# Data Reduction of Digital Twin Simulation Experiments Using Different Optimisation Methods
## 2 Literature Background
### Optimisation Methods
#### 2.2.9 Evolution Strategy  
Evolution strategies (ES) introduced by [PERSON] [49] are a heuristic optimisation technique based on the ideas of adaptation and evolution, a special form of the evolutionary algorithm [50; 51].  
Evolution Strategies have the following features:  
* They usually use vectors of real numbers as solution candidates. In other words, both the search and the problem space are fixed-length strings of floating-point numbers, such as the real-encoded genetic algorithms;
* Mutation and selection are the primary operators and recombination is less common;
* Mutation most often changes the solution candidate gene to a number drawn from a normal distribution;
* The values of standard deviations are governed by self-adaptation [52; 53; 54] such as covariance matrix adaptation [55; 56; 57; 58; 59];
* In all other aspects, they perform exactly as basic evolutionary algorithms.  
The algorithm can use different population strategies: (1 + 1)--the population only consists of a single individual, which is reproduced. From the elder and the offspring, the better individual will survive and form the next population; (\(\mu+1\))--the population contains \(\mu\) individuals from which one is drawn randomly. This individual is reproduced from the joint set of its offspring and the current population, the least fit individual is removed; (\(\mu+\lambda\))--using the reproduction operations, from \(\mu\) parent individuals \(\lambda\geq\mu\) offspring are created. From the joint set of offspring and parents, only the \(\mu\) fittest ones are kept; (\(\mu\), \(\lambda\))--in (\(\mu\), \(\lambda\)) evolution strategies, introduced by [PERSON] [60], again \(\lambda\geq\mu\) children are created from \(\mu\) parents. The parents are subsequently deleted, and from the \(\lambda\) offspring individuals, only the \(\mu\) fittest are retained [60; 61];  
(\(\mu/\rho\), \(\lambda\))--evolution strategies named (\(\mu/\rho\), \(\lambda\)) are basically (\(\mu\), \(\lambda\)) strategies. The additional parameter \(\rho\) is added, denoting the number of parent individuals of one offspring. As already mentioned, we normally only use mutation (\(\rho=1\)). If recombination is also used as in other evolutionary algorithms, \(\rho=2\) holds. A special case of (\(\mu/\rho\), \(\lambda\)) algorithms is the (\(\mu/\mu\), \(\lambda\)) evolution strategy [1] (\(\mu/\rho+\lambda\))--analogously to (\(\mu/\rho\), \(\lambda\))--evolution strategies,the (\(\mu/\rho+\lambda\))--evolution strategies are (\(\mu,\lambda\)) approaches where \(\rho\) denotes the number of parents of an offspring individual; (\(\mu^{\prime}\),\(\lambda^{\prime}\) (\(\mu\),\(\lambda\))\(\gamma\))--[PERSON] et al. [2; 3; 4] have developed nested evolution strategies [62], where \(\lambda^{\prime}\) offspring are created and isolated for \(\gamma\) generations from a population of the size \(\mu^{\prime}\). In each of the \(\gamma\) generations, \(\lambda\) children are created from which the fittest \(\mu\) are passed on to the next generation. After \(\gamma\) generations, the best individuals from each of the \(\gamma\) isolated solution candidates are propagated back to the top-level population, i.e., selected. Then, the cycle starts again with \(\lambda^{\prime}\) new child individuals. This nested evolution strategy can be more efficient than the other approaches when applied to complex multimodal fitness environments [36; 63; 64].  
Our implemented algorithm uses steady-state evolution where the number of successes (the offspring is better than the parent) is monitored by the relative frequency of success. An offspring is generated by the mutation of its parent using a normal distribution with defined deviations for each gene (axes of the search space--gen in genotype) which is affected by Rechenberg 1/5 th-rule. A detailed description of our implemented method of evolution strategy is described in [65].