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| − | ECE QE AC-3 August 2011 | + | = ECE QE AC-3 August 2011 = |
| − | 1. (20 pts) Consider the optimization problem, | + | ===== 1. (20 pts) Consider the optimization problem, ===== |
| − | maximize | + | maximize <math>-x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}</math> |
| − | subject to | + | subject to <math>x_{1}\geq0, x_{2}\geq0</math> |
| − | (i) Characterize feasible directions at the point | + | ===== (i) Characterize feasible directions at the point <math>x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]</math> ===== |
| − | (ii) Write down the second-order necessary condition for . Does the point satisfy this condition? | + | <span class="texhtml">''d''</span> is a feasible direction at <math>x^{*}(d\in\Re_{2}, d\neq0)</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> |
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| + | ===== (ii) Write down the second-order necessary condition for . Does the point satisfy this condition? ===== | ||
Revision as of 16:16, 21 June 2012
Contents
ECE QE AC-3 August 2011
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] $
d is a feasible direction at $ x^{*}(d\in\Re_{2}, d\neq0) $, 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} $
