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| − | = ECE QE AC-3 August 2011 | + | = ECE QE AC-3 August 2011 Solusion = |
===== 1. (20 pts) Consider the optimization problem, ===== | ===== 1. (20 pts) Consider the optimization problem, ===== | ||
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===== (i) Characterize feasible directions at the point <math>x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]</math> ===== | ===== (i) Characterize feasible directions at the point <math>x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]</math> ===== | ||
| − | <span class="texhtml"> | + | <span class="texhtml"</span><math>d\in\Re_{2}, d\neq0</math> is a feasible direction at <math>x^{*}</math>, if <math>\exists\alpha_{0}</math> that <math>\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right] + \alpha\left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \in\Omega \right]</math> for all <math>0\leq\alpha\leq\alpha_{0}</math><br> |
| − | | + | <math>\because x_{1}\geq0, x_{2}\geq0</math> |
| − | <br> | + | <math>\therefore d= |
| + | \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? ===== | ===== (ii) Write down the second-order necessary condition for . Does the point satisfy this condition? ===== | ||
Revision as of 17:21, 21 June 2012
Contents
ECE QE AC-3 August 2011 Solusion
1. (20 pts) Consider the optimization problem,
maximize $ -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2} $
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] $
<span class="texhtml"</span>$ d\in\Re_{2}, d\neq0 $ is a feasible direction at $ x^{*} $, if $ \exists\alpha_{0} $ that $ \left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right] + \alpha\left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \in\Omega \right] $ for all $ 0\leq\alpha\leq\alpha_{0} $
$ \because x_{1}\geq0, x_{2}\geq0 $
$ \therefore d= \left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right], d_{1}\in\Re_{2}, d_{2}\neq0 $
