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= ECE QE AC-3 August 2011 Solusion = | = ECE QE AC-3 August 2011 Solusion = | ||
| − | ==== < | + | ==== <font face="serif"><math>\text{1. } \left( \text{20 pts} \right) \text{ Consider the optimization problem, }</math><br></font> ==== |
<math>\text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}</math> | <math>\text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}</math> | ||
| − | <math>\text{subject to } x_{1}\geq0, x_{2}\geq0</math> | + | <math>\text{subject to } x_{1}\geq0, x_{2}\geq0</math><font color="#ff0000" face="serif" size="4"><br></font> |
| − | < | + | '''<math>\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]</math>''' |
| − | ===== <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><br> | <math>\text{We need to find a direction }d\text{, such that } \exists\alpha_{0}>0,</math><br> | ||
| − | < | + | <br> |
| − | <math> | + | <math>\color{blue}\text{Solution 2:}</math> |
| − | <math>\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}</math> <br> | + | <math>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}</math> <br> |
'''<math>\because \begin{Bmatrix}x\in\Omega: x_{1}\geq0, x_{2}\geq0\end{Bmatrix}</math>''' | '''<math>\because \begin{Bmatrix}x\in\Omega: x_{1}\geq0, x_{2}\geq0\end{Bmatrix}</math>''' | ||
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\left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right], d_{1}\in\Re^{2}, d_{2}\neq0</math><br> | \left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right], d_{1}\in\Re^{2}, d_{2}\neq0</math><br> | ||
| − | ===== (ii) Write down the second-order necessary condition for . Does the point satisfy this condition? ===== | + | ===== <math>\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} {. Does the point } x^{*} { satisfy this condition?}</math><br> ===== |
Revision as of 21:42, 21 June 2012
Contents
ECE QE AC-3 August 2011 Solusion
$ \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 $
$ \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, $
$ \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 $
