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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> ===== | ||
| − | ===== Solusion 1: ===== | + | ===== Solusion 1: ===== |
| − | We need to find a direction < | + | We need to find a direction <span class="texhtml">''d''</span>, such that <math>\exists\alpha_{0}>0,</math>, |
| − | ===== Solusion 2: ===== | + | ===== Solusion 2: ===== |
<math>d\in\Re_{2}, d\neq0</math> is a feasible direction at <span class="texhtml">''x''<sup> * </sup></span>, 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} \right] \in\Omega</math> for all <math>0\leq\alpha\leq\alpha_{0}</math><br> | <math>d\in\Re_{2}, d\neq0</math> is a feasible direction at <span class="texhtml">''x''<sup> * </sup></span>, 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} \right] \in\Omega</math> for all <math>0\leq\alpha\leq\alpha_{0}</math><br> | ||
| − | + | '''<math>\begin{Bmatrix}x\in\Omega: x_{1}\geq0, x_{2}\geq0\end{Bmatrix}</math>''' | |
| − | <math>\therefore d= | + | <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> | \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 17:33, 21 June 2012
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] $
Solusion 1:
We need to find a direction d, such that $ \exists\alpha_{0}>0, $,
Solusion 2:
$ 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} \right] \in\Omega $ for all $ 0\leq\alpha\leq\alpha_{0} $
$ \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 $
