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= ECE QE AC-3 August 2011 Solusion = | = ECE QE AC-3 August 2011 Solusion = | ||
| − | + | ==== 1. (20 pts) Consider the optimization problem, ==== | |
maximize <math>-x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}</math> | maximize <math>-x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}</math> | ||
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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: ===== | |
| − | <math>\because x_{1}\geq0, x_{2}\geq0</math> | + | We need to find a direction <math>d</math>, such that <math>\exists\alpha_{0}>0,</math>, |
| + | |||
| + | ===== 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>\because \left{x\in\Omega: x_{1}\geq0, x_{2}\geq0\right}</math> | ||
<math>\therefore d= | <math>\therefore d= | ||
| − | \left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right], d_{1}\in\ | + | \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 16:32, 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} $
$ \because \left{x\in\Omega: x_{1}\geq0, x_{2}\geq0\right} $
$ \therefore d= \left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right], d_{1}\in\Re^{2}, d_{2}\neq0 $
