## 4 Trajectory of the numerical parameters
### Trajectory of the parameters in the [PERSON]'s convergence domain  
[PERSON] has considered a simplified case: a deterministic system (_i.e_. with \(r_{0}\)=1, \(r_{1}\)= \(r_{2}\)=1/2) composed by a single particle. In such a case, he used the theory of dynamical systems to analyse the conditions of convergence of a particle towards an equilibrium point. He has found the following convergence necessary and sufficient conditions:  
\[a<1,\ \ \ b>0,\ \ \ 2a-b+2>0, \tag{8}\]  
where  
\[a=c_{0},\ \ \ b=\frac{c_{1}+c_{2}}{2}. \tag{9}\]  
These conditions, that determine a triangular domain in the plane (\(a\), \(b\)), ensure that the magnitude of the roots of the characteristic equation  
\[\lambda^{2}-(a-b+1)\lambda+a=0 \tag{10}\]  
of the single particle dynamical system is less than one, which is the necessary and sufficient condition for being the equilibrium point an attractor. In addition, the study of the roots of eqn. (10) gives also the quality of the convergence:harmonic oscillations around the equilibrium point happen if the roots \(\lambda_{1}\) and \(\lambda_{2}\) are complex; this happens whenever \[a^{2}+b^{2}-2 ab-2a-2b+1<0;\] (11) * zigzagging around the equilibrium point happens if at least one of the roots (real or complex) has negative real part; this happens whenever \[a<0\quad\text{or}\quad a-b+1<0.\] (12)  
More precisely, we can subdivide the above mentioned triangular domain into five regions (see Fig. 2):  
* R1: region where only condition (11) is satisfied (complex roots with positive real parts): oscillatory convergence, with a decay which depends upon the point (\(a\), \(b\));
* R2: region where conditions (11) and (12) are satisfied (complex roots with negative real parts): zigzag superposed to an oscillatory convergence;
* R3: region where only conditions (12) are satisfied and the real roots are both negative: symmetric zigzagging convergence;
* R4: region where only conditions (12) are satisfied and the two real roots have opposite signs: asymmetric zigzagging convergence;
* R5: region where conditions (11) and (12) are not satisfied (two real positive roots): non-oscillatory convergence. Oscillations and/or zigzagging allows the particle well explore the domain around its equilibrium point, so a choice of the parameters determining a point (\(a\), \(b\)) that allows this kind of dynamical behaviour (regions R2 and R3) are well suited for an early stage of the computation; at a final stage, it is better to tend towards conditions allowing quick convergence (_i.e._, to tend towards regions R4 or R5). So, the update of the parameters should be determined in such a way that their evolution along iterations satisfies this criterion. Figure 2: [PERSON]’s deterministic convergence domain. Actually, each choice of the coefficients \(c_{0}\), \(c_{1}\) and \(c_{2}\) determines a point (\(a\), \(b\)) in the convergence region above (or outside it); if the values of \(c_{0}\), \(c_{1}\) and \(c_{2}\) change during iterations, the position of the point (\(a\), \(b\)) changes too, describing, with iterations, _a trajectory_ in the plane (\(a\), \(b\)). The basic idea is to change the coefficients in order to respect a criterion like the one suggested above (start from region R2 or R3 and evolve towards region R4 or R5) and following a trajectory that belongs to the convergence domain. In this way, convergence conditions are respected at each iteration but the changes of the parameters carries the computation from an initial phase when exploration is preferred to a final phase when exploitation, and the rapidity of the convergence, is preferred. This constitutes a criterion for determining the power laws defining the parameters \(c_{0}\), \(c_{1}\) and \(c_{2}\) update in ALE-PSO. The analysis of Trelea is simplified and it is based upon a deterministic model; the introduction of the random coefficients \(r_{0}\), \(r_{1}\) and \(r_{2}\) alters the results, and it is questionable if the results found for the deterministic case are still valid.