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{\Large \bf Optimization methods \\

HW  5. Due December 1 2010.}

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\begin{enumerate}
\item  Penalty method

Use the penalty function and the congugate gradient method to find
$$
\min ( x^3+ y^3 - x y) \quad
/
x<0, y< 0, x^2+ y^2 \leq  2$$

Use the quadratic penalty function absolute value penalty, and penalty function
$$p(x, A, \epsilon)=A \sqrt{x^2+ \epsilon^2}$$
where $A$ and $\epsilon$ are parameters that you may assign.

\item Solve by the Augmented Lagrangian Method
$$
\min (x + y) \quad
 /y \geq \frac{3}{1+x^2}, ~~x \geq 0.
$$

 \item Solve by the barrier method and the congugate gradient method
$$
\min (\exp(x + y^2)) \quad
 / x>2, y> 4
$$


\item Solve the quadratic program by any method
$$ \min ( x^2 + y^2 +  z)
$$
subject to
$$ x+ 2 z=3, \quad x+ y + z\geq 2
$$
\end{enumerate}
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