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 $
