## 3 Methodology
\begin{array}{cc}q_{ij,\,(t+1)}&\text{if }(rand_{j}(0,1)\leq C_{r_{i},t})\text{ or }(j\ =\ j_{rand})\\ x_{ij,\,t}&\text{otherwise}\end{array} \right. \tag{6}\]

for \(i=1,2,\ldots N_{p}\) and \(j=1,2,\ldots D\). Index \(j_{rand}\in\{1,N_{p}\}\) is a randomly chosen integer that is responsible for the trial vector containing at least one component from the mutant vector. In the selection process, a greedy selection scheme for a minimization is used and it is shown in Equation (7). \[x_{i,\,(t+1)}\ =\ \left\{ \begin{array}{cc}p_{i,\,(t+1)}&\text{if }f(p_{i,\,(t+1)}) \leq f(x_{i,t})\\ x_{i,\,t}&\text{otherwise}\end{array} \right. \tag{7}\]

In each iteration \(t\), the best position that has been found by individual \(i\), \(\mathit{Pbest}_{i}=[\mathit{Pbest}_{i1},\mathit{Pbest}_{i2},\ldots,\mathit{ Pbest}_{iD}]\) and the best position that has been found by the whole swarm \(\mathit{Gbest}=[\mathit{Gbest}_{1},\mathit{Gbest}_{2},\ldots,\mathit{Gbest}_{D}]\) guide individual \(i\) to update its velocity and position by Equations (1) and (2). The FIPSaDE is effective in traversing through swarm spaces while avoiding getting stuck inside local search using the information collected from individuals' neighborhood. The individuals of each population will contain a control parameter adapted from SaDE and also a traversing parameter from FIPS in order to exploit the swarm space that is searching for the best global point, while avoiding getting trapped inside a local point. If some population suffers from slow and premature convergence or if the population becomes stagnant, the velocity parameter and position parameter from each individual will be able to help the population to evolve out into a better point. Experimental results validated the effectiveness and strength of our proposed model. ## 4 Experimental Setup

The performance of FIPSaDE is evaluated against the 25 benchmark functions from \"CEC 2005 Special Session on Real-Parameter Optimization\" (CEC 2005) [32], namely, F1-F25. The benchmark functions consist of 5 unimodal functions, 7 basic multimodal functions, 2 expanded multimodal functions and 11 composition functions. The details of the functions are described in Table 1. The number of variables for each function is represented by \(D\) and the ranges of variable search are represented by \(S\). All functions are a minimization of problems with their best minimum solutions can refer to [32]. All experiments were conducted in the Eclipse software by using the Java programming language and on a laptop featuring an Intel(R) Core (TM) i5-7200U CPU @ 2.50 GHz processor with 4 GB of RAM. In this experiment, the performance of FIPSaDE is studied by comparing the algorithm with four known algorithms, which are FIPS by [PERSON] et al. [1], DE by [PERSON] and [PERSON] [10], SaDE by [PERSON] et al. [8] and DE-PSO by [PERSON] et al. [31]. FIPS, DE and SaDE were selected as these algorithms can be considered as the parent or the ancestor of FIPSaDE. As for DE-PSO, it was selected as the algorithm is a hybrid algorithm that features a similar structure and concept as FIPSaDE. The parameters for the algorithms are shown in Tables 2 and 3. The optimization test for all algorithms is restricted to a maximum number of function evaluation (_MAX_FEs_), which are \(5\times 10^{4}\) for both \(10D\) and \(50D\) problem, and \(3\times 10^{4}\) for \(30D\) problem. A success-threshold of \(10^{-14}\) was administered for the experiments. This means the evolutionary processes are terminated if the best-fitness, \(f_{best}<f_{target}\) is reached, with \(f_{target}~{}=~{}f(x*)+10^{-14}\). Otherwise, the processes continue until they reach _MAX_FEs_. The evolutions for all models of DEs are run until they meet the stopping criterion and repeated by using the \(k\)-th seed numbers, \(k~{}=~{}1,2,\ldots,K\), with \(K\) referring to the maximum number of runs.