Although the proposed COEA is capable of adapting to the different requirements of the MO problems, it has to be adapted for dynamic optimization. In particular, the issues of diversity and outdated archived solutions have to be addressed before the dynamic COEA is capable of dealing with the environmental variations. Section 6.3.1 describes a diversity scheme which allow diversity to be introduced while exploiting useful past information. Section 6.3.2 describes a simple temporal memory which stores outdated nondominated solution and introduces these solutions back into the archive at the appropriate moment.
6.3.1 Introducing Diversity Via Stochastic Competitors
The diversity necessary for the tracking of the dynamic PS∗t by COEA can be either be introduced explicitly through mechanisms such as random restart and hypermutation or maintained by means of niching methods and other elaborate diversity preservation schemes. Note that the third approach of using multiple populations to explore the different regions of the search space is not applicable because the application of subpopulations in COEA serves in another purpose of optimizing a specific subcomponent of the problem. Explicit generation of diversity will enable the algorithm to react faster to severe environmental changes but it is limited by failure to utilize any past information. On the other hand, the potential for information exploitation in diversity preservation schemes is attained at the expense of slower convergence rates. This is also known as the exploration-exploitation dilemma for dynamic optimization [19].
In order to solve this problem, a diversity scheme which exploits the competitive mech- anism of COEA is implemented. In every generation, a fixed number of archived solutions are re-evaluated and the current objective values are checked against previous values for discrepancies. Any environmental variation will result in the subsequent activation of the competitive mechanism, in addition to its fixed schedule. The rationale of this strategy
is to allow the algorithm to assess the potential of existing information within the various subpopulations for exploitation in the new problem landscape.
Furthermore, the competitive process provides a natural conduit in which the intro- duction of diversity into the subpopulations can be regulated. Instead of reinitialization or subjecting the entire subpopulation to hypermutation, a set of stochastic competitors is introduced together with the competitors from the other subpopulations where the ratio between the two types of competitors is given by the parameter, SCratio. The idea is to
compare the potential of new regions in the search space and past information to decide whether the subpopulation should be initialized. Latin hypercube sampling is applied to generate individuals uniformly along each dimension. In the case where the stochastic com- petitor emerges as the winner, the particular subpopulation is reinitialized in the region that the winner is sampled from. Therefore, diversity is introduced into the subpopulations only if it presents an advantage over the current information at hand.
6.3.2 Handling Outdated Archived Solutions
After an environmental change, it is unlikely that the archived solutions will remain non- dominated. If left unchecked, these solutions will keep out the true nondominated solutions at that particular time instance. Therefore appropriate measures must be taken to minimize the detrimental effects of outdated archived solutions. One simple approach is to re-evaluate all the outdated solutions and remove only the dominated solutions from the archive. Since most MOEAs are elitist in nature, this approach may have the disadvantage of misleading the optimization process with these nondominated but outdated archived solutions. Further- more, re-evaluation results in additional computation cost. Another approach is to simply discard all archived solutions. While this approach will not incur any extra computation cost and there is no risk of misguiding the evolutionary process, the information about past PFt cannot be exploited in the case where the PS∗t is cyclic in nature.
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In order to store these potentially useful information in dCOEA, an additional external population which is denoted as the temporal memory is used in conjunction with the archive. In the ideal situation, the temporal memory will be a repository of all the nondominated so- lutions prior to any environmental variation. However, in the light of limited computational resources, decision must be made on what solutions and how solutions are to be stored. On the latter issue of how outdated solutions are stored, a fixed number Rsize of the archive is
added to the temporal memory upon a landscape change. When the upper bound of the temporal memory is reached, the oldest set of Rsize outdated solutions is removed to make
room for newer solutions. As for the former issue of selecting Rsize outdated solutions, the
dCOEA stores the extreme solutions along each dimension in the objective space. In the case where Rsize is greater than the number of extreme solutions, the rest of the solutions
to be stored are randomly selected from the archive. On the other hand, if Rsize is smaller
than the number of extreme solutions, Rsizeextreme solutions will be randomly selected into
the temporal memory. Note that Rsize actually controls the tradeoff between the storage
of information across the different environmental changes and information for a particular instance of landscape change. In particular, a smaller Rsize allows for a more diverse range
of past solutions.
After the Rsize outdated archived individuals have been added to the temporal memory,
all archived solutions will be discarded. Subsequently, the temporal memory will be re- evaluated and archive updating is conducted on this external population. The computational cost incurred by this re-evaluation process is necessary to exploit any possible information regarding the current PS∗
t. However, to address the possibility that solutions updated
into the archive through this scheme may misguide the optimization process, no archived solutions will be reinserted back to the subpopulations in the generation immediately after the environmental change.
Table 6.1: Parameter setting for different algorithms
Parameter Settings
Populations Population size 100 in NSGAII, SPEA2, PAES, and IMOEA; Subpopulation size 10 in COEA and CCEA;
Archive (or secondary population) size 100. Chromosome Binary coding; 30 bits per decision variable. Selection Binary tournament selection
Crossover operator Uniform crossover Crossover rate 0.8
Mutation operator Bit-flip mutation
Mutation rate L1 for DTLZ3 where Lis the chromosome length;
1
B for FON and KUR whereB is the bit size per decision variable;
Niche Radius Dynamic sharing.