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| + | [[Category:QE]] | ||
| + | [[Category:CNSIP]] | ||
| + | [[Category:problem solving]] | ||
| + | [[Category:automatic control]] | ||
| + | [[Category:optimization]] | ||
| − | + | <center> | |
| + | <font size= 4> | ||
| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
| + | </font size> | ||
| − | ( | + | <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> | ||
| + | \begin{align} | ||
| + | 1- \rho_{N-1} = \frac{F_{2}}{F_{3}} = \frac{2}{3}, | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | '''where <span class="texhtml">''N'' − 1</span> is the number of steps performed in the uncertainty range reduction process. ''' | ||
| + | |||
| + | <br> | ||
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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> | ||
| + | :'''Click [[QE2012_AC-3_ECE580-1|here]] to view [[QE2012_AC-3_ECE580-1|student answers and discussions]]''' | ||
| + | ---- | ||
| − | + | '''Problem 2. (20pts) Employ the DFP method to construct a set of Q-conjugate directions using the function''' | |
| − | 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. | ||
| − | + | :'''Click [[QE2012_AC-3_ECE580-2|here]] to view [[QE2012_AC-3_ECE580-2|student answers and discussions]]''' | |
| − | + | ||
| + | ---- | ||
| − | '''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} | ||
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| − | 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> ''' |
| − | + | ||
| − | + | ||
| − | + | ||
| + | :'''Click [[QE2012_AC-3_ECE580-3|here]] to view [[QE2012_AC-3_ECE580-3|student answers and discussions]]''' | ||
| + | ---- | ||
'''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. ''' | ||
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<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> '''Problem 5.(20pts) Solve the following problem:''' | ||
| + | <span class="texhtml">''Minimize''</span>''' <math>-x_1^2 + 2x_2</math>''' | ||
| + | <span class="texhtml">''Subject to''</span>''' <math>x_1^2+x_2^2 \le 1</math>''' | ||
| + | <math> x_1 \ge 0</math> | ||
| + | <math>x_2 \ge 0</math> | ||
'''(i)(10pts) Find the points that satisfy the KKT condition.''' | '''(i)(10pts) Find the points that satisfy the KKT condition.''' | ||
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<br> '''(ii)(10pts)Apply the SONC and SOSC to determine the nature of the critical points from the previous part.''' | <br> '''(ii)(10pts)Apply the SONC and SOSC to determine the nature of the critical points from the previous part.''' | ||
| + | |||
| + | :'''Click [[QE2012_AC-3_ECE580-5|here]] to view [[QE2012_AC-3_ECE580-5|student answers and discussions]]''' | ||
| + | ---- | ||
| + | [[ECE_PhD_Qualifying_Exams|Back to ECE QE page]] | ||
Latest revision as of 10:17, 13 September 2013
Automatic Control (AC)
Question 3: Optimization
August 2012
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
