## 4 Applications in Optimization
### Differential Evolution (DE)
#### 4.2.1 Overspeed Protection System of a Gas Turbine  
In [58], dos Santos Coehlo considered the reliability-redundancy optimization problem of an overspeed protection system for a gas turbine. The fundamental idea behind DE is a scheme whereby it generates the trial parameter vectors. A DE population is subject to mutation, crossover, and selection operators. At each time step, DE mutates vectors by adding weighted random vector differentials to these operators. If the cost of the trial vector is better than that of the target, the target vector is replaced by trial vector in the next generation.  
Among the many different strategies possible for DE, dos Santos Coehlo implemented the \"DE/rand/1/bin\" strategy, meaning that the target vector was randomly selected and only one difference vector was used. The bin acronym indicates that recombination was controlled by a binomial decision rule. Their strategy consisted of seven steps [58]  
_Step 1: Choice of the control parameters_  
_Step 2: Initialization of population with M individuals_  
_Step 3: Evaluation of individuals_  
_Step 4: Mutation operation (or differential operation)_  
_Step 5: Recombination operation_  
_Step 6: Selection operation_  
_Step 7: Verification of the stopping criterion_  
The mutation operation in _Step 4_ is an operation that adds a vector differential to a population vector of individuals according to  
\[z_{i}(t+1)=x_{i,r_{1}}(t)+f_{m}(t)\big{[}x_{i,r_{2}}(t)-x_{i,r_{3}}(t)\big{]}, \tag{47}\]  
where \(i=1,2,\ldots,M\) is the individual's index of population; \(t\) is the generation (time), \(x_{i}(t)=\big{[}x_{i_{1}}(t),x_{i_{2}}(t),\ldots,x_{i_{n}}(t)\big{]}^{T}\) stands for the \(i-th\) individual of a population of real-valued \(N\)-dimensional vectors, \(z_{i}(t)=\big{[}z_{i_{1}}(t),z_{i_{2}}(t),\ldots,z_{i_{n}}(t)\big{]}^{T}\) stands for the \(i-th\) individual of a mutant vector, and \(r_{1}\), \(r_{2}\), and \(r_{3}\) are mutually different integers that are different from the running index \(i\), which is randomly selected with a uniform distribution from the set \(\{1,2,\ldots,i-1,i+1,\ldots N\}\). The mutation factor \(f_{m}(t)>0\) is a real parameter which controls the amplification of the difference between two individuals with indexes \(r_{2}\) and \(r_{3}\).  
It is well known that DE is particularly sensitive to its control parameters, particularly the mutation factor \(f_{m}(t)\). In this context, due to the ergodic and dynamic properties of the Lozi map variables, dos Santos Coehlo introduced Lozi map-based DE (LDE), which can be useful in mutation factor design for escaping from local optima in optimization problems. The LDE approach was employed to increase the diversity of individuals' population and prevent the premature convergence. The LDE approach is a modification of (47) provided by  
\[z_{i}(t+1)=x_{i,r_{i}}(t)+L(t)\big{[}x_{i,r_{2}}(t)-x_{i,r_{3}}(t)\big{]}, \tag{48}\]  
where \(L(t)\) is the Lozi map (3) normalized following (30) in the range \([0.1,0.9]\).  
In each case study, fifty independent runs were carried out for each of the optimization methods involving fifty different initial trial solutions for each optimization method. Moreover, the DE and LDE approaches were adopted using 2250 cost function evaluations in each run. Both the DE and the LDE optimization methods were successfully applied to estimate the parameters of the overspeed protection system of a gas turbine in a reliability-redundancy optimization problem including discrete and continuous variables. Among seven variants of each of these methods, three DE variants and LDE five variants succeeded in finding the best solution for the tested methods. The author concluded that the LDE strategies have potential to be further applied in constrained optimization problems.