# The mutation scheme

## EPSO recombination operator - the mutation scheme

Given a population with a set of particles, the general scheme of EPSO is the following:

- REPLICATION - each particle is replicated r times
- MUTATION - each particle has its strategic parameters mutated
- REPRODUCTION - each mutated particle generates an offspring through recombination, according to the particle movement rule, described below
- EVALUATION - each offspring has its fitness evaluated
- SELECTION - by stochastic tournament or other selection procedure, the best particles survive to form a new generation, composed of a selected descendant from every individual in the previous generation

Mutation of a parameter w into w* is ruled by multiplicative Lognormal random numbers such as in **w _{i}^{*}=w_{i}[logN(0,1)]^{τ}** , multiplicative Gaussian numbers such as in

**w**or by additive Gaussian distributed random numbers such as in

_{i}^{*}=w_{i}(1+σN(0,1))**w**not reccomended.The learning parameter (

_{i}^{*}=w_{i}+σN(0,1) -**τ**or

**σ**) must be fixed externally.

The recombination rule for EPSO becomes (see Figure 1):

where:

**X _{i}^{(k)}** - location of particle i at generation k

**V _{i}^{(k)}=X_{i}^{(k)}-X_{i}^{(k-1)}** - is the velocity of particle i at generation k

**b _{i}** - best point found by particle i in its past life up to the current generation

**b _{G}** - best overall point found by the swarm of particles in their past life up to the current generation

**w _{i1}** - weight conditioning the

*inertia*term

**w _{i2}** - weight conditioning the

*memory*term

**w _{i3}** - weight conditioning the

*cooperation*or

*information exchange*term

**P** - communication factor matrix (discussed below)

Figure 1 – Illustration of the EPSO movement rule. Notice that the vector associated with the *cooperation* factor does not point exactly to the global optimum b_{g} but to a mutated location.

The symbol * indicates that the parameter will undergo mutation. In the most effective EPSO variant, not only the weights affecting the components of movement are mutated but also the global best is randomly disturbed to give

where w_{i4} is the forth strategic parameter (target weight) associated with particle i. It controls the “size” of the neighborhood of b_{G} where it is more likely to find the real global best solution.

The communication factor **P** is a diagonal matrix whose diagonal elements are 0 or 1 depending on a threshold *communication probability. *each time an element is needed, this threshold is compared against a random number Rnd() - uniform distribution - and the value of 0 or 1 is selected depending on Rnd() being smaller or greater than the fixed threshold.

Selection is modeled from the Stochastic Tornament concept: among the ofspring of each particle, one compares the best one with another particle randomly sampled, and the best is selected with probability (1 – luck), where the *luck *parameter is defined in [0,1] but is usually small. If luck = 0 we have elitist selection.