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===== <math>\color{blue}\text{Solution 1:}</math> ===== | ===== <math>\color{blue}\text{Solution 1:}</math> ===== | ||
| − | <math>\text{We need to find a direction }d\text{, such that } \exists\alpha_{0}>0,</math> | + | <math>\text{We need to find a direction }d\text{, such that } \exists\alpha_{0}>0,</math> |
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| + | <math>\left( \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right) + \alpha d \text{ for all } \alpha\in\Omega \left[0,\alpha_{0}\right]</math><br> | ||
<math>\text{As } x_{1}\geq0, x_{2}\geq0, d= \left( \begin{array}{c} x \\ y \end{array} \right)\text{where } x\in\Re, \text{ and } y\geq0.</math> | <math>\text{As } x_{1}\geq0, x_{2}\geq0, d= \left( \begin{array}{c} x \\ y \end{array} \right)\text{where } x\in\Re, \text{ and } y\geq0.</math> | ||
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<math>\color{blue}\text{Solution 1:}</math> | <math>\color{blue}\text{Solution 1:}</math> | ||
| − | <math>\text{ | + | <font face="serif"><math>\text{It is equivalent to minimize } f\left(x\right) \text{, }</math> </font><math>\text{ subject to } g_{1}(x)\leq0, g_{2}(x)\leq0</math> |
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| − | <math>\text{ subject to } g_{1}(x)\leq0, g_{2}(x)\leq0 | + | |
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| − | <font color="#ff0000" | + | <font color="#ff0000" style="font-size: 17px;">'''<math>\left{ \begin{array}{c} l\left(x,\mu \right) = \nabla f(x)+\mu_{1}\nabla g_{1}(x)+ \mu_{2}\nabla g_{2}(x)=\left( \begin{array}{c} 2x_{1}-1+x_{2} \\ 1+x_{1} \end{array} \right) + \left( \begin{array}{c} -\mu_{1} \\ 0 \end{array} \right) +\left( \begin{array}{c} 0 \\ -\mu_{2} \end{array} \right) =0 \\ -\mu_{1}x_{1}-\mu_{2}x_{2}=0 \end{array}</math><br>'''</font> |
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Revision as of 21:35, 26 June 2012
ECE Ph.D. Qualifying Exam: Automatic Control (AC)- Question 3, August 2011
$ \color{blue}\text{1. } \left( \text{20 pts} \right) \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}\text{Solution 1:} $
$ \text{We need to find a direction }d\text{, such that } \exists\alpha_{0}>0, $
$ \left( \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right) + \alpha d \text{ for all } \alpha\in\Omega \left[0,\alpha_{0}\right] $
$ \text{As } x_{1}\geq0, x_{2}\geq0, d= \left( \begin{array}{c} x \\ y \end{array} \right)\text{where } x\in\Re, \text{ and } y\geq0. $
$ \color{blue}\text{Solution 2:} $
$ d\in\Re_{2}, d\neq0 \text{ is a feasible direction at } x^{*} \text{, if } \exists \alpha_{0} \text{ that } \left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right] + \alpha\left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right] \in\Omega \text{ for all } 0\leq\alpha\leq\alpha_{0} $
$ \because \begin{Bmatrix}x\in\Omega: x_{1}\geq0, x_{2}\geq0\end{Bmatrix} $
$ \therefore d= \left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right], d_{1}\in\Re^{2}, d_{2}\neq0 $
$ \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?} $
$ \color{blue}\text{Solution 1:} $
$ \text{It is equivalent to minimize } f\left(x\right) \text{, } $ $ \text{ subject to } g_{1}(x)\leq0, g_{2}(x)\leq0 $
$ \left{ \begin{array}{c} l\left(x,\mu \right) = \nabla f(x)+\mu_{1}\nabla g_{1}(x)+ \mu_{2}\nabla g_{2}(x)=\left( \begin{array}{c} 2x_{1}-1+x_{2} \\ 1+x_{1} \end{array} \right) + \left( \begin{array}{c} -\mu_{1} \\ 0 \end{array} \right) +\left( \begin{array}{c} 0 \\ -\mu_{2} \end{array} \right) =0 \\ -\mu_{1}x_{1}-\mu_{2}x_{2}=0 \end{array} $
