One of the research contributions made is the new-found stability curve Equation 62: MSAPSO Stability Line (MSL) hypothesis over the set of convergence curves which is defined via Equation 57: Order-2 convergence system for generic probability distributions. It is named from now on MSAPSO-Stability-Formula (MSF). This curve describes the general relation between inertia weight, social & cognition in the case of the optimal balance point between exploration and exploitation of the MSAPSO algorithm. In a dynamic algorithm either inertia weight is variated or the standard deviation of the probability distribution. Also in this case the stability curve still describes the parameter relations correctly. It is important to mention here that the stability equation is stochastic in nature and is not an 100% exact prediction of the searched parameter set. The prerequisite of the stability curve/equation is, that it requires a probability distribution which do have an average and a standard deviation. There are probability distributions such as Cauchy distribution, where the stability curve then is not a valid approach.
The proof that the stability curve is a property of the set of the convergence curves was made in the previous chapter. Also, the hypothesis was proved that order-2 convergence room collapses into the order-1 convergence room under certain conditions. This is exactly the case when randomness was taken out in the order-2 convergence analysis. This was
zero. In the next chapter it is analyzed how the variation options of the MSAPSO algorithm can look like.
4.3 Variation options of MSAPSO Stability Line
Figure 15: Variation options of MSAPSO Stability Line
As mentioned before the set of convergence curves and the resulting stability curve can be influenced via the probability distribution parameters such as average value and
options on how to influence the behavior of the MSAPSO algorithm are outlined. The first thing to mention here is, that the form of the order-2 convergence curve is fully dependent on the average value and the appropriate standard deviation of the probability distribution. In the graph, the convergence curve triggered by a uniform distribution lies in the “middle” of all convergence curves possible. Secondly when a normal distribution with N (1
2, 1 2√3 )
is used, the convergence curve form equals exactly that of the convergence curves caused by a uniform distribution.
The average value of a uniform distribution in the range of [0,1] is 1
2 and the standard
deviation of the uniform distribution equals to 1
2√3. The hypothesis is that when the different
flavors of probability distributions do own the same average value and standard deviations that the form and the equations of the corresponding convergence lines are equal. In Figure 15: Variation options of MSAPSO Stability Line it is visualized that it is an almost exact overlay of the convergence curves triggered by uniform as well as the normal distribution with N (1
2, 1
2√3 ). If Equation 59: Order-2 convergence system for MSAPSO with uniform
distribution and Equation 60: Order-2 convergence system for MSAPSO with normal distribution is recalled, it is obvious that both probability distributions do lead to the same mathematical specific convergence equation.
As the two specific probability distributions triggers the “same” order-2 convergence curve for MSAPSO and also lie in the middle of all possible convergence curves it can be concluded that these levels of convergence curve are ideal over all possible benchmarks
Figure 16: Lowered convergence and stability curve with N(0.5,0.475)
In more detail, the above figure shows how to manipulate the behavior of MSAPSO with different levels of normal distributions. In this case a normal distribution with N(0.5,0.475) was used to influence the general behavior of the MSAPSO algorithm.
First of all the broader standard deviation of the normal distribution does have an influence on the height of the triggered convergence curve. Secondly the saddle point has shifted to the left in this case. This new location of the saddle point also has a corresponding inertia
weight, social and cognition value then. In general, if the inertia weight value is lowered and left-shifted the algorithms converge “faster”. The same can be achieved when the sum of social and cognition is lowered. In this case both is true, so having a low sum of social and cognition as well as low inertia weight makes the algorithm to accelerate to find solutions more quickly.
This knowledge can be used to design algorithms in order to perform faster in unimodal problems, where benchmarks do have simple problem surfaces. Whereas in multimodal benchmark problems to fast convergence speed can be a problem as there is a higher risk to be stuck into local optima and then afterwards losing the capability to find better local optima. In general, the described behavior is useful to put the algorithm into exploitation mode.
Figure 17: Raised convergence and stability curve with N(0.5,0.075)
In the other case, which means that the standard deviation of the normal distribution with N(0.5,0.075) is decreased, then the corresponding convergence curve moves up to the upper right corner. In fact, this means that the exploration behavior of the algorithm is improved, which is useful in multimodal functions but not so useful in unimodal functions.
4.3.1 Variation with success-based inertia-weight strategy
Figure 18: Balanced convergence and stability curve with N(0.5,0.288675),UNIF
Another way to let the MSAPO algorithm self-adapt is the creation of a dynamic inertia weight strategy around the saddle point location. In this case MSAPSO uses the probability distributions with uniform and normal distribution of N (1
2, 1
2√3). It is important to mention
with regard to performance of the algorithm. This is especially true when the benchmark implies a lot of dimensions. Actually, all the options described before are being used in the real MSAPSO algorithm implementation. The method described along with N (1
2, 1
2√3) , UNIF probability distributions is the self-adaptive method used along the
regular convergence of the algorithm. When the algorithm is stuck into local optima then the self-adaptive method N (1
2, narrow value) is used to avoid premature convergence.
For the last iterations during the algorithmic run, when global optimum search stabilizes then N (1
2, broad value) is applied in order to accelerate convergence by end of the
algorithm or when in between a unimodal area is being seen by the algorithm.