## 2 Preliminaries
### Evolutionary Optimization  
Evolutionary algorithms (EAs) [32] are stochastic search mechanisms based on natural selection notions. EAs have been applied to a broad range of problems, including search problems [52], optimization problems [53], and in many areas as in economics, engineering, biology, etc. The primary idea is to maintain a population of chromosomes, that represent valid solutions to the problem and which evolve over time through a process of competition and targeted variation. CHC [54], is a well-known evolutionary model that introduces different techniques to achieve a trade-off between exploration and exploitation; such as incest prevention, reinitialization of the search process when the population converges or the search stops making progress and the competition among parents and offsprings into the replacement process. Prototype Selection can be considered as a search problem where EAs can be applied. The search space consists of all the subsets of the training set. This can be represented by using a binary chromosome with two possible states for each gene. If the gene value is 1 then the associated instance is included in the subset, if the value is 0, this does not occur. The chromosome will have as many genes as the number of instances in the training set. EAs have been used to solve the prototype selection problem with promising results [55; 56; 57; 58; 59]. Label-specific feature selection is a very complex task. If the original set contains \(q\) features and \(c\) labels, the objective is to find a \(q\times c\) binary matrix where each \(ij\) position indicates if the individual feature \(x_{i}\) will be taken into consideration to predict the label \(d_{j}\). Many search strategies can be used to find a solution to this problem but finding the optimal subset can be a huge time-consuming task. Therefore, it is justifiable to use an evolutionary algorithm that gives us an approximate solution in an acceptable time. Several studies have successfully applied feature reduction to multi-label problems using evolutionary algorithms [60; 61; 62; 63], but as of today no such technique has been used to solve the label-specific feature learning problem. ProLSFEO-LDL: Prototype Selection and Label-Specific Feature Evolutionary Optimization for Label Distribution Learning  
ProLSFEO-LDL is an evolutionary algorithm proposal adapted to LDL specificities that combines prototype selection and label-specific feature learning. It uses the framework of CHC where the search space will be represented by a chromosome (or individual) in which we will code the prototype selection and label-specific feature as detailed in the next Section 3.1. Through the evolutionary algorithm described in Algorithm 1, we will optimize this initial solution until we reach a solution that meets our expectations. We start from a parent population \(P\) of size \(N\), randomly initialized, to generate a new population \(P^{\prime}\) obtained by crossing the individuals of the parent population. The recombination method to obtain the offsprings is detailed in the Section 3.2. Then, a survival competition is held where the best \(N\) individuals from the parent population \(P\) and the offspring population \(P^{\prime}\) are selected to compose the next generation. To determine if one individual is better than another we need to evaluate the quality of each chromosome, for this we use the fitness method described in the Section 3.3. When the population converges or the search no longer progresses, the population is reinitiated to introduce a new diversity into the search, for this purpose we use a threshold \(t\) to control when the reinitialization will take place, this step is explained in the Section 3.4. The process of evolution iterates over \(G\) generations and finally returns the best solution \(B\) found in the search. In the following sections, we explain in depth each part of the proposed method.