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| − | = [[ECE PhD Qualifying Exams|ECE Ph.D. Qualifying Exam]]: Automatic Control (AC)- Question 3, August 2011 = | + | = [[ECE PhD Qualifying Exams|ECE Ph.D. Qualifying Exam]]: Automatic Control (AC)- Question 3, August 2011 = |
<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{5. } \left( \text{20 pts} \right) \text{ Consider the optimization problem, }</math></span></font> | <font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{5. } \left( \text{20 pts} \right) \text{ Consider the optimization problem, }</math></span></font> | ||
| − | <font color="#ff0000"> | + | <font color="#ff0000"> </font><math>\text{optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2}</math> |
| − | <math>\text{subject to } x_{2}- x_{1}^{2}\geq0</math> | + | <math>\text{subject to } x_{2}- x_{1}^{2}\geq0</math> |
| − | <math>2-x_{1}-x_{2}\geq0, x_{1}\geq0.</math> | + | <math>2-x_{1}-x_{2}\geq0, x_{1}\geq0.</math> |
<math>\color{blue} \text{The point } x^{*}=\begin{bmatrix} | <math>\color{blue} \text{The point } x^{*}=\begin{bmatrix} | ||
0 & 0 | 0 & 0 | ||
| − | \end{bmatrix}^{T} \text{ satisfies the KKT conditions.}</math> | + | \end{bmatrix}^{T} \text{ satisfies the KKT conditions.}</math> |
| − | <math>\color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?}</math> | + | <math>\color{blue}\left( \text{i} \right) \text{Does } x^{*} \text{ satisfy the FONC for minimum or maximum? Where are the KKT multipliers?}</math> |
<math>\color{blue}\text{Solution 1:}</math> | <math>\color{blue}\text{Solution 1:}</math> | ||
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| + | <math>\text{ Standard form: optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2}</math><br> | ||
| + | |||
| + | <math>\text{subject to } g_{1}\left( x \right) x_{1}^{2}-x_{2}\leq0</math> | ||
| + | |||
| + | <math>g_{2}\left( x \right) x_{1}+x_{2}-2\leq0</math> | ||
| + | |||
| + | <math>g_{3}\left( x \right) -x_{1}\leq0</math> | ||
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| − | Automatic Control (AC)- Question 3, August 2011<br>Problem | + | Automatic Control (AC)- Question 3, August 2011<br>Problem 1: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion<br>Problem 2: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion-2<br>Problem 3: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion-3<br>Problem 4: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion-4<br> |
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Revision as of 18:10, 27 June 2012
ECE Ph.D. Qualifying Exam: Automatic Control (AC)- Question 3, August 2011
$ \color{blue}\text{5. } \left( \text{20 pts} \right) \text{ Consider the 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}\text{Solution 1:} $
$ \text{ Standard form: optimize} \left(x_{1}-2\right)^{2}+\left(x_{2}-1\right)^{2} $
$ \text{subject to } g_{1}\left( x \right) x_{1}^{2}-x_{2}\leq0 $
$ g_{2}\left( x \right) x_{1}+x_{2}-2\leq0 $
$ g_{3}\left( x \right) -x_{1}\leq0 $
$ \color{blue}\text{Solution 2:} $
Automatic Control (AC)- Question 3, August 2011
Problem 1: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion
Problem 2: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion-2
Problem 3: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion-3
Problem 4: https://www.projectrhea.org/rhea/index.php/ECE-QE_AC3-2011_solusion-4
