The variable table

Assume now that you have chosen a mixed bisection in which the bisection is applied on $n>1$ variables. The procedure will choose the bisected variables using, for example, the smear function. But in some cases it may be interesting to guide the bisection: for example if we know that subsets of the variables have a strong influence on the extremal value of the optimum function it may be interesting to indicate that as soon as the smear function has led to bisecting one variable in a given subset it may be good to bisect also the other variables in the subset. For example consider the following functions:

$$\begin{eqnarray*} &&f_i = u_x +a\cos\theta_i+b\cos(\theta_i+\psi_i) -x_i\\ &&g_i = u_y +a\sin\theta_i+b\sin(\theta_i+\psi_i) -y_i \end{eqnarray*}$$

where $u_x, u_y, a, b, \theta_i, \psi_i$ are unknowns and $x_i, y_i$ are given. Consider now the optimum function $F$:

$$F = \sum_{i =1}^{i =10} f^2_i+g^2_i$$

which has 24 unknowns. But clearly each subset $(\theta_i, \psi_i)$ has a strong influence on the minimum of $F$. Hence if one of the $\theta_i$ is bisected it may be interesting to bisect also $\psi_i$. This may be done by setting the flag ALIAS_Guide_Bisection to 1 and using the variables table: for a problem with $m$ unknowns the variables table $V$ is an array of size $m \times m$ and a 1 in $V(i,j)$ indicates that if the variable $i$ is bisected then the variable $j$ should be also bisected. In ALIAS the variables table is implemented under the name ALIAS_Bisecting_Table. It is the responsibility of the user to clear this array and update it as in the following example:

 
	Resize(ALIAS_Bisecting_Table,24,24);
	Clear(ALIAS_Bisecting_Table);
	ALIAS_Bisecting_Table(1,2)=1;
	ALIAS_Bisecting_Table(2,1)=1;
	ALIAS_Bisecting_Table(3,4)=1;
	ALIAS_Bisecting_Table(4,3)=1;