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'''Part 5.''' (20 pts) | '''Part 5.''' (20 pts) | ||
| + | <font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ Consider the following optimization problem, }</math></span></font> | ||
| + | |||
| + | <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>2-x_{1}-x_{2}\geq0</math> | ||
| + | |||
| + | <math>x_{1}\geq0.</math> | ||
| + | |||
| + | <math>\color{blue} \text{The point } x^{*}=\begin{bmatrix} | ||
| + | 0 & 0 | ||
| + | \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{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 04:34, 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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