ECE Ph.D. Qualifying Exam in Automatic Control (AC), Question 3, August 2011

Question

Part 1. 20 pts

$\color{blue} \text{ Consider the optimization problem, }$

$\text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}$

$\text{subject to } x_{1}\geq0, x_{2}\geq0$

$\color{blue}\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]$

$\color{blue}\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} \text{. Does the point } x^{*} \text{ satisfy this condition?}$