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\title{Optimization Method 2016}
\author{HW 7 }
\date{Due November 3}
\maketitle
\begin{enumerate}
\item Solve
\begin{equation}
\min_{x\in R^2} \left( x_1^2 + x_2^2 + 2 a x_1 x_2 \right) \mbox{ subject to: } |x_1| + |x_2| \leq 1
\end{equation}
where $a$ is a parameter, $-\infty < a < \infty$
\item Find
\begin{equation}
\min_{x\in R^n} \left( x^T A \,x  \right) \mbox{ subject to: } x^T x = 1
\end{equation}
where $A$ is a symmetric positively defined $n \times n$ matrix. 
\item 
a. Solve
\begin{equation}
\min_{x\in R^3} \sum_{i=1}^3 \left( x_i^2  +x_i \right) 
\end{equation}
subject to 
\begin{equation}
x_1-x_2+ 2 x_3 =1,
\end{equation}
\begin{equation}
2 x_1 + x_2 - 3 x_3 \leq a,
\end{equation}
 where $a$ is a parameter, $-\infty < a < \infty$
 
b. Formulate and solve a dual problem. 
 
 
\end{enumerate}


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