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{\large\bf C\&O 367 \\ 
Assignment 5 
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{\large  Due on Tuesday, Dec. 7,~~~~   Instructor H. Wolkowicz
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\begin{enumerate}
\item
Find an optimal solution $x^*$ of the problems
\begin{enumerate}
\item
\[  \begin{array}{cccc}
       \max & x_1 + x_2  \\
\mbox{ subject to~} &  1- x_1^2 - x_2^2 &\geq &0\\
       &   x_1 - x_2^2 &\geq& 0
     \end{array}
\]
\item
\[  \begin{array}{cccc}
       \max & x_1 - (x_2 + x_3)^2  \\
\mbox{ subject to~} &  x_1 - x_2^2 - 2e^{x_3} &\geq &0\\
       & 10-  x_1 - x_3^4 &\geq& 0
     \end{array}
\]
\end{enumerate}
by the barrier method. (Compare with matlab.)
\item
Solve the problem
\begin{enumerate}
\item
\[  \begin{array}{cccc}
       \max & x_1 + x_2  \\
\mbox{ subject to~} &   x_1^2 + x_2^2 -1 &= &0
     \end{array}
\]
\item
\[  \begin{array}{cccc}
       \max & x_1 - (x_2 + x_3)^2  \\
\mbox{ subject to~} &  x_1 + x_2^2 + x_3^2 &= &0\\
         &  x_1 + x_2 + x_3 &\leq &0\\
     \end{array}
\]
\end{enumerate}
by the penalty function method.
(Compare with matlab.)
\item
\begin{enumerate}
\item
Let $A$ be an $n \times n$ nonsingular matrix. Prove that
\[   A^{-1} = \lim_{\lambda \rightarrow \infty}
  ( \frac 1{\lambda} I + A^t A)^{-1} A^t.
\]
\item
The {\em Moore-Penrose generalized inverse} of an $m \times n$ matrix $A$
is the unique $n \times m$ matrix $A^\dagger$ that satisfies the four
equations
\[ A A^\dagger A = A,~ A^\dagger A A^\dagger = A^\dagger,~
(AA^\dagger)^t = AA^\dagger,~
(A^\dagger A)^t = A^\dagger A.
\]
If $A$ is nonsingular, then $A^\dagger = A^{-1}.$ Prove that
\[   A^{\dagger} = \lim_{\lambda \rightarrow \infty}
  ( \frac 1{\lambda} I + A^t A)^{-1} A^t.
\]
\end{enumerate}
\end{enumerate}
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