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\title{Optimization 2016. List of projects}
\author{Overlook of modern optimization problems and methods}
\date{}
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Research the following topics, using online sources. Write an short (2-4 pages) report explaining the principle of the approach and illustrate it with an example. Prepare a ten minutes class presentation.
Address the following: What kind of problems can these methods solve? 


\begin{enumerate}
\item Discrete optimization. Traveling salesman problem: statement and
  methods. 
(You may use \verb|http://www.csd.uoc.gr/~hy583/papers/ch11.pdf)|
\item Pareto optimization: Concept of Pareto efficiency, Pareto front. Compare with Lagrange multipliers techniique
\item Genetic algorithms: 	Initialization, Selection, Genetic operators, Termination, The building block hypothesis
\item Stimulated annealing. (You could use examples from \\ http://csg.sph.umich.edu/abecasis/class/2006/615.19.pdf and\\  http://www.theprojectspot.com/tutorial-post/simulated-annealing-algorithm-for-beginners/6)
\item Stochastic algorithms. Use https://arxiv.org/pdf/0704.3780.pdf.


 \item Find minimum of $f(x,y,z)$ for any method:


$$
f=|y|^{3/2} |x|^{2/3}\left(1.5-\cos\sqrt{100 |x|+\frac{z}{y+2.5}}\right)-.2\cos(6x)\cos(9y)e^{-z}
$$
$$x\in [-2, 2], y\in [-2, 2], z\in [-1, 0] $$
Justify the choice of the method. 


\item Solve (numerically) a boundary value problem 
$$
(e^{x^2 u''})''= 1, ~u(0)=u'(0)=0, ~u(1)=1, u'(1) =2
$$
by the shooting method: 
(a)  solve (numerically) the above DE with the initional conditions:  $u(0)=u'(0)=0,~u{''}=a, ~u^{'''}=b$,
 and (b) choose $a$ and $b$ to minimize the function 
$$
f=(u(1)-1)^2 +(u'(1) -2)^2
$$
{\bf Comment}: You may want to repace the fourth-order DE with an equivalent system
of first-order DEs.

\end{enumerate}
Please indicate your preferences. The projects will be assigned in the class. 


You can also use the project you are working on for the dergee, if you
have one. 


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