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\title{Optimization methods m 5770-001 /6640-001 }
\author{HW 6}

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\begin{enumerate}
\item  Write a code for  Nelder-Mead method. Use 
  http://www.scholarpedia.org/article/Nelder-Mead\_algorithm. Test the
  code by minimizing $$f(x)= x_1^2 + x_2^2 + x_1x_2 + x_1 x_3 + x_3^2$$

\item  Test  Nelder-Mead  method, a quasi-Newton  method, and a conjugate gradient method  minimizing
the following functions: (see https://en.wikipedia.org/wiki/Test\_functions\_for\_optimization)


2a. Beale's function 
$$b(x,y)=(1.5-x+xy)^2 + (2.25 -x + xy^2)^2 + (2.625 -x+xy^3)^2 $$
2b. Rosenbrock function, n=4

$$ f(x)=\sum_{i=1}^{n-1} \left[100(x_{i+1} - x_i^2 )^2 + (x_i-1)^2
  \right]
$$ 
Compare the results obained by these methods.

\item  Using Lagrange multipliers method, find
$$
\min_{x \in R^n}\left( \sum_{i=1}^n a_i^2 \,x_i \right)~~\mbox{subject to a constraint:} ~~ \sum_{i=1}^n \frac{1}{x_i -c} = \frac{1}{b}
$$
\end{enumerate}
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