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\title{Introduction to Optimization \\
Math  5770-001,  Math 6640-001,  ME EN 6025-001 
\\Midterm exam 
\\ Search Methods for Unconstrained Optimization}
\author{Your name and class section ..................................................................}
\date{Due Day: Thursday,  October 17.}
\maketitle
Test the following methods
\begin{enumerate}
\item Conjugate gradient (5.1) or, for nonquadratic functions,  (5.2)
\item Quasi-Newton BFGS (6.1)
\item Quasi-Newton BFGS (6.2)
\item Nedler-Mead (9.5)
\end{enumerate}
Write the programs using the discussed algorithms. For non-quadratic functions, use any line search method (see Chapter 4), explain.
Find minima of the following functions

I. Quadratic A
\begin{equation}
Q_1(x)=  x^T A x - b^T x, \quad A=\left(\begin{array}{rrrr}
4 & 2 & 2& 4 \\ 2& 6& 10& -1 \\ 2& 10& 40& 3 \\ 4& -1& 3& 10
\end{array}
\right), ~~ 
b=\left(\begin{array}{r}1 \\ 0 \\ -2 \\ 3
\end{array}
\right)
\end{equation}
Starting point $x_0= (0, 0, -1, 3)$
Compute minimum of $Q_1$ by Maple or Matlab and plot distance from the minimum to the approximations at five iterations. Discuss.

~


 Quadratic B
\begin{equation}
Q_2(x) =  \sum_{i=1}^6  \sum_{j=1}^6  x_i x_j + \sum_{i=1}^6 x_i^2 - \sum_{i=1}^6 \left(\frac{x_i}{i}\right)
\end{equation}
Starting point $x_0= 0$
Compute minimum of $Q_2$ by Maple or Matlab and plot  distance from the minimum to the approximations at seven iterations. Discuss.

~

II. Rosenblock function (see problem 2.1)
\begin{equation}
F_R(x)= 100(x_2-x_1^2)^2 + (1-x_1)^2
\end{equation}
 Starting point $x_0= (0, 0)$. Plot 20 iterations in the $x_1, x_2$ plane. Discuss.

~

III. Exotic functions:

a. Flat surface:
\begin{equation}
F_a(x)= \exp{\left( - x_1^2 - x_2^2 \right)}
\end{equation}
 Two starting points $x_0= (1, 1)$ and  $x_0= (-1, 1)$ . Plot 30 iterations in the $x_1, x_2$ plane. Discuss.
 
 ~
 
 b. Small Hessian 
\begin{equation}
F_b(x) = \sqrt{ 0.2 + x_1^2 + 2 x_2^2}
\end{equation}
 Starting point $x_0= (2, 2)$. Plot 20 iterations in the $x_1, x_2$ plane. Discuss. 
 
 ~
 
 {\bf Note:} At Tuesday, October 15, the class  is cancelled due to the SES Annual Technical Meeting. 
 
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