# Non-Stationary Stochastic Global Optimization Algorithms
## 4 Results
### Recombination Scheme Formalization
These methods can be characterized by the kernels \(K_{FlatC1}:(\Omega^{n}\times\Omega^{n})\times\Sigma^{\otimes n}\rightarrow[1,0]\), \(K_{ArithmeticC1}:(\Omega^{n}\times\Omega^{n})\times(\Sigma^{\otimes n} \times\Sigma^{\otimes n})\rightarrow[1,0]\) where the definitions are:

\[K_{FlatC1}=\oplus_{i=1}^{n}[\mathbf{1}_{+_{1}}\circ[\pi_{i}(A)\otimes\pi_{i}(B)]]\]

\[K_{\mathrm{SPC}_{Q_{r}}}=\pi_{\{1\ldots d\}}(B)\oplus\pi_{\{d+1\ldots n\}}(A)\]0

**Proof. **\(K_{ArithmeticC1}\) is defined in terms of projection kernel, join-kernels and kernel composition. \(\square\)

FlatC1(\(A,B\))
```
1:\(Q=\{\}\)
2:for\(i=1\) to \(n\)do
3:\(\alpha\sim U[0,1]\)
4:\(Q_{l}=\alpha*\pi_{i}(A)+(1-\alpha)*\pi_{i}(B)\)
5:return(\(Q\))
```
**Algorithm 6** Flat and Arithmetic Crossover-ShuffleC1, ArithmeticC1
```
1:\(Q=\{\}\)
2:for\(i=1\) to \(n\)do
3:\(\alpha\sim U[0,1]\)
4:\(Q_{l}=\alpha*\pi_{i}(A)+(1-\alpha)*\pi_{i}(B)\)
5:return(\(Q\))
```

```
1:\(Q=\{\}\)
2:for\(i=1\) to \(n\)do
3:\(x_{min}=min(\pi_{i}(A),\pi_{i}(B))\)
4:\(x_{max}=max(\pi_{i}(A),\pi_{i}(B))\)
5:\(dx=x_{max}-x_{min}\)
6:\(\beta\sim[0,1]\)
7:\(Q_{i}=\beta*(x_{min}-\alpha*dx)+(1-\beta)*(x_{max}+\alpha*dx)\)
8:return(\(Q\))
```
**Algorithm 7** Blended Crossover-BlendedC1
7. A **Linear Crossover** scheme, (LinearC1 : \(\Omega^{n}\times\Omega^{n}\rightarrow\Omega^{n}\times\Omega^{n}\times\Omega^{n}\)). The definitions can be seen in Algorithm 8. These methods can be characterized by the kernels \(K_{LinearC1}\): \((\Omega^{n}\times\Omega^{n})\times(\Sigma^{\otimes n}\times\Sigma^{\otimes n} \times\Sigma^{\otimes n})\rightarrow[1,0]\), defined by:

\[K_{\mathrm{SPC}_{Q_{r}}}=\pi_{\{1\ldots d\}}(B)\oplus\pi_{\{d+1\ldots n\}}(A)\]1

**Proof. **\(K_{LinearC1}\) is defined in terms of kernel, join-kernels, kernel composition and addition kernel. \(\Box\)
```
1:\(Q_{1}=\left\{\right\}\)
2:\(Q_{2}=\left\{\right\}\)
3:\(Q_{3}=\left\{\right\}\)
4:for\(i=1\) to \(n\)do
5:\(Q_{1_{i}}=(1/2)*\pi_{i}(A)+(1/2)*\pi_{i}(B)\)
6:\(Q_{2_{i}}=(3/2)*\pi_{i}(A)-(1/2)*\pi_{i}(B)\)
7:\(Q_{3_{i}}=(-1/2)*\pi_{i}(A)+(3/2)*\pi_{i}(B)\)
8:return(\(Q_{1},Q_{2},Q_{3}\))
```
**Algorithm 8** Linear Crossover-LinearC1

### Simulated Annealing (sa)

#### 4.3.1 Concept

The Simulated Annealing algorithm (sa) considers the idea behind the process of heating and cooling a material to recrystallize it, see Algorithm 9. When the temperature decreases, the material settles into a more ordered state, and the state into which they settle is not always the same. This state tends to have low energy compared when the material is in the presence of high temperature ([25]). If we consider energy as a cost function, we can use this approach to minimize cost functions. Therefore, SA is a stochastic algorithm that works with a single-individual that generates a single candidate-solution \(x\) (parent) and sets a high temperature to explore the search space. Then, a variation mechanism generates a new candidate-solution \(y\) (child) and measures its cost. A replacement policy, that fitness function and the temperature, picks one individual between the father and the child. Finally, a process decreases the temperature looking for each new solution having less energy. Clearly, the replacement policy in Algorithm 9 (lines 6,..., 11) is not elitist. This allows sa to expand the search but can lead to the loss of some good candidate-solutions. In practice, it is normal to keep track of the best solution found so far [25]. If this is done, the replacement policy is an elitist one. ```
**Algorithm 9** Simulated Annealing [25]