problem1

 <img _fckfakelement="true" _fck_mw_math="\color{blue}\text{1. } \left( \text{20 pts} \right) \text{ Consider the optimization problem, }" src="/rhea/images/math/a/8/8/a88c02625a766bb14bcec7ab81a7e43e.png" />

               <img _fckfakelement="true" _fck_mw_math="\text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}" src="/rhea/images/math/2/e/0/2e046f41918f5380c244ca2e543746c1.png" />

               <img _fckfakelement="true" _fck_mw_math="\text{subject to } x_{1}\geq0, x_{2}\geq0" src="/rhea/images/math/7/9/1/791b98e1dff2e27113963f1df17f3d03.png" />

<img _fckfakelement="true" _fck_mw_math="\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]" src="/rhea/images/math/c/0/7/c0738584b253bfa00734c08cc8992a85.png" />

<img _fckfakelement="true" _fck_mw_math="\color{blue}\text{Solution 1:}" src="/rhea/images/math/a/c/d/acd917ee0c5101a0580012701a3e4ec3.png" />

<img _fckfakelement="true" _fck_mw_math="\text{We need to find a direction }d\text{, such that } \exists\alpha_{0}>0," src="/rhea/images/math/1/7/c/17c47e55bcfe34d7ba52d92beac10285.png" /> ...

<img _fckfakelement="true" _fck_mw_math="\color{blue}\text{Solution 2:}" src="/rhea/images/math/f/8/9/f8958531ab15db78cd7585f04c7dbb85.png" />

<img _fckfakelement="true" _fck_mw_math="d\in\Re_{2}, d\neq0 \text{ is a feasible direction at } x^{*} \text{, if } \exists \alpha_{0} \text{ 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 \text{ for all } 0\leq\alpha\leq\alpha_{0}" src="/rhea/images/math/5/1/9/51943ad435b6eaa36150e4aa075148df.png" /> 

<img _fckfakelement="true" _fck_mw_math="\because \begin{Bmatrix}x\in\Omega: x_{1}\geq0, x_{2}\geq0\end{Bmatrix}" src="/rhea/images/math/4/b/6/4b60a1b588e5fd24283c4e3acf50920f.png" />

<img _fckfakelement="true" _fck_mw_math="\therefore d= \left[ \begin{array}{c} d_{1} \\ d_{2} \end{array} \right], d_{1}\in\Re^{2}, d_{2}\neq0" src="/rhea/images/math/3/c/9/3c9cfe683d02e17b0a5fee185f045f88.png" />

<img _fckfakelement="true" _fck_mw_math="\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?}" src="/rhea/images/math/a/4/a/a4ab915376a82698de32c2ee65efdf28.png" />

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