimization problem, }</math></span></font>
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<font color="#ff0000">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</font><math>\text{optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2}</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>\text{subject to  }  x_{2}- x_{1}^{2}\geq0</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>2-x_{1}-x_{2}\geq0</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>x_{1}\geq0.</math>
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<math>\color{blue} \text{The point }  x^{*}=\begin{bmatrix}
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0 & 0
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\end{bmatrix}^{T} \text{ satisfies the KKT conditions.}</math>
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<math>\color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?}</math>
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<math>\color{blue}\left( \text{ii} \right) \text{Does } x^{*} \text{ satisfy SOSC? Carefully justify your answer.}</math><br>
  
 
:'''Click [[ECE-QE_AC3-2011_solusion-5|here]] to view student [[ECE-QE_AC3-2011_solusion-5|answers and discussions]]'''
 
:'''Click [[ECE-QE_AC3-2011_solusion-5|here]] to view student [[ECE-QE_AC3-2011_solusion-5|answers and discussions]]'''
 
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Revision as of 05:34, 28 June 2012


ECE Ph.D. Qualifying Exam in Automatic Control (AC), Question 3, August 2011


Question

Part 1. 20 pts


 $ \color{blue} \text{ Consider the optimization problem, } $

               $ \text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2} $

               $ \text{subject to } x_{1}\geq0, x_{2}\geq0 $

$ \color{blue}\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right] $

$ \color{blue}\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} \text{. Does the point } x^{*} \text{ satisfy this condition?} $

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Part 2.


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Part 3.


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Part 4.


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Part 5. (20 pts)

 $ \color{blue} \text{ Consider the following optimization problem, } $

                            $ \text{optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2} $

                        $ \text{subject to } x_{2}- x_{1}^{2}\geq0 $

                                                 $ 2-x_{1}-x_{2}\geq0 $

                                                 $ x_{1}\geq0. $

$ \color{blue} \text{The point } x^{*}=\begin{bmatrix} 0 & 0 \end{bmatrix}^{T} \text{ satisfies the KKT conditions.} $

$ \color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?} $

$ \color{blue}\left( \text{ii} \right) \text{Does } x^{*} \text{ satisfy SOSC? Carefully justify your answer.} $

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