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\pagestyle{myheadings}
\markright{(Evolutionary Computations)}
\begin{document}

\title{An Introduction to Evolutionary Computations \\
       {\large (14 April 2003 --- one day in winter 2004)}}
\author{Akira Imada\\
        \hspace*{1in}\\
        Brest State Technical University \\
        e-mail: akira@bstu.by \\
        \hspace*{1in}\\
        \hspace*{1in}\\
        \hspace*{1in}\\
last modified on:
}
\maketitle

\begin{LARGE}

%---------------------------------------------------------------------
\clearpage
\begin{large}
\begin{center} {\bf Bibliography} \end{center}
\hspace*{1in}\\
\noindent
The followings are 
the most popular, frequently cited
and hopefully still available books in the field of
Evolutionary Computations (ECs).\\

\begin{itemize}
\item[(1)]
Goldberg D.E. (1989)
"Genetic Algorithm in Search, Optimization and Machine 
Learning", Addison-Wesley.

\begin{itemize}
\item[$\cdot$] 
Though it covers only Genetic Algorithm (GA), we might 
say it was this book that made Evolutionary Computations 
so popular nowadays.
\end{itemize}

\item[(2)]
Fogel D.B. (1995)
"Evolutionary Computation: Toward a New Philosophy of Machine 
Intelligence", IEEE Press.

\begin{itemize}
\item[$\cdot$] 
This book covers all the main Evolutionary Computations, that is,
Genetic Algorithm, Evolution Strategy (ES) and Evolutionary
Programming (EP) but emphasis is on EP.
\end{itemize}

\item[(3)]
Back T (1996)
"Evolutionary Algorithms in Theory and Practice", 
Oxford University Press.

\begin{itemize}
\item[$\cdot$] 
This also covers GA, ES, and EP but emphasis is on ES.
Description is a little more mathematical than the previous two.
\end{itemize}

\item[(4)]
Michalevwicz Z. (1994)
"Genetic Algorithms + Data Structures = Evolutionary Programs",
Springer.

\begin{itemize}
\item[$\cdot$] 
This again covers only GA, but from a different aspect than 
the first one. Also this used to be a very popular book among 
the community.
\end{itemize}

\item[(5)]
Koza J.R.
"Genetic Programming: On the Programming of Computers by Means
of Natural Selection", MIT Press.

\begin{itemize}
\item[$\cdot$] 
This book explains only Genetic Programming (GP), that is evolution
of tree structures, while previous four treat binary or real number
of string structures. This book includes a lot of examples of 
its application as well as source codes in LISP language.
\end{itemize}
\end{itemize}
\end{large}

%---------------------------------------------------------------------
\clearpage
\noindent
$\Box$ {\bf Contents:}
\begin{Large}
\begin{itemize}
\item  Introduction: \\ 
       \hspace*{.4in}---  What are Evolutionary Computations, what for and how?
   \begin{itemize}
   \item[$\star$] A Simple Example: Feed-forward Neural Networks.
      \begin{itemize}
      \item[$\cdot$] Neural Network as a black box: 
         \begin{itemize}
         \item[] `But how we determine its weights?'
         \end{itemize}
      \end{itemize}
   \item[$\star$] The simplest Neural Networks.
      \begin{itemize}
      \item[$\cdot$] AND, OR.
      \item[$\cdot$] XOR: still simple enough, 
         \begin{itemize}
         \item[] `but can we determine even only 6 weights by hand?'
         \end{itemize}
      \end{itemize}
   \end{itemize}
\item Again, what are ECs?  --- A little more in detail.
   \begin{itemize}
   \item[$\star$] All what we should design.
      \begin{itemize}
      \item[$\cdot$] How we represent the problem by chromosomes?
      \item[$\cdot$] How do we evaluate fitness?
      \end{itemize}
   \item[$\star$] Various schemes of selection.
      \begin{itemize}
      \item[$\cdot$] Truncating, Roulette-wheel, and Tournament Selection.
      \end{itemize}
   \item[$\star$] Various schemes of crossover.
      \begin{itemize}
      \item[$\cdot$] One-point, Multi-point, and Uniform Crossover.
      \end{itemize}
   \end{itemize}
\item Combinatorial Optimization Problem.
   \begin{itemize}
   \item[$\star$] What is Combinatorial Optimization?
   \item[$\star$] NP-complete.
   \item[$\star$] Examples:
      \begin{itemize}
      \item[$\cdot$] Knapsack Problem.
      \item[$\cdot$] Traveling Salesperson Problem (TSP).
      \end{itemize}
   \end{itemize}
\end{itemize}
\end{Large}

%%-------------------------------------------------------------------------------------------
\clearpage
\hspace*{1in}\\
\begin{center}
{\bf
The 1st day: \\
\hspace*{1in}\\
Monday, 14 April 2003\\
(13:20 -- 14:40)}

\vspace{1in}
\noindent
Today's Keywords:\\
\hspace*{1in}\\
{\it 
chromosome,
gene,
selection,
fitness,
recombination,
crossover,
mutation,
population,
generation}
\end{center}

%%-------------------------------------------------------------------------------------------
\clearpage
\noindent
$\Box$ {\bf What are Evolutionary Computations?}\\
\hspace*{1.2in} --- Then what for, and how? \\

\begin{itemize}
\item[](What?)
\item Algorithms inspired by Biological Evolution, i.e.,
   \begin{itemize}
   \item[]`Survival of the Fittest.'
   \end{itemize}
\item[]
\item[](What for?)
\item To obtain  (Near-optimum) solution(s) to
   \begin{itemize}
   \item[$\cdot$] not-solvable-analytically
   \item[]        \hspace*{1in}and/or
   \item[$\cdot$] very difficult problems
   \end{itemize}
\item[]
\item[] (But how?)
\item All what we need to design are?
   \begin{itemize}
   \item[$\cdot$] How we represent the problem by chromosomes?
   \item[$\cdot$] How do we evaluate fitness?
   \end{itemize}
\end{itemize}


%%-------------------------------------------------------------------------------------------
\clearpage
\noindent
$\Box$ {\bf How we evolve candidate solutions?} \\
\hspace*{1.2in} --- From stupid solutions to an excellent one! \\

\begin{center}
%Determine how we represent candidate solutions ({\it phenotype}) \\
%as a {\it chromosomes} candidate solutions ({\it phenotype}) \\
Design the form of {\it chromosome}\footnote{The name is in the sense that our chromosome
give instruction or is interpreted on how to build a feasible phenotype like
biological chromosome when phenotype is created.}{\it (genotypes/individuals)} \\
so that it respresents a candidate solution ({\it phenotype}). \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
Create a {\it population} of random chromosomes \\
to construct the 1st {\it generation}.\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
Evaluate {\it fitness} (how good they are?)  of each of these individuals. \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\it Select} two chromosomes according to the fitness such that\\
``the higher the fitness the more likely to be selected.'' \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\it Recombine} the two chromosomes to produce \\ one child chromosome 
by {\it crossover} and {\it mutation}. \\
\hspace*{1in}\\
$\Downarrow$\\
\clearpage
\hfill (cont'd)\\
$\Downarrow$\\
\hspace*{1in}\\
Create the next generation by repeating \\ selection \& recombination. \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
Thus, we expect better individuals \\ from generation to generation. \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
The cycle of reconstructing the new population with better fitness individuals
and restarting the search is repeated 
until one of the global optima is found or 
a set maximum number of generation has been reached.
                     
                     
\end{center} 
%%\end{Large}

%%-------------------------------------------------------------------------------------------
\clearpage
\noindent
{\bf $\Box$ A Simple Example} \\
\hspace*{0.4in} --- Evolution of Weights of Feed-forward Neural Networks.\\

\begin{itemize}
\item NN as a black box.
   \begin{itemize}
   \item[$\star$] E.g. for Pattern Recognition.
   \end{itemize}
\end{itemize}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=feed-f-nn.eps,width=6cm,height=4cm}
 \end{center}
\end{figure}
\begin{itemize}
\item But how we determine its weights?
   \begin{itemize}
   \item[$\star$] A proposal is by using ECs.
   \end{itemize}
\end{itemize}

%%-------------------------------------------------------------------------------------------
\clearpage
\noindent
{\bf $\Box$ $J_{ij}$ by Genetic Algorithm?} \\

\begin{itemize}
\item[1.] Represent a series of $J_{ij}$ as a {\it population} of strings ({\it chromosome}).
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=chromosome.eps,width=15cm,height=1cm}
 \end{center}
\end{figure}
\item[2.] Define a {fitness} evaluation. \\ (How good is each individual?)
\item[3.] Generate an initial {\it population} at random.
\item[4.] Evolve them with
   \begin{itemize}
   \item {\it Selection}
   \item {\it Crossover}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=crossover.eps,width=15cm,height=4cm}
 \end{center}
\end{figure}
   \item {\it Mutation} \footnote{A commonly used mutation rate is 
                                  one over the length of chromosome.}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=mutation.eps,width=15cm,height=2cm}
 \end{center}
\end{figure}
   \end{itemize}
\item[5.] Better Solutions from {\it generation} to {\it generation.}
\end{itemize}

%%-------------------------------------------------------------------------------------------
\clearpage
\hspace*{1in}\\
\begin{center}
{\bf
The 2nd day: \\
\hspace*{1in}\\
Monday, 21 April 2003\\
(13:20 -- 14:40)}

\vspace{1in}
\noindent
Today's Keywords:\\
\hspace*{1in}\\
{\it 
AND, OR, XOR, neuron, synapse, weight, transfer function,
threshold, neural networks, \\ input/hidden/output layer}
\end{center}

%%-------------------------------------------------------------------------------------------
\clearpage
\noindent
{\bf $\Box$ Probably a Simplest Examples.}

\begin{itemize}
\item Output $Y$ of the neuron which receives weighted-sum of the signals $X_i$
      from other $N$ neurons is:
\begin{displaymath}
Y = \mbox{sgn} (\sum_{i=1}^N w_{i}X_i - \theta)
\end{displaymath}
\item[] where $\mbox{sgn}(x) = 1$ if $x \ge 0$ and 0 otherwise, and $w_{i}$ and $\theta$
        are called {\it weight} and {\it threshold}, respectively. \\
\item NN to solve AND \& OR. \\
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=and.eps,width=10cm,height=5cm}
 \end{center}
\end{figure}
\item[$\cdot$] Well, how about NN to solve XOR?
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=xor.eps,width=12cm,height=6cm}
 \end{center}
\end{figure}
\end{itemize}

\newtheorem{exercise}{Exercise}
\begin{exercise}
Obtain six weights values so that the above NN function as XOR.
\end{exercise}
\begin{exercise}
Create Pseudo code for EC to obtain the six weights above.
\end{exercise}

%%-------------------------------------------------------------------------------------------
\clearpage
\hspace*{1in}\\
\begin{center}
{\bf
The 3rd day: \\
\hspace*{1in}\\
Monday, 28 April 2003\\
(13:20 -- 14:40)}

\vspace{1in}
\noindent
Today's Keywords:\\
\hspace*{1in}\\
{\it Trancate/Roulette-Wheel(Fitness-proportionate)/Tornament Selection \\
one-point/two-point/multipoint Crossover, uniform-crossover 
}
\end{center}

%%-------------------------------------------------------------------------------------------
\clearpage
\hspace*{1in}\\
\begin{center}
{\bf
The 4th day: \\
\hspace*{1in}\\
Monday, 19 May 2003\\
(13:20 -- 14:40)}

\vspace{1in}
\noindent
Today's Keywords:\\
\hspace*{1in}\\
{\it Combinatorial Optimization Problem \\
Polynomial time, Combinatorial explosion,
NP-complete, Knap-sack-problem
}
\end{center}

%% ---------------------------------------------------------------------------------------------
\clearpage
\noindent
$\Box$ {\bf Combinatorial Optimization Problem.}\\

\begin{center}
When the problem is of size $N$ \footnote{In Traveling Salesperson Problem (TSP)
the size is the number of city to which a salesperson must visit in a tour.}\\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
the number of possible candidate solutions is typically \footnote{In TSP the number of 
possible candidate solutions is $(N-1)!$.}\\
\hspace*{1in}\\
$N!$, $N^N$, $e^N$, $\cdots$ etc\\
\hspace*{1in}\\
(called {\it combinatorial explosion}) \\
(Imagine how huge is the solution space for a large $N$.)\\
(See also the graph of next page.)

\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
If we want the one that minimizes the cost function \\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
we call these problems \\
\hspace*{1in}\\
{\it Combinatorial Optimization Problems.}
\end{center}

%% ---------------------------------------------------------------------------------------------
\clearpage
\noindent
$\Box$ {\bf NP-Complete.}\\
\begin{center}
If there exists an algoithm that solves the problem 
in a time that grows only polynomially (or slower) w.r.t. N \\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
then it is said to be polynomial class: P.
\end{center}

\hspace*{1in}\\
\begin{itemize}
\item See examples of how the number grows w.r.t. $N$.
\end{itemize}
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=np-hard.eps,width=16cm,height=13cm}
 \end{center}
\end{figure}

%%-------------------------------------------------------------------------------------------
\clearpage
\hfill (cont'd)\\
\begin{center}
If one can verify in polynomial time \\ whether any guess of the solution is right or not\\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
we call it non-deterministic  polynomial class: {\it NP}.

\hspace*{1in}\\
\underline{\hspace*{5in}}\\
\hspace*{1in}\\

If it's impossible to verify in polynomial time \\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
we call it {\it NP-hard}.

\hspace*{1in}\\
\underline{\hspace*{5in}}\\
\hspace*{1in}\\

Then \\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
{\it NP} + {\it NP-hard} = {\it NP-complete} \footnote{If one could find 
a deterministic algorithm that solves one NP-complete problem in polynomial time  
then all other NP problems could be solved in polynomial time.}
\end{center}

%%-------------------------------------------------------------------------------------------
\clearpage
\hfill (cont'd)\\
$\bullet$ Examples:
\begin{itemize}
\item[$\star$] 0-1 Knapsack Problem
   \begin{itemize}
   \item[$\cdot$] When  $n$ items whose $i$-th item has weight $w_i$ and profit $p_i$, 
                  search for a binary vector 
                     \begin{displaymath}
                     \mbox{{\bf x}} = (x_1, x_2, \cdots, x_n)
                     \end{displaymath}
                   that maximizes  
                      \begin{displaymath}
                      \sum_{i=1}^n {x_ip_i}.
                      \end{displaymath}
                   such that 
                      \begin{displaymath}
                      \sum_{i=1}^n {x_iw_i} \le C
                      \end{displaymath} 
                   where $C$ is capacity of the knapsack. \\
   \item[] \begin{center} \underline{\hspace*{4.5in}} \\ \end{center}
   \item[]
   \item[$\star$] Assume {\bf x} be chromosome of our EC. \\
   \item[] \begin{center} \underline{\hspace*{4.5in}} \\ \end{center} 
   \item[]
   \item[$\star$] We sort all items in decreasing order of $p_i/w_i$ then we take
                  $i$-th item iff $i$-th gene of chromosome {\bf x} is 1 
                  until knapsack is full or there are no items left.  
   \end{itemize}

%%-------------------------------------------------------------------------------------------
\clearpage
\hspace*{1in}\\
\begin{center}
{\bf
The 5th day: \\
\hspace*{1in}\\
Monday, 26 May 2003\\
(13:20 -- 14:40)}

\vspace{1in}
\noindent
Today's Keywords:\\
\hspace*{1in}\\
{\it Traveling Salesperson Problem (TSP)}
\end{center}

\clearpage
\hfill (cont'd)
\item[$\star$] Traveling Salesman Problem \\
   \begin{itemize}
   \item[$\cdot$] A salesman must visit a number of cities, and then return home.
                  In which order should the cities be visited to minimize 
                  the distance traveled? \\
   \item[$\cdot$] If we represent a candidate solution with a list of cities 
                  to be visited in order, the results of crossover and mutation
                  are feasible? \\
   \item[$\cdot$] In order for the result of crossover and mutation to be feasible
                  what representations are possible?
   \end{itemize}
\end{itemize}

\clearpage
\hfill (cont'd)\\
An example:
\begin{figure}[h]
\begin{center}
\epsfile{file=tsp.eps,width=6cm,height=6cm}
\end{center}
\end{figure}

\hspace*{1in}\\
\noindent
Chromosome is to be designed so that it allows us to form a tour
by following the procedure:\\

\begin{itemize}
\item[]
   \begin{itemize}
   \item[]
      \begin{itemize}
      \item[Step-1.] Set $i=1$.
      \item[Step-2.] If $i$-th gene is $n$ then $n$-th city in the list is the city 
                     to be currently visited.
      \item[Step-3.] Remove the city from the list.
      \item[Step-4.] Set $i=i+1$ and repeat Step-2 to Step-4 while $i \le n$.
      \end{itemize}
   \end{itemize}
\end{itemize} 
\hspace*{1in}\\
\noindent
For example, when the list of cities is
\begin{center} \{A, B, C, D, E, F, G, H, I \} \end{center} 
\noindent
chromosome: $(1 1 2 1 4 1 3 1 1)$ is the tour:
\begin{center} A-B-D-C-H-E-I-F-G. \end{center}

\begin{exercise}
Try some one-point crossover on two parents $(1 1 2 1 4 1 3 1 1)$ and $(5 1 5 5 5 3 3 2 1)$ 
\end{exercise}

\clearpage
\hfill (cont'd) \\
\begin{itemize}
\item[$\star$] Then how we design mutation?
   \begin{itemize}
   \item[$\cdot$] How about specifying two points at random and reverse the gene order?
   \end{itemize}
\item[]
\item[$\star$] When to stop a run?
   \begin{itemize}
   \item[$\cdot$] If we don't know the optimum, then observe the evolution of fitness.
      \begin{itemize}
      \item[-] We can expect the run converges to the optimum or near-optimum.
      \item[-] See an example bellow.
      \end{itemize}
   \end{itemize}
\end{itemize}
\begin{figure}[h]
\begin{center}
\epsfile{file=fitness-example.eps,width=9cm,height=9cm}
\end{center}
\end{figure}

%%-------------------------------------------------------------------------------------------
\clearpage
\hspace*{1in}\\
\begin{center}
{\bf
Let's open the course again! \\
\hspace*{1in}\\
The 1st day of this fall semester \\
(6th day since the last spring semester): \\
\hspace*{1in}\\
Tuesday, 03 November 2003\\
(10:00 -- 11:20)}

\vspace{1in}
\noindent
Today's Keywords:\\
\hspace*{1in}\\
{\it Commonly used testfunctions, Sphere Model, \\ Ackley's Function}
\end{center}

%% ---------------------------------------------------------------------------------------------
\clearpage
\noindent
$\Box$ {\bf Let's try EC on a testfunction.}\\

\indent $\bullet$ Commonly Used Test Functions \\
\hspace*{1in} --- to learn ``How an EC works?'' 

\subsection*{Sphere Model $F_1$:}
\begin{displaymath}
      y = \sum_{i=1}^n {x_i^2}, \hspace{.2in}
      x_i \in [-5.12, 5.12].
\end{displaymath}

\subsection*{Ackley's Function $F_9$:}

\begin{displaymath}
y = -20 \sum_{i=1}^n \exp {(-0.2 \sqrt{x_i^2 /n})} 
    - \exp{((\sum_{i=1}^n {\cos2 \pi x_i})/n)} + 20, 
\end{displaymath}
\begin{displaymath}
\hspace{5.1in} x_i \in [-30, 30].
\end{displaymath}

\begin{itemize}
\item[$\cdot$] A two dimensional example:
\begin{center}
$y = -20 \exp{(-0.2 \sqrt{x^2})} - \exp{(\cos 2 \pi x)} + 20. $
\end{center}
\end{itemize}

\begin{figure}[h]
\begin{center}
\epsfile{file=ackley.eps,width=18cm,height=9cm}
\end{center}
\end{figure}


\clearpage
\hfill (cont'd) \\
\indent $\bullet$ Other Test Functions

\subsubsection*{Rastrigin's Function $F_6$:}
\begin{displaymath}
   y = nA + \sum_{i=1}^n {(x_i^2 - A \cos(2\pi x_i))},  \hspace{.2in}
   x_i \in [-5.12-5.12].
\end{displaymath}

\subsubsection*{Schwefel's Function $F_7$:}
\begin{displaymath}
      y = \sum_{i=1}^n {(x_i \sin(\vert x_i \vert))},  \hspace{.2in}
      x_i \in [-500, 500].
\end{displaymath}
                            
\subsubsection*{Griewangk's Function $F_8$:}

\begin{displaymath}
      y = \sum_{i=1}^n {x_i^2/4000} - \prod_{i=1}^n {\cos(x_i/\sqrt{i})} + 1, \hspace{.2in}
      x_i \in [-600, 600].
\end{displaymath}

\hspace*{1in} \\
\hspace*{1in} \\
\hspace*{1in} \\
\underline{\hspace*{3in}}

\newtheorem{excersise}{Excersise}
\begin{excersise}
Choose one out of four functions above, and explore the function.
That is, draw surface of its 3-D version, 
check the location of the global minimum,
learn distribution of local minima, etc... 
\end{excersise}

\begin{excersise}
Apply Evolutionary Computation to locate the global minimum.
Try various parameters and/or selection and crossover system, for example,
try with $N=100, p_m = 0.05$, tornament selection and one-point crossover, etc. 
\end{excersise}



\end{LARGE}
\end{document}

%% =====================================================================================



\item Exploitation of Diploidy Chromosomes.
   \begin{itemize}
   \item[$\star$] Sorting Network Problem.
      \begin{itemize}
      \item[$\cdot$] What is the minimum number of comparisons?
      \item[$\cdot$] Does EC work better than human? \\
                     \hspace*{0.5in}--- A competition: Human vs EC.
      \end{itemize}
   \end{itemize}
\item Coevolution --- Predator vs Prey.
\item Evolutionary Game Theory.
   \begin{itemize}
   \item[$\star$] Prisoner's Dilemma. 
   \end{itemize}
\clearpage
\hfill (cont'd)
\item A Visualization of High-D space --- Summon Mapping by EC
   \begin{itemize}
   \item[]
      \begin{itemize}
      \item[] `EC Reveals a Geometry in High-dimensional Space.'
      \end{itemize}
   \item[$\star$] Sammon Mapping by EC.
   \end{itemize}
\item NN Revisited --- Evolving its Architecture.\\
\hspace*{0.4in}--- Can we evolve not only weights but also its architecture?
\item Lamarckian Inheritance \& Baldwin Effect \\
\hspace*{0.4in}--- Not biologically plausible but ...
   \begin{itemize}
   \item[$\star$] Lamarckian Inheritance.
      \begin{itemize}
      \item[] `Why giraffe's neck is so long?'
      \end{itemize}
   \item[$\star$] Baldwin Effect.
      \begin{itemize}
      \item[] `Would hard effort of one's own be inherited?'
      \end{itemize}
   \end{itemize}
\item Search for Multiple Peaks Simultaneously
   \begin{itemize}
   \item[$\star$]Multi-modal Optimization.
      \begin{itemize}
      \item[] `Can we search for multiple peaks at a time?'
      \item[$\cdot$] Niching Method
         \begin{itemize}
         \item[-] Fitness sharing.
         \item[-] Crowding.
         \end{itemize}
      \item[$\cdot$] Speciation Method.
      \end{itemize} 
   \end{itemize} 
\item Multi-objectives Optimization
   \begin{itemize}
   \item[]
      \begin{itemize}
      \item[] `What if we have multiple criteria that are trade-off?'
      \end{itemize}
   \end{itemize}
\clearpage
\hfill (cont'd)
\item Variations of EC's
      \begin{itemize}
      \item[$\cdot$] GA, EP, ES and GP
      \end{itemize}
\item Commonly Used Test Functions --- to learn more about EC's: 
   \begin{itemize}
   \item[$\star$] Ten commonly used test functions.
   \item[$\star$] NK landscape: to control ruggedness of a fitness landscape.
   \item[$\star$] A problem that tries to cheat us: Royal Road Function.
   \item[$\star$] Hitch-hiker's Problem.
   \end{itemize}
\item There's no free lunch
   \begin{itemize}
   \item[] No Free Lunch Theorem.
      \begin{itemize}
      \item[] `There's no free lunch: \\
               \hspace{1in} --- Be careful when we compare two methods.'
      \end{itemize}
   \end{itemize}
\item Summary and Conclusions.
\item Related Web-pages.
