Self-adaptation

Real-encoded GAs can be thought of as being similar to ESs, depending on the details of the operators, where ESs are able to adapt their control parameter (or strategy parameter as it was called). The GA described in this work adopts this extra capability to adapt the control parameters but in a self-adaptive manner.

The mechanism used by ESs originally was not self-adaptive as they used a meta-mutation rate to control the rate of change of the mutation rate control itself. The term self-adaptive used here is meant in the sense of that coined by

Eiben et al. (2006b), to indicate control parameters of the GA that are encoded in the chromosome along with the problem definition parameters applying to the objective functions (the main parameters), and that these control parameters are subject to change along with themain parameters due to mutation and crossover. This is different from a purelyadaptive control parameter strategy as in that case the change is instigated algorithmically by some feedback at the higher level of the GA rather than the lower level of each chromosome/solution in the population. The deterministic approach is rule-based and is not considered adaptive.

B¨ack et al.(2000) used an initial random mutation rate in which the mutation rate of each solution was initialised to a random number in the range (0.001, 0.25), however they suspected that this randomness slowed down convergence to some extent. Problems can be highly sensitive to the mutation rate, so starting off from ones which are ‘far’ from the ideal would indeed likely inhibit progress.

Eiben et al. (2006a) showed how population size and tournament selection size can be made to be self-adaptive, although in the former case to the detriment of performance of the optimization. Nonetheless, the latter case was shown to improve performance, and the method by which a parameter whose context is the population can be set through the aggregation of its representation at the individuals within the population, can be extended to other parameters having the same high-level context. However, the above work only uses mutation to affect each self-adapting parameter gene, rather than including the parameter genes in

the crossover of the chromosome as a whole, and the model used is a steady-state GA (SSGA) with relatively low replacement strategy rather than a generational one (GGA). This work uses a fixed tournament size in order to restrict all the self- adaptation occurring to the level of the individual, rather than by aggregation, since this is the focus of the work.

Smith and Fogarty(1996) Discuss their steady-state GA having adaptive mu- tation, in which mutation first occurs to the gene encoding the mutation rate and then the new mutation rate is applied to themain genome. This is similar to the (real-encoded) steady-state ESs described by Beyer and Schwefel (2002).

Zhang and Sanderson (2008), (Zhang and Sanderson, 2007) describe differ- ential evolution (DE) algorithms that use self-adaptation, including their multi- objective (MO) JADE2 and JADE algorithms, that generate new values for muta- tion factors and crossover probabilities based on probability distributions governed by self-adaptingmeans. DEs (Storn and Price, 1997) are similar to GAs but new solutions are produced by adding the weighted difference of two population vectors to a third, to create a new donor vector which is recombined (crossover) with a target (parent) vector to produce the trial (child) vector. Differences between GAs and DEs, both algorithmic and from a performance perspective, are discussed in

Tusar and Filipic (2007). In a DE scheme, the mutation factor is a weight rather than a probability as in a GA, and notably crossover acts on whole parameters (the genes in a GA) rather than parts of parameters (Holland’s schemas).

Sareni et al. (2004) describe self-adaptation in a multi-objective genetic al- gorithm (MOGA) in which there is a self-adaptive choice between three different crossover operators for crossover, and in which mutation is self-adapted by the standard deviation of the amount of perturbation applied to a gene. Both of these mechanisms are different to the ones employed by the MOOEA in this work.

Tan et al. (2006) expounded their binary MOGA in which their mutation operator (adoptive mutation or AMO) modifies the mutation rate deterministically as a function of time. Their AMO is a modified bit flip operator, and acts upon a binary-encoded chromosome with each decision variable (DV) having the same

probability of undergoing change. This probability is set so that on average 1 DV is changed per chromosome, and the mutation rate is the probability of a bit of the DV changing if selected for change. The mutation rate is set by a deterministic function proportionally to the generation number, such that it tends to start off relatively high and diminishes over time. The mutation rate is thus a global one applying to all chromosomes equally. Their scheme has the property that the start rate is much higher than normal (0.8) and that it is switched to a lower one (0.05 or less) suddenly rather than having a smooth decrease over time. Also of interest is their ‘enhanced exploration strategy’ (EES), which is essentially a deterministically adaptive diversity promoter that acts by creating a sub-population whose purpose is to explore lesser populated areas. The size of the sub-population is determined at each generation deterministically and is a function of the rate of progress towards convergence, where progress is measured by the ratio (epr) of new non-dominated solutions in g + 1 to the total number of non-dominated solutions in g. The number of solutions in the exploring sub-

population is set higher when epr is low and conversely lower when epr is higher. Their results showed both improved performance. The EES approach is probably applicable to real-encoded algorithms too.

Tan et al. (2009) discuss a binary MOGA in which they define an ‘adaptive variation operator’ having two elements: (1) a deterministic control to modulate the degree of exploration or exploitation; (2) an adaptive control that coordinates crossover and mutation operators. Each gene (DV) of each chromosome has its own control parameter for the rates of crossover and mutation, which are deter- ministically assigned by (1) with the intention that they change linearly over time (starting higher and decreasing). They have a scheme to modify bits relative to their position in the gene, which is specific to the binary chromosome; that bits towards the high end (most signigicant bit (MSB)) change more at the beginning and less at the end due to the size of the effect they have, and other bits change proportionally to their position. Their adaptive control (2) is predicated on the idea that mutation and crossover should not be independent operations as mu- tation can destroy valuable schema swapped in crossover (remembering that it

is a binary chromosome), and that the performance of crossover, and mutation, varies throughout the life-cycle. They suppose in general that dissimilar chromo- some structures are often more dominant in the initial stages while similar ones dominate in the later stages as fitness selection or drift direct. Thus the con- trol considers the chromosome content and only allows mutation to occur when crossover occurs and only for regions in each chromosome which are similar. Their test results showed these approaches would work well.

It seems reasonable to suppose that having two control parameters per gene is effectively extending the search space by a possibly signigicant factor, leading to the adaptive behaviour needing more time in which to improve itself. It is arguable whether there is a theoretical equivalent scheme for control (2) in a real-encoded chromosome.

Ho et al. (1999) used a binary GA for single objective optimisation in which sub-population groups adapted their mutation or crossover rates based on feedback from average fitness increase, while Li et al. (2004) investigated diversity-guided mutation and deterministically adaptive mutation and crossover rates in a binary single-objective GA. These works all found their implementations of the various adaptive methods provided advantages on mathematically based benchmark prob- lems.