# A Metaheuristic Framework with Experience Reuse for Dynamic Multi-Objective Big Data Optimization
## 3 The Proposed Framework
### Problems  
The test suites in the framework contain several types of test problems, making it easy for researchers to evaluate the performance of their algorithms. Different problems could help researchers improve the robustness and generalization capability of their algorithms. The problems included in test suites are listed in Table 2.  
The test suites contain a variety of MOPs, DMOPs, and DMDDOPs. The MOPs are the classic ZDT1-ZDT4, ZDT6, UF1-UF7, and BigOpt2015 from the Big Data 2015 Competition. Among the DMOPs are FDA1-FDA3 and DF1-DF9. Furthermore, DMDDOPs problems are developed based on some of the above problems, including big data FDA (BDFDA) problems derived from the FDA series, big data DF (BDDF) problems derived from the DF series, and Dynamic BigOpt2015 (DBigOpt2015), which is a dynamic version of BigOpt2015. In total, there are 13 DMDDOPs. For example, DBigOpt2015 is a dynamic version of BigOpt2015. BigOpt2015 is abstracted from the electroencephalographic (EEG) signal processing problem. EEG signal processing is a dynamic problem that has a high real-time requirement. The competition only intercepts fragments of data to turn them into an MOP containing a large number of variables. This big data MOP could be mathematically defined as  
\[ \begin{split}\text{min}&\mathbf{f}_{1}=\frac{1}{N \times M}\sum_{ij}(S_{ij}-S1_{ij})^{2}\\ \text{min}&\mathbf{f}_{2}=\frac{1}{N^{2}-N}\sum_{i, j\
eq i}(C_{ij}^{2})+\frac{1}{N}\sum_{i}(1-C_{ii})^{2}\end{split}  \tag{2}\] \begin{table}
\begin{tabular}{l c} \hline \hline
**Problem** & **Description** \\ \hline \multicolumn{3}{c}{**Multi-Objective Optimization Problems**} \\ \hline ZDT1-ZDT4, ZDT6 & Unconstrained MOPs \\ \hline UF1-UF7 & Unconstrained MOPs for the CEC2009 \\ \hline BigOpt2015 & MOP of Big Data 2015 Competition \\ \hline \multicolumn{3}{c}{**Dynamic Multi-Objective Optimization Problems**} \\ \hline FDA1-FDA3 & DMOPs with continuous search space \\ \hline DF1-DF9 & DMOPs for CEC2018 \\ \hline \multicolumn{3}{c}{**Dynamic Multi-Objective Big Data Optimization Problems**} \\ \hline BDFDA1-BDFDA3 & DMBDOPs derived from FDA1-FDA3 \\ \hline BDDF1-BDDF9 & DMBDOPs derived from DF1-DF9 \\ \hline DBigOpt2015 & Dynamic version of BigOpt2015 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Problems integrated into the proposed framework. Figure 2: Source code for parallelized function evaluationwith  
\[ \begin{split}\mathbf{S}&=\mathbf{S1}+\mathbf{S2}\\ \mathbf{X}&=\mathbf{A}\times\mathbf{S1}+\mathbf{A} \times\mathbf{S2}\\ \mathbf{C}&=\frac{covar(\mathbf{X},\mathbf{A} \times\mathbf{S1})}{var(\mathbf{X})\times var(\mathbf{A}\times\mathbf{S1})}\\ S1_{ij}&\in[-8,8],i=1,\dots,N,j=1,\dots,M\end{split}  \tag{3}\]  
where \(\mathbf{A}\), \(\mathbf{X}\), and \(\mathbf{S}\) are three matrices. \(\mathbf{A}\) is a linear transformation matrix with dimension \(N\times N\). Matrix \(\mathbf{X}\) and matrix \(\mathbf{S}\) separately contain \(N\) inter-dependent and independent time series. The length of each time series is \(M\). \(covar(. )\) represents the covariance matrix, \(var(. )\) is the variance, and \(\mathbf{C}\) is the matrix of Pearson correlation coefficients between \(\mathbf{X}\) and \(\mathbf{A}\times\mathbf{S1}\). The goal is to find \(\mathbf{S1}\) that is as similar as possible to \(\mathbf{S}\) and maximize the diagonal elements of \(\mathbf{C}\) while minimizing the off-diagonal elements to zeros. In DBigOpt2015, the time variable \(t\) is introduced to simulate the dynamic changes in time series data, turning the original static problem into a DMBDOP. Specifically, the original \(\mathbf{S}\) dataset is made to vary with time \(t\) based on Equation (4) to obtain different \(\mathbf{S}^{*}\) matrices, and at the same time, different matrices \(\mathbf{X}^{*}\) are obtained while the matrix \(\mathbf{A}\) remains unchanged. The definition of the time variable \(t\) is referenced in [2; 51]. \(n_{t}\) is the number of distinct steps in \(t\), representing the severity of the change.