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\begin{document}

\title{The Polynomial Fitting Problem}
\date{}
\author{}

\maketitle

The Polynomial Fitting Problem has its roots in electronic filter design \cite{rabiner75theory}, and challenges an
optimization procedure by forcing it to find parameter values with grossly different magnitudes, something very common
in technical systems. However, the first metaheuristic used to deal with this problem have been presented in
\cite{storn95differential}. The problem lies in finding the coefficients of the following polynomial in $z$:

\begin{eqnarray}
f(z) = \sum_{j=0}^{2k} c_j \times z^j,\ \;\;\;\; k \in Z, k > 0 \nonumber
\end{eqnarray}


\noindent such that

\begin{eqnarray}
f(z) \in [-1,1], \;\;\;\; z \in [-1,1] \;\;\;\; and \nonumber \\
f(1.2) \geq T_{2k}(1.2), \;\; f(-1.2) \geq T_{2k}(-1.2) \nonumber
\end{eqnarray}

\noindent where $T_{2k}(z)$ is a Chebychev polynomial of degree $2k$. The Chebychev Polynomials are defined recursively
according to the difference \sloppy equation \mbox{$T_{n+1}(z)=2z \cdot T_n(z)-T_{n-1}(z)$}, $n \in Z$ and  $n > 0$,
with the initial conditions $T_0(z)=1$ and $T_1(z)=z$.

The solution to the Polynomial Fitting Problem consists of the coefficients of $T_{2k}(z)$. This polynomial oscillates
between -1 and 1 when its argument $z$ is between -1 and 1. Outside this region, the polynomial rises steeply in the
direction of high positive ordinate values.

If $C=(c_0, \cdots, c_{2k})$ and $f_C(z)=\sum_{j=0}^{2k}c_j \times z^j$, with $k \in Z$, $k > 0$, then the pseudocode
algorithm shown below can be used in order to transform the constraints of this problem into an objective function to
be minimized, called $f_{Cheb}$ \cite{herrera00gradual,alba03parallel}. Each parameter of this function (coefficient of
the polynomial to fit to) is in the range $[-5.12,5.12]$. The objective function value of the optimum in $f_{Cheb}(x^*)
= 0$.



%\begin{quotation}
\begin{figure}
\begin{code}
\>\>\>\texttt{Choose} $p_0,p_1, \cdots, p_{100}$ \texttt{from} [-1,1] ; \\
\>\>\>$R = 0$ ; \\
\>\>\> \texttt{For} $i = 0, \cdots, 100$ \texttt{do}                      \\
\>\>\>    \> \texttt{If} $(-1 > f_C(p_i)$ \texttt{or} $f_C(p_i) > 1)$       \\
\>\>\>    \>    \> \texttt{then} $R \leftarrow R + (1 - f_C(p_i))^2$ ;        \\

\>\>\> \texttt{If} $(f_C(1.2) - T_{2k}(1.2) < 0)$                         \\
\>\>\>    \> \texttt{then} $R \leftarrow R + (f_C(1.2) - T_{2k}(1.2))^2$ ;     \\

\>\>\> \texttt{If} $(f_C(-1.2) - T_{2k}(-1.2) < 0)$                      \\
\>\>\>    \> \texttt{then} $R \leftarrow R + (f_C(-1.2) - T_{2k}(-1.2))^2$ ;   \\

\>\>\> \texttt{Return} $R$ ; \\
\end{code}
\end{figure}

%\end{quotation}

\newpage

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