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== '''AC - 3 August 2012 QE'''  ==
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[[Category:ECE]]
 +
[[Category:QE]]
 +
[[Category:CNSIP]]
 +
[[Category:problem solving]]
 +
[[Category:automatic control]]
 +
[[Category:optimization]]
  
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'''Problem 1. (20 pts)'''
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<center>
 +
<font size= 4>
 +
[[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]]
 +
</font size>
  
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(i) (10 pts) Find the factor by which the uncertainty range is reduced when using the Fibonacci method. Assume that the last step has the form
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<font size= 4>
 +
Automatic Control (AC)
  
 +
Question 3: Optimization
 +
</font size>
 +
 +
August 2012
 +
</center>
 +
----
 +
----
 +
:Student answers and discussions for [[QE2012_AC-3_ECE580-1|Part 1]],[[QE2012_AC-3_ECE580-2|2]],[[QE2012_AC-3_ECE580-2|3]],[[QE2012_AC-3_ECE580-4|4]],[[QE2012_AC-3_ECE580-5|5]]
 +
----
 +
'''1.(20 pts)'''
 +
<br>
 +
'''(i)(10 pts) Find the factor by which the uncertainty range is reduced when using the Fibonacci method. Assume that the last step has the form'''
 
<math>  
 
<math>  
 
\begin{align}
 
\begin{align}
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</math>  
 
</math>  
  
−
where <span class="texhtml">''N'' − 1</span> is the number of steps performed in the uncertainty range reduction process.  
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'''where <span class="texhtml">''N'' − 1</span> is the number of steps performed in the uncertainty range reduction process. '''
  
 
<br>  
 
<br>  
  
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<br> Solution:
 
−
 
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    The reduction factor is <span class="texhtml">(1 − ρ<sub>1</sub>)(1 − ρ<sub>2</sub>)(1 − ρ<sub>3</sub>)...(1 − ρ<sub>''N'' − 1</sub>)</span>
 
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Since
 
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<math> 1- \rho_{N-1} = \frac{F_{2}}{F_{3}} = \frac{2}{3}, </math>
 
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we have
 
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<math> 1- \rho_{N-2} = \frac{F_{3}}{F_{4}}</math>    and so on.
 
−
 
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    Then, we have
 
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<math> (1- \rho_{1})(1- \rho_{2})(1- \rho_{3})...(1- \rho_{N-1}) = \frac{F_{N}}{F_{N+1}}  \frac{F_{N-1}}{F_{N}}  ...  \frac{F_{2}}{F_{3}} = \frac{F_{2}}{F_{N+1}}</math>
 
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Therefore, the reduction factor is
 
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<math>\frac{2}{F_{N+1}}</math>
 
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<br>
 
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Solution 2:
 
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The uncertainty interval is reduced by <math> (1- \rho_{1})(1- \rho_{2})(1- \rho_{3})...(1- \rho_{N-1}) = \frac{F_{N}}{F_{N+1}}  \frac{F_{N-1}}{F_{N}}  ...  \frac{F_{2}}{F_{3}} = \frac{F_{2}}{F_{N+1}}</math>
 
−
 
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<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}
 
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\text{In this specific case, we use only N-1 iterations. }
 
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</math></span></font>
 
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<br>
 
  
 
<br> '''(ii)(10 pts)''' It is known that the minimizer of a certain function f(x) is located in the interval[-5, 15]. What is the minimal number of iterations of the Fibonacci method required to box in the minimizer within the range 1.0? Assume that the last useful value of the factor reducing the uncertainty range is 2/3, that is  
 
<br> '''(ii)(10 pts)''' It is known that the minimizer of a certain function f(x) is located in the interval[-5, 15]. What is the minimal number of iterations of the Fibonacci method required to box in the minimizer within the range 1.0? Assume that the last useful value of the factor reducing the uncertainty range is 2/3, that is  
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<math> 1- \rho_{N-1} = \frac{F_{2}}{F_{3}} = \frac{2}{3}, </math>  
 
<math> 1- \rho_{N-1} = \frac{F_{2}}{F_{3}} = \frac{2}{3}, </math>  
  
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<br> Solution:  
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:'''Click [[QE2012_AC-3_ECE580-1|here]] to view [[QE2012_AC-3_ECE580-1|student answers and discussions]]'''
 +
----
  
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      Final Range: 1.0; Initial Range: 20.
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'''Problem 2. (20pts) Employ the DFP method to construct a set of Q-conjugate directions using the function'''  
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      <math> \frac{2}{F_{N+1}} \le \frac{1.0}{20}</math>, or <math> F_{N+1} \ge 40</math>
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−
 
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      So, <span class="texhtml">''N'' + 1 = 9</span>
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      Therefore, the minimal iterations is N-1 or 7.
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−
 
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<br>
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−
 
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Solution 2:
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Since the final range is 1 and the initial range is 20, we can say
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<math>
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\frac{2}{F_{N+1}} \le \frac{1.0}{20} \text{or equivalently } F_{N+1} \ge 40
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</math>
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−
 
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From the inequality, we get <math> F_{N+1} \ge 40 , \text{ so } N+1=9 </math>. Therefore the minimum number of iteration is N-1=7
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−
 
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<font color="#0000FF "><span style="font-size: 19px;"><math>\color{blue}
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F_9=55 \text{ and } F_8=34
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</math></span></font>
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−
 
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<br>
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−
 
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Problem 2. (20pts) Employ the DFP method to construct a set of Q-conjugate directions using the function'''  
+
  
 
       <math>f = \frac{1}{2}x^TQx - x^Tb+c </math>
 
       <math>f = \frac{1}{2}x^TQx - x^Tb+c </math>
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Where <span class="texhtml">''x''<sup>(0)</sup></span> is arbitrary.  
 
Where <span class="texhtml">''x''<sup>(0)</sup></span> is arbitrary.  
  
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<br>
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:'''Click [[QE2012_AC-3_ECE580-2|here]] to view [[QE2012_AC-3_ECE580-2|student answers and discussions]]'''
  
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Solution:
+
----
−
 
+
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      <math>f = \frac{1}{2}x^TQx - x^Tb+c </math>
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    Use initial point x<sup>(0)</sup> = [0,0]<sup>T</sub> and H<sub>0</sub> = I<sub>2</sub>
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−
   
+
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In this case
+
−
    <math>g^{(k)} = \begin{bmatrix}
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  1 & 1 \\
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  1 & 2
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\end{bmatrix} x^{(k)} - \begin{bmatrix}
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  2  \\
+
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  1
+
−
\end{bmatrix}</math>
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−
 
+
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      Hence
+
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    <math>g^{(0)} = \begin{bmatrix}
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−
  -2  \\
+
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  -1
+
−
\end{bmatrix},</math>  <math>d^{(0)} = -H_0g^{(0)} =- \begin{bmatrix}
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  1 & 0 \\
+
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  0 & 1
+
−
\end{bmatrix}\begin{bmatrix}
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  -2  \\
+
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  -1
+
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\end{bmatrix} = \begin{bmatrix}
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  2  \\
+
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  1
+
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\end{bmatrix}</math>
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+
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<br>
+
−
 
+
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      Because f is a quadratic function
+
−
 
+
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      <math>\alpha_0 = argminf(x^{(0)} + \alpha d^{(0)}) = - \frac{g^{(0)^T}d^{(0)}}{d^{(0)^T}Qd^{(0)}} = - \frac{\begin{bmatrix}
+
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  -2 & -1
+
−
\end{bmatrix}\begin{bmatrix}
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−
  2  \\
+
−
  1
+
−
\end{bmatrix}}{\begin{bmatrix}
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  2 & 1\end{bmatrix}\begin{bmatrix}
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−
  1 & 1 \\
+
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  1 & 2
+
−
\end{bmatrix}\begin{bmatrix}
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  2  \\
+
−
  1
+
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\end{bmatrix}} = \frac{1}{2}</math>
+
−
 
+
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      <math>x^{(1)} = x^{(0)} + \alpha d^{(0)} = \frac{1}{2} \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
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−
  1  \\
+
−
  \frac{1}{2}
+
−
\end{bmatrix}</math>
+
−
 
+
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      <math>\Delta x^{(0)} = x^{(1)}- x^{(0)} = \begin{bmatrix}
+
−
  1  \\
+
−
  \frac{1}{2}
+
−
\end{bmatrix}</math>
+
−
    <math>g^{(1)} =\begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix} x^{(1)} - \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix}= \begin{bmatrix}
+
−
  -\frac{1}{2}  \\
+
−
  1
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <math>\Delta g^{(0)} = g^{(1)} - g^{(0)} = \begin{bmatrix}
+
−
  -\frac{3}{2}  \\
+
−
  2
+
−
\end{bmatrix} </math>
+
−
 
+
−
<br>
+
−
 
+
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      Observe that
+
−
    <math>\Delta x^{(0)} \Delta x^{(0)^T} = \begin{bmatrix}
+
−
  1  \\
+
−
  \frac{1}{2}
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  1  & \frac{1}{2}
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  1 & \frac{1}{2}  \\
+
−
  \frac{1}{2}  & \frac{1}{4}
+
−
\end{bmatrix} </math>
+
−
    <math> \Delta x^{(0)^T} \Delta g^{(0)} = \begin{bmatrix}
+
−
  1  & \frac{1}{2}
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  \frac{3}{2}  \\
+
−
  2
+
−
\end{bmatrix}  = \frac{5}{2}</math>
+
−
    <math>H_0 \Delta g^{(0)} = \begin{bmatrix}
+
−
  1 & 0 \\
+
−
  0 & 1
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  \frac{3}{2}  \\
+
−
  2
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  \frac{3}{2}  \\
+
−
  2
+
−
\end{bmatrix},</math> <math>(H_0 \Delta g^{(0)})(H_0 \Delta g^{(0)})^T = \begin{bmatrix}
+
−
  \frac{9}{4}  & 3 \\
+
−
  3 & 4
+
−
\end{bmatrix}</math>
+
−
    <math>\Delta g^{(0)^T}H_0 \Delta g^{(0)} = \begin{bmatrix}
+
−
  \frac{3}{2}  & 2
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  1 & 0 \\
+
−
  0 & 1
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  \frac{3}{2}  \\ 2
+
−
\end{bmatrix} = \frac{25}{4}</math>
+
−
    <font face="serif" size="3">''Using the above, now we have''</font>
+
−
    <math>H_1 = H_0 + \frac{\Delta x^{(0)} \Delta x^{(0)^T}}{\Delta x^{(0)^T} \Delta g^{(0)}} - \frac{(H_0 \Delta g^{(0)})(H_0 \Delta g^{(0)})^T}{\Delta g^{(0)^T}H_0 \Delta g^{(0)} }  = \begin{bmatrix}
+
−
  1 & 0 \\
+
−
  0 & 1
+
−
\end{bmatrix} + \begin{bmatrix}
+
−
  \frac{2}{5} & \frac{1}{5} \\
+
−
  \frac{1}{5} & \frac{1}{10}
+
−
\end{bmatrix} - \frac{25}{4}\begin{bmatrix}
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−
  \frac{9}{4} & 3 \\
+
−
  3 & 4
+
−
\end{bmatrix} = \begin{bmatrix}
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−
  \frac{26}{25} & -\frac{7}{25} \\
+
−
  -\frac{7}{25} & \frac{23}{50}
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <span class="texhtml">''T'hen we have''</span><span class="texhtml">''','''</span>
+
−
    <math>d^{(1)} = -H_1 g^{(0)} = - \begin{bmatrix}
+
−
  \frac{26}{25} & -\frac{7}{25} \\
+
−
  -\frac{7}{25} & \frac{23}{50}
+
−
\end{bmatrix} \begin{bmatrix}
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−
  -\frac{1}{2}  \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <math>\alpha_1 = argminf(x^{(1)} + \alpha d^{(1)}) = - \frac{g^{(1)^T}d^{(1)}}{d^{(1)^T}Qd^{(1)}} = - \frac{\begin{bmatrix}
+
−
  -2 & 1
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix}}{\begin{bmatrix}
+
−
  \frac{4}{5} & -\frac{3}{5}\end{bmatrix}\begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix}} = \frac{5}{2}</math>
+
−
 
+
−
      <math>x^{(2)} = x^{(1)} + \alpha_1 d^{(1)} = \begin{bmatrix}
+
−
  1  \\
+
−
  \frac{1}{2}
+
−
\end{bmatrix} + \frac{5}{2}\begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  3  \\
+
−
  -1
+
−
\end{bmatrix} </math>
+
−
 
+
−
      <math>\Delta x^{(1)} = x^{(2)} - x^{(1)} = \begin{bmatrix}
+
−
  2  \\
+
−
  -\frac{3}{2}
+
−
\end{bmatrix}</math>
+
−
    <math>g^{(2)} = \begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix} x^{(0)} - \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  0  \\
+
−
  0
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <span class="texhtml">''Note that we have''</span> <math>d^{(0)^T}Qd^{(0)} = 0;</math>
+
−
    <span class="texhtml">''that is''</span>, <math>d^{(0)} = \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix}</math> and  <math>d^{(1)}  = \begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix}</math> <span class="texhtml">''are Q-conjugate directions''</span><span class="texhtml">'''.'''</span>
+
−
 
+
−
 
+
−
Solution 2:
+
−
 
+
−
<math>
+
−
\text{Let the initial point be } x^{(0)}= \begin{bmatrix} 0\\ 0 \end{bmatrix}
+
−
\text{and initial Hessian be } H_0=\begin{bmatrix} 1 & 0\\ 0 & 1\end{bmatrix}
+
−
 
+
−
</math>
+
−
 
+
−
<math>g^{(k)} = \begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix} x^{(k)} - \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix} , \text{so}</math>
+
−
 
+
−
<math>g^{(0)} = \begin{bmatrix}
+
−
  -2  \\
+
−
  -1
+
−
\end{bmatrix},</math>  <math>d^{(0)} = -H_0 g^{(0)} =- \begin{bmatrix}
+
−
  1 & 0 \\
+
−
  0 & 1
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  -2  \\
+
−
  -1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>\alpha_0 = - \frac{g^{(0)^T}d^{(0)}}{d^{(0)^T}Qd^{(0)}} = - \frac{\begin{bmatrix}
+
−
  -2 & -1
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix}}{\begin{bmatrix}
+
−
  2 & 1\end{bmatrix}\begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix}} = \frac{1}{2}</math>
+
−
 
+
−
<math>x^{(1)} = x^{(0)} + \alpha d^{(0)} = \frac{1}{2} \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  1  \\
+
−
  \frac{1}{2}
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>\Delta x^{(0)} = x^{(1)}- x^{(0)} = \begin{bmatrix}
+
−
  1  \\
+
−
  \frac{1}{2}
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>g^{(1)} =\begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix} x^{(1)} - \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix}= \begin{bmatrix}
+
−
  -\frac{1}{2}  \\
+
−
  1
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>\Delta g^{(0)} = g^{(1)} - g^{(0)} = \begin{bmatrix}
+
−
  -\frac{3}{2}  \\
+
−
  2
+
−
\end{bmatrix} </math>
+
−
 
+
−
<math>
+
−
\text{If we plug in the above numbers in the formula, we can get}
+
−
</math>
+
−
 
+
−
<math>H_1 = H_0 + \frac{\Delta x^{(0)} \Delta x^{(0)^T}}{\Delta x^{(0)^T} \Delta g^{(0)}} - \frac{(H_0 \Delta g^{(0)})(H_0 \Delta g^{(0)})^T}{\Delta g^{(0)^T}H_0 \Delta g^{(0)} }  = \begin{bmatrix}
+
−
  1 & 0 \\
+
−
  0 & 1
+
−
\end{bmatrix} + \begin{bmatrix}
+
−
  \frac{2}{5} & \frac{1}{5} \\
+
−
  \frac{1}{5} & \frac{1}{10}
+
−
\end{bmatrix} - \frac{25}{4}\begin{bmatrix}
+
−
  \frac{9}{4} & 3 \\
+
−
  3 & 4
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  \frac{26}{25} & -\frac{7}{25} \\
+
−
  -\frac{7}{25} & \frac{23}{50}
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>d^{(1)} = -H_1 g^{(1)} = - \begin{bmatrix}
+
−
  \frac{26}{25} & -\frac{7}{25} \\
+
−
  -\frac{7}{25} & \frac{23}{50}
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  -\frac{1}{2}  \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>\alpha_1 = - \frac{g^{(1)^T}d^{(1)}}{d^{(1)^T}Qd^{(1)}} = - \frac{\begin{bmatrix}
+
−
  -2 & 1
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix}}{\begin{bmatrix}
+
−
  \frac{4}{5} & -\frac{3}{5}\end{bmatrix}\begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix}} = \frac{5}{2}</math>
+
−
 
+
−
<math>x^{(2)} = x^{(1)} + \alpha_1 d^{(1)} = \begin{bmatrix}
+
−
  1  \\
+
−
  \frac{1}{2}
+
−
\end{bmatrix} + \frac{5}{2}\begin{bmatrix}
+
−
  \frac{4}{5}  \\
+
−
  -\frac{3}{5}
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  3  \\
+
−
  -1
+
−
\end{bmatrix} </math>
+
−
 
+
−
<math>\Delta x^{(1)} = x^{(2)} - x^{(1)} = \begin{bmatrix}
+
−
  2  \\
+
−
  -\frac{3}{2}
+
−
\end{bmatrix}
+
−
</math>
+
−
 
+
−
<math>
+
−
g^{(2)} = \begin{bmatrix}
+
−
  1 & 1 \\
+
−
  1 & 2
+
−
\end{bmatrix} x^{(2)} - \begin{bmatrix}
+
−
  2  \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  0  \\
+
−
  0
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>
+
−
\text{When the gradient is 0, we reach the minimum point, which is } x^{(2)}=\begin{bmatrix}
+
−
  3 \\
+
−
  -1
+
−
\end{bmatrix}
+
−
</math>
+
−
 
+
−
<br>
+
  
−
'''Problem 3. (20pts) For the system of linear equations,<math> Ax=b </math> where  
+
'''Problem 3. (20pts) For the system of linear equations,<math> Ax=b </math> where '''
  
 
<math>A = \begin{bmatrix}
 
<math>A = \begin{bmatrix}
Line 445: Line 76:
  
  
−
Find the minimum length vector <math>x^{(\ast)}</math> that minimizes <math>\| Ax -b\|^{2}_2</math>  
+
'''Find the minimum length vector <math>x^{(\ast)}</math> that minimizes <math>\| Ax -b\|^{2}_2</math> '''
  
−
<br>
+
:'''Click [[QE2012_AC-3_ECE580-3|here]] to view [[QE2012_AC-3_ECE580-3|student answers and discussions]]'''
−
 
+
----
−
<br> Solutions:  
+
−
 
+
−
      <math>A = BC = \begin{bmatrix}
+
−
  1 & 0  \\
+
−
  0 & 1  \\
+
−
  0 & -1
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  1 & 0 &-1  \\
+
−
  0 & 1 & 0 
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <math>B^{\dagger} = (B^T B)^{-1}B^T = \begin{bmatrix}
+
−
  1 & 0 \\
+
−
  0 & 2 
+
−
\end{bmatrix}^{-1} \begin{bmatrix}
+
−
  1 & 0 & 0  \\
+
−
  0 & 1 & -1 
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  1 & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2} 
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <math>C^{\dagger} = C^T(CC^T)^{-1} =\begin{bmatrix}
+
−
  1 & 0  \\
+
−
  0 & 1  \\
+
−
  -1 & 0
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  2 & 0 \\
+
−
  0 & 1 
+
−
\end{bmatrix}^{-1} = \begin{bmatrix}
+
−
  \frac{1}{2} & 0  \\
+
−
  0 & 1  \\
+
−
  -\frac{1}{2} & 0
+
−
\end{bmatrix} </math>
+
−
 
+
−
      <math>A^{\dagger} = C^{\dagger}B^{\dagger} =\begin{bmatrix}
+
−
  \frac{1}{2} & 0  \\
+
−
  0 & 1  \\
+
−
  -\frac{1}{2} & 0
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  1 & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2} 
+
−
\end{bmatrix} =  \begin{bmatrix}
+
−
  \frac{1}{2} & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2}  \\
+
−
  -\frac{1}{2} & 0 & 0
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <math> x^{\ast} = A^{\dagger} b = \begin{bmatrix}
+
−
  \frac{1}{2} & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2}  \\
+
−
  -\frac{1}{2} & 0 & 0
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  0 \\
+
−
  \frac{1}{2} \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  0 \\
+
−
  \frac{1}{2} \\
+
−
  0
+
−
\end{bmatrix}</math>
+
−
 
+
−
<br>
+
−
 
+
−
Solution 2:
+
−
 
+
−
<math>x^{(\ast)}=A^{\dagger}b</math>
+
−
 
+
−
Since <math>A = BC = \begin{bmatrix}
+
−
  1 & 0  \\
+
−
  0 & 1  \\
+
−
  0 & -1
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  1 & 0 &-1  \\
+
−
  0 & 1 & 0 
+
−
\end{bmatrix}</math>
+
−
 
+
−
<math>B^{\dagger} = (B^T B)^{-1}B^T = \begin{bmatrix}
+
−
  1 & 0 \\
+
−
  0 & 2 
+
−
\end{bmatrix}^{-1} \begin{bmatrix}
+
−
  1 & 0 & 0  \\
+
−
  0 & 1 & -1 
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  1 & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2} 
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <math>C^{\dagger} = C^T(CC^T)^{-1} =\begin{bmatrix}
+
−
  1 & 0  \\
+
−
  0 & 1  \\
+
−
  -1 & 0
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  2 & 0 \\
+
−
  0 & 1 
+
−
\end{bmatrix}^{-1} = \begin{bmatrix}
+
−
  \frac{1}{2} & 0  \\
+
−
  0 & 1  \\
+
−
  -\frac{1}{2} & 0
+
−
\end{bmatrix} </math>
+
−
 
+
−
      <math>A^{\dagger} = C^{\dagger}B^{\dagger} =\begin{bmatrix}
+
−
  \frac{1}{2} & 0  \\
+
−
  0 & 1  \\
+
−
  -\frac{1}{2} & 0
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  1 & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2} 
+
−
\end{bmatrix} =  \begin{bmatrix}
+
−
  \frac{1}{2} & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2}  \\
+
−
  -\frac{1}{2} & 0 & 0
+
−
\end{bmatrix}</math>
+
−
 
+
−
      <math> x^{\ast} = A^{\dagger} b = \begin{bmatrix}
+
−
  \frac{1}{2} & 0 & 0  \\
+
−
  0 & \frac{1}{2} & -\frac{1}{2}  \\
+
−
  -\frac{1}{2} & 0 & 0
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  0 \\
+
−
  \frac{1}{2} \\
+
−
  1
+
−
\end{bmatrix} = \begin{bmatrix}
+
−
  0 \\
+
−
  \frac{1}{2} \\
+
−
  0
+
−
\end{bmatrix}</math>
+
−
 
+
−
<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}
+
−
\text{ The pseudo inverse of a matrix has the property }
+
−
(BC)^{\dagger}=C^{\dagger}B^{\dagger}
+
−
</math></span></font>
+
−
 
+
−
<br>
+
 
'''Problem 4. (20pts) Use any simplex method to solve the following linear program. '''  
 
'''Problem 4. (20pts) Use any simplex method to solve the following linear program. '''  
  
Line 591: Line 88:
 
                         <math>x_1 \ge 0, x_2 \ge 0.</math>
 
                         <math>x_1 \ge 0, x_2 \ge 0.</math>
  
−
<br>
+
:'''Click [[QE2012_AC-3_ECE580-4|here]] to view [[QE2012_AC-3_ECE580-4|student answers and discussions]]'''
−
 
+
----
−
<br> Solution:  
+
−
 
+
−
        We can transfer the problem to the following standard form:
+
−
        <span class="texhtml">''Mnimize''</span>'''    <span class="texhtml"> − ''x''<sub>1</sub> − 2''x''<sub>2</sub></span>'''
+
−
        <span class="texhtml">''Sbject to''</span>'''    <span class="texhtml"> − 2''x''<sub>1</sub> + ''x''<sub>2</sub> + ''x''<sub>3</sub> = 2</span>'''
+
−
                        <span class="texhtml"> − ''x''<sub>1</sub> + ''x''<sub>2</sub> + ''x''<sub>4</sub> = 3</span>
+
−
                        <span class="texhtml">''x''<sub>1</sub> + ''x''<sub>5</sub> = 3</span>
+
−
                        <math>x_1, x_2, x_3, x_4, x_5 \ge 0.</math>
+
−
 
+
−
        We form the corresponding tableau for the problem
+
−
      <math>\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  -2 & 1& 1 &0& 0& 2 \\
+
−
  -1& 1& 0 &1 &0 &3\\
+
−
  1 &0& 0& 0& 1& 3\\
+
−
  -1& -2 &0& 0& 0& 0
+
−
\end{matrix}</math>
+
−
     
+
−
Since it is already in canonical form. Because <span class="texhtml">''r''<sub>2</sub> =  − 7</span>, we bring <span class="texhtml">''a''<sub>2</sub></span> into the basis.
+
−
 
+
−
      After computing the ratios, we pivot about the (1,2) element of the tableau to obtain
+
−
      <math>\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  -2 & 1& 1 &0& 0& 2 \\
+
−
  1& 0& -1 &1 &0 &1\\
+
−
  1 &0& 0& 0& 1& 3\\
+
−
  -5& 0 &2& 0& 0& 4
+
−
\end{matrix}</math>
+
−
 
+
−
      Similarly, we pivot about the (2,1) element to obtain
+
−
      <math>\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  0 & 1& -1 &2& 0& 4 \\
+
−
  1& 0& -1 &1 &0 &1\\
+
−
  0 &0& 1& -1& 1& 2\\
+
−
  0& 0 &-3& 5& 0& 9
+
−
\end{matrix}</math>
+
−
 
+
−
      Similarly, we pivot about the (3,3) element to obtain
+
−
      <math>\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  0 & 1& 0 &1& 1& 6 \\
+
−
  1& 0& 0 &0 &1 &3\\
+
−
  0 &0& 1& -1& 1& 2\\
+
−
  0& 0 &0& 2& 3& 15
+
−
\end{matrix}</math>
+
−
 
+
−
      Therefore, <span class="texhtml">''x''<sub>1</sub> = 3,</span> <span class="texhtml">''x''<sub>2</sub> = 6,</span> maximize the function.
+
−
 
+
−
Solution 2:
+
−
 
+
−
The standard form for this problem is
+
−
        <span class="texhtml">''Mnimize''</span>'''    <span class="texhtml"> − ''x''<sub>1</sub> − 2''x''<sub>2</sub></span>'''
+
−
        <span class="texhtml">''Sbject to''</span>'''    <span class="texhtml"> − 2''x''<sub>1</sub> + ''x''<sub>2</sub> + ''x''<sub>3</sub> = 2</span>'''
+
−
                        <span class="texhtml"> − ''x''<sub>1</sub> + ''x''<sub>2</sub> + ''x''<sub>4</sub> = 3</span>
+
−
                        <span class="texhtml">''x''<sub>1</sub> + ''x''<sub>5</sub> = 3</span>
+
−
                        <math>x_1, x_2, x_3, x_4, x_5 \ge 0.</math>
+
−
 
+
−
<math>\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  -2 & 1& 1 &0& 0& 2 \\
+
−
  -1& 1& 0 &1 &0 &3\\
+
−
  1 &0& 0& 0& 1& 3\\
+
−
  -1& -2 &0& 0& 0& 0
+
−
\end{matrix}\Rightarrow
+
−
 
+
−
\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  -2 & 1& 1 &0& 0& 2 \\
+
−
  1& 0& -1 &1 &0 &1\\
+
−
  1 &0& 0& 0& 1& 3\\
+
−
  -5& 0 &2& 0& 0& 4
+
−
\end{matrix}
+
−
\Rightarrow
+
−
\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  0 & 1& -1 &2& 0& 4 \\
+
−
  1& 0& -1 &1 &0 &1\\
+
−
  0 &0& 1& -1& 1& 2\\
+
−
  0& 0 &-3& 5& 0& 9
+
−
\end{matrix}
+
−
\Rightarrow
+
−
\begin{matrix}
+
−
  a_1 & a_2 & a_3 & a_4 & a_5 & b \\
+
−
  0 & 1& 0 &1& 1& 6 \\
+
−
  1& 0& 0 &0 &1 &3\\
+
−
  0 &0& 1& -1& 1& 2\\
+
−
  0& 0 &0& 2& 3& 15
+
−
\end{matrix}</math>
+
−
 
+
−
From the last tableau, we can see that <math>\begin{bmatrix}
+
−
  x^{1} = 3\\ x^{2} = 6 
+
−
\end{bmatrix}</math>
+
−
maximizes the object function. The maximum value is 15
+
−
 
+
−
<font color="#0000FF "><span style="font-size: 19px;">
+
−
<math>\color{blue}
+
−
\text{ It should be noted that, }
+
−
\begin{bmatrix}
+
−
  x^{1} = 3\\ x^{2} = 6 
+
−
\end{bmatrix} \text{actually minimizes the standard form,}
+
−
</math>
+
−
<math>\color{blue}
+
−
\text{which equivalently maximizes the original problem}
+
−
</math>
+
−
</span></font>
+
  
 
<br> '''Problem 5.(20pts) Solve the following problem:'''  
 
<br> '''Problem 5.(20pts) Solve the following problem:'''  
Line 709: Line 100:
 
'''(i)(10pts) Find the points that satisfy the KKT condition.'''  
 
'''(i)(10pts) Find the points that satisfy the KKT condition.'''  
  
−
Solution:
 
  
−
        Standard Form:
 
−
        <span class="texhtml">''Minimize''</span>'''    <math>-x_1^2 + 2x_2</math>'''
 
−
        <span class="texhtml">''Su''</span>'''  <math>g_1(x) = x_1^2+x_2^2 - 1 \le 0</math>'''
 
−
                        <math>g_2(x) = -x_2 \le 0</math>
 
  
−
        The KKT Condition.
+
<br> '''(ii)(10pts)Apply the SONC and SOSC to determine the nature of the critical points from the previous part.'''
−
      <math>1. \mu _1, \mu _2 \ge 0</math> 　
+
−
      <span class="texhtml">2. − 2''x''<sub>1</sub> + 2''x''<sub>1</sub>μ<sub>1</sub> = 0</span> 　
+
−
        <span class="texhtml">2.2 + 2''x''<sub>2</sub>μ<sub>1</sub> − μ<sub>2</sub> = 0</span> 　
+
−
      <math>3. (x_1^2+x_2^2-1)\mu_1 - x_2\mu _2 = 0</math> 　
+
−
      <math>4. g_1(x),g_2(x) \le 0</math> 　
+
−
 
+
−
<br>
+
−
 
+
−
        Case 1:
+
−
      <span class="texhtml">''I''''f'''''</span>''''' <span class="texhtml">μ<sub>1</sub> = 0</span> and <span class="texhtml">μ<sub>2</sub> = 0,</span>'''''
+
−
            <span class="texhtml">''t''''h''''e''''n'''''<b>,2 = 0</b></span>'''which is impossible.'''
+
−
     
+
−
      Case 2:
+
−
      <span class="texhtml">''I''''f'''''</span>''''' <span class="texhtml">μ<sub>1</sub> = 0</span> and <math>\mu_2 \not= 0,</math>'''''
+
−
            <span class="texhtml">''t''''h''''e''''n'''''<b>,μ<sub>2</sub> = 2,''x''<sub>1</sub> = 0,''x''<sub>2</sub> = 0.</b></span>'''.'''
+
−
 
+
−
        Case 3:
+
−
      <span class="texhtml">''I''''f'''''</span>''''' <math>\mu_1 \not= 0</math> and <span class="texhtml">μ<sub>2</sub> = 0,</span>'''''
+
−
            <math>then, x_1^2 + x_2^2 = 1 </math>
+
−
                      <math>if x1 \not= 0, then \mu_1 = 1, x_2 = -1 </math>which is impossible.
+
−
                      <span class="texhtml">''i'''f '''x''1 = 0,''then x''<sub>2</sub> = 1,μ<sub>1</sub> =  − 1</span>which is impossible.
+
−
 
+
−
        Case 4:
+
−
      <span class="texhtml">''I''''f'''''</span>''''' <math>\mu_1 \not= 0</math> and <math>\mu_2 \not= 0,</math>'''''
+
−
            <math>then, x_2 = 0, x_1^2 + x_2^2 = 1. So,  x_1 = 1, \mu_1 = 1, \mu_2 = 2. </math>.
+
−
 
+
−
        Therefore, there are two points that satisfy KKT condition.
+
−
+
−
      <math>\begin{bmatrix}
+
−
  \mu_1 ^{\ast}= 0 \\ \mu_2 ^{\ast}= 2 \\ x_1^{\ast}= 0 \\x_2^{\ast}= 0
+
−
\end{bmatrix}and \begin{bmatrix}
+
−
  \mu_1 ^{\ast}= 1 \\ \mu_2 ^{\ast}= 2 \\ x_1^{\ast}= 1 \\x_2^{\ast}= 0
+
−
\end{bmatrix} </math>
+
−
 
+
−
Solution 2:
+
−
 
+
−
The standard form of the optimizing problem is
+
−
&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; &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; &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; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;
+
−
<math>\text{minimize } -x_1^2 + 2x_2</math>
+
−
 
+
−
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>\text{subject to  }  g_1(x)=x_1^2+x_2^2-1 \le 0  \text { and }  g_2(x)=-x_2 \le 0</math>
+
−
 
+
−
The KKT Condition.
+
−
      <math>1. \mu _1, \mu _2 \ge 0</math> 　
+
−
      <math>2.\begin{bmatrix}
+
−
  -2x_1+2\mu_1 x_1 \\ 2+2\mu_1 x_2-\mu_2
+
−
\end{bmatrix}=0</math>
+
−
      <math>3. (x_1^2+x_2^2-1)\mu_1 - x_2\mu _2 = 0</math> 　
+
−
      <math>4. g_1(x),g_2(x) \le 0</math> 　
+
−
 
+
−
<math>\text {Case 1: } \mu_1=\mu_2=0 \text{, which implies } 2=0 \Rightarrow \text{Impossible} </math>
+
−
 
+
−
<math>\text {Case 2: } \mu_1 \neq 0,  \mu_2=0 \Rightarrow
+
−
\left\{\begin{matrix}
+
−
 
+
−
x_1^2+x_2^2-1=0\\
+
−
-2x_1+2\mu_1x_1=0\\
+
−
2+2\mu_1 x_1=0
+
−
\end{matrix}\right. \Rightarrow
+
−
 
+
−
\left\{\begin{matrix}
+
−
 
+
−
\mu_1=1\\
+
−
\mu_2=0\\
+
−
x_1=0\\
+
−
x_2=-1
+
−
\end{matrix}\right.
+
−
 
+
−
\Rightarrow
+
−
 
+
−
x_2<0,
+
−
\text{ Impossible }
+
−
 
+
−
</math>
+
−
 
+
−
 
+
−
 
+
−
<math>\text {Case 3: } \mu_1 = 0,  \mu_2 \neq 0 \Rightarrow
+
−
\left\{\begin{matrix}
+
−
 
+
−
-x_2\mu_2=0\\
+
−
-2x_1=0\\
+
−
2-\mu_2=0
+
−
\end{matrix}\right. \Rightarrow
+
−
 
+
−
\left\{\begin{matrix}
+
−
 
+
−
\mu_1=0\\
+
−
\mu_2=2\\
+
−
x_1=0\\
+
−
x_2=0
+
−
\end{matrix}\right.
+
−
 
+
−
</math>
+
−
 
+
−
<math>\text {Case 4: } \mu_1 \neq 0,  \mu_2 \neq 0 \Rightarrow
+
−
\left\{\begin{matrix}
+
−
 
+
−
x_1^2+x_2^2-1=0\\
+
−
x_2=0\\
+
−
\end{matrix}\right. \Rightarrow
+
−
 
+
−
\left\{\begin{matrix}
+
−
 
+
−
\mu_1=1\\
+
−
\mu_2=2\\
+
−
x_1=1\\
+
−
x_2=0
+
−
\end{matrix}\right.
+
−
 
+
−
</math>
+
−
 
+
−
<font color="#0000FF "><span style="font-size: 19px;">
+
−
It's a lot more easier as the student notice that ignoring the constraint 
+
−
<math>\color{blue}
+
−
x_1>0
+
−
</math>
+
−
doesn't have any effect on the optimizing problem.This will save the time of discussing 4 more cases. But the student should have explain why this is true. </span></font>
+
−
 
+
−
<font color="#0000FF "><span style="font-size: 19px;">
+
−
Each coefficient <math>\color{blue}\mu </math> can be either 0 or not. When solving the equations, a systematic way would be to discuss all cases.
+
−
<math>\color{blue}
+
−
 
+
−
</math>
+
−
</span></font>
+
−
 
+
−
<br> '''(ii)(10pts)Apply the SONC and SOSC to determine the nature of the critical points from the previous part.'''  
+
−
 
+
−
Solution:
+
−
 
+
−
        Apply SOSC to the first point.
+
−
      <math>\nabla g_2(x)^t y = 0</math>
+
−
      <math>\begin{bmatrix}
+
−
  0 & -1
+
−
\end{bmatrix} y = 0</math>
+
−
      <math>y = \begin{bmatrix}
+
−
  a \\ 0
+
−
\end{bmatrix},  a\not= 0</math>
+
−
 
+
−
        <math>L(x,\mu) = F(x) + \mu_2 G_2(x) =\begin{bmatrix}
+
−
  -2 & 0 \\ 0 & 0
+
−
\end{bmatrix} </math>
+
−
      <span class="texhtml">''Check ''</span><span class="texhtml">''y''<sup>''t''</sup>''L''(''x'',μ)''y''</span>
+
−
      <math>\begin{bmatrix}
+
−
  a & 0
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  -2 & 0 \\ 0 & 0
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  a \\ 0
+
−
\end{bmatrix} = -2a^2 \lneq 0</math> which means SOSC fails.
+
−
 
+
−
<br>
+
−
 
+
−
        Apply SOSC to the second point.
+
−
      <math>\nabla g_1(x)^t y = 0, </math>
+
−
 
+
−
        <math>\begin{bmatrix}
+
−
  2x_1 & 2x_2
+
−
\end{bmatrix} y = 0</math>
+
−
      <math>y = \begin{bmatrix}
+
−
  0 \\ a
+
−
\end{bmatrix} where a\not= 0</math>
+
−
 
+
−
        <math>L(x,\mu) = F(x) + \mu_1 G_1(x) + \mu_2 G_2(x) =\begin{bmatrix}
+
−
  -2 & 0 \\ 0 & 0
+
−
\end{bmatrix}  + \begin{bmatrix}
+
−
  2 & 0 \\ 0 & 2 
+
−
\end{bmatrix}  = \begin{bmatrix}
+
−
  0 & 0 \\ 0 & 2
+
−
\end{bmatrix} </math>
+
−
      <span class="texhtml">''Check ''</span><span class="texhtml">''y''<sup>''t''</sup>''L''(''x'',μ)''y''</span>
+
−
      <math>\begin{bmatrix}
+
−
  0 & a
+
−
\end{bmatrix}\begin{bmatrix}
+
−
  0 & 0 \\ 0 & 2
+
−
\end{bmatrix} \begin{bmatrix}
+
−
  0 \\ a
+
−
\end{bmatrix} = 2a^2 \gneq 0</math> that implies
+
−
      <math>\begin{bmatrix}
+
−
  x^{\ast} = 1\\ x^{\ast} = 0
+
−
\end{bmatrix}</math> is the minimized point.
+
−
 
+
−
Solution 2:
+
−
 
+
−
<math>\text{For } \left\{\begin{matrix}
+
−
 
+
−
\mu_1=0\\
+
−
\mu_2=2\\
+
−
x_1=0\\
+
−
x_2=0
+
−
\end{matrix}\right. \text{, let } \nabla g_2(x)^T y=0, \text{We get } y=\begin{bmatrix}
+
−
  a\\  0
+
−
\end{bmatrix} \text{, where } y \text { are all the points on the tangent plane}</math>
+
−
 
+
−
<math>
+
−
L(x,\mu)=F(x,\mu)+\mu_2 G_2 (x)=\begin{bmatrix}
+
−
  -2 & 0\\ 0 & 0
+
−
\end{bmatrix}
+
−
</math>
+
−
 
+
−
<math>
+
−
y^T L(x,\mu)y=-2a^2 \le 0 \Rightarrow \text{It doesn;t satisfy SOSC}
+
−
</math>
+
−
 
+
−
 
+
−
<math>\text{For } \left\{\begin{matrix}
+
−
 
+
−
\mu_1=1\\
+
−
\mu_2=2\\
+
−
x_1=1\\
+
−
x_2=0
+
−
\end{matrix}\right. \text{, let } \nabla g_2(x)^T y=0, \text{We get } y=\begin{bmatrix}
+
−
  0\\ a
+
−
\end{bmatrix} \text{, where } y \text { are all the points on the tangent plane}</math>
+
−
 
+
−
<math>
+
−
L(x,\mu)=F(x,\mu)+\mu_2 G_2 (x)=\begin{bmatrix}
+
−
  0 & 0\\ 0 & 2
+
−
\end{bmatrix}
+
−
</math>
+
−
 
+
−
<math>
+
−
y^T L(x,\mu)y=2a^2 \geq 0 \Rightarrow \text{It doesn;t satisfy SOSC}
+
−
</math>
+
−
 
+
−
<math>
+
−
\text{So, }
+
−
\begin{bmatrix}
+
−
x^{\ast} = 1\\ x^{\ast} = 0
+
−
\end{bmatrix}
+
−
\text{is the minimum point.}
+
−
</math>
+
  
−
<font color="#0000FF "><span style="font-size: 19px;">
+
:'''Click [[QE2012_AC-3_ECE580-5|here]] to view [[QE2012_AC-3_ECE580-5|student answers and discussions]]'''
−
When checking the SOSC condition, we need to find out the tangent plane. Only the subject function with a non-zero 
+
----
−
<math>\color{blue}
+
[[ECE_PhD_Qualifying_Exams|Back to ECE QE page]]
−
\mu
+
−
</math>
+
−
is used to find out the tangent plane.
+
−
</span></font>
+

Latest revision as of 09:17, 13 September 2013


ECE Ph.D. Qualifying Exam

Automatic Control (AC)

Question 3: Optimization

August 2012

Student answers and discussions for Part 1,2,3,4,5

1.(20 pts)
(i)(10 pts) Find the factor by which the uncertainty range is reduced when using the Fibonacci method. Assume that the last step has the form $ \begin{align} 1- \rho_{N-1} = \frac{F_{2}}{F_{3}} = \frac{2}{3}, \end{align} $

where N − 1 is the number of steps performed in the uncertainty range reduction process.




(ii)(10 pts) It is known that the minimizer of a certain function f(x) is located in the interval[-5, 15]. What is the minimal number of iterations of the Fibonacci method required to box in the minimizer within the range 1.0? Assume that the last useful value of the factor reducing the uncertainty range is 2/3, that is

$ 1- \rho_{N-1} = \frac{F_{2}}{F_{3}} = \frac{2}{3}, $

Click here to view student answers and discussions

Problem 2. (20pts) Employ the DFP method to construct a set of Q-conjugate directions using the function 

      $ f = \frac{1}{2}x^TQx - x^Tb+c  $
     $   =\frac{1}{2}x^T  \begin{bmatrix}   1 & 1 \\   1 & 2  \end{bmatrix}x-x^T\begin{bmatrix}   2  \\   1  \end{bmatrix} + 3. $

Where x(0) is arbitrary.

Click here to view student answers and discussions

Problem 3. (20pts) For the system of linear equations,$ Ax=b $ where

$ A = \begin{bmatrix} 1 & 0 &-1 \\ 0 & 1 & 0 \\ 0 & -1& 0 \end{bmatrix}, b = \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix} $


Find the minimum length vector $ x^{(\ast)} $ that minimizes $ \| Ax -b\|^{2}_2 $

Click here to view student answers and discussions

Problem 4. (20pts) Use any simplex method to solve the following linear program. 

           Maximize    x1 + 2x2
        S'ubject to    $ -2x_1+x_2 \le 2 $
                       $ x_1-x_2 \ge -3 $
                       $ x_1 \le -3 $
                       $ x_1 \ge 0, x_2 \ge 0. $
Click here to view student answers and discussions


Problem 5.(20pts) Solve the following problem: 

           Minimize    $ -x_1^2 + 2x_2 $
        Subject to    $ x_1^2+x_2^2 \le 1 $
                       $  x_1 \ge 0 $
                       $ x_2 \ge 0 $

(i)(10pts) Find the points that satisfy the KKT condition.



(ii)(10pts)Apply the SONC and SOSC to determine the nature of the critical points from the previous part.

Click here to view student answers and discussions

Back to ECE QE page

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