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= ECE QE AC-3 August 2011 Solusion = | = ECE QE AC-3 August 2011 Solusion = | ||
| − | ==== < | + | ==== <span class="texhtml">1. (20 pts) Consider the optimization problem,</span><br> ==== |
<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> | ||
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<math>\text{subject to } x_{1}\geq0, x_{2}\geq0</math> | <math>\text{subject to } x_{1}\geq0, x_{2}\geq0</math> | ||
| − | < | + | <span class="texhtml">(i) Characterize feasible directions at the point</span><math>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> ===== | ||
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<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> | ||
| − | + | <math>\color{blue}\text{Solution 2:}</math> | |
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| + | <math>d\in\Re_{2}, d\neq0 \text{ is a feasible direction at } x^{*} \text{, if}</math> <math>\exists\alpha_{0}</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>\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> | ||
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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? ===== |
Revision as of 21:32, 21 June 2012
Contents
ECE QE AC-3 August 2011 Solusion
1. (20 pts) 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 $
(i) 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 $
