# A Knowledge-Guided Competitive Co-Evolutionary Algorithm for Feature Selection
## 3 Knowledge-Guided Evolutionary Algorithm
### Feature Grouping Based on Correlation
In contrast, the calculation of IGD faces unique challenges in feature selection problems, which are discrete and complex, rendering the exact Pareto front nearly impossible to ascertain. Unlike in traditional benchmark test problems such as DTLZ [39] or WFG [40], this study adopts a novel approach for fairness. We aggregate the Pareto fronts collected from multiple runs of different algorithms into a unified population. This collective set is then subjected to non-dominated sorting, retaining only the foremost non-dominated front as the reference point set for IGD. The IGD metric, therefore, provides a comprehensive assessment by reflecting both the proximity of solutions to the Pareto front and their distribution. Lower IGD values signify superior algorithm performance, an inverse relationship compared to HV values. #### 5.2.1 Hypervolume

The Hypervolume indicator measures the volume covered by the members of a solution set in the objective space [37]. It quantifies both the convergence and diversity of the solutions by calculating the volume enclosed between the solutions and a reference point. Formally, the HV of a solution set \(S\) with respect to a reference point \(r=(r_{1},r_{2},...,r_{m})\) in an \(m\)-dimensional objective space is defined as the volume of the union of the hypercubes formed by each solution in \(S\) and \(r\). The reference point is typically chosen to be worse than the nadir point of the Pareto front. For normalization purposes, all objective values are scaled to the interval [0, 1], with ideal and nadir points set to (0, 0) and (1.1, 1.1), respectively. The HV calculation can be represented as follows:

\[\text{HV}(S) =\lambda\Bigg{(}\bigcup_{x\in S}[f_{1}(x),r_{1}]\times\cdots\times [f_{m}(x),r_{m}]\Bigg{)}\] \[=\lambda\Bigg{(}\bigcup_{x\in S}\prod_{k=1}^{m}[f_{k}(x),r_{k}] \Bigg{)} \tag{5}\]

where \(\lambda\) denotes the Lebesgue measure, and \(f_{i}(x)\) is the \(i\)-th objective function value of solution \(x\). The larger the HV, the better, indicating that the solution set covers a larger portion of the objective space up to the reference point. #### 5.2.2 Inverted Generational Distance

The Inverted Generational Distance (IGD) measures both the convergence and the diversity of the solutions [38]. For each point on the true Pareto front, the IGD finds the minimum distance to a point on the Pareto front obtained by the algorithm and then averages these minimum distances. A lower IGD value indicates that the algorithm's solutions are both closer to the true Pareto front and more evenly distributed. The specific formula for calculating IGD is as follows:\[IGD =\frac{\sum_{j=1}^{|P^{*}|}d_{\min}(x_{j}^{*},P)}{|P^{*}|}, \tag{6}\] \[d_{\min}(x_{j}^{*},P) =\min\Biggl{\{}\sqrt{\sum_{k=1}^{m}(f_{k}(x_{i})-f_{k}(x_{j}^{*}))^ {2}}\Biggr{\}},\] \[i=1,2,\ldots,|P|.\]

where \(P\) is the set of Pareto optimal solutions obtained by the multi-objective evolutionary algorithm, \(P^{*}\) is the set of true Pareto optimal solutions, \(|P|\) is the number of obtained optimal solutions, \(|P^{*}|\) is the number of true optimal solutions, \(x_{i}\) represents the \(i\)-th solution in \(P\), \(x_{j}^{*}\) is the \(j\)-th solution in \(P^{*}\), \(m\) is the number of objectives, and \(f_{k}(x)\) is the function value of individual \(x\) on the \(k\)-th objective. For instance, in a two-dimensional objective space, as shown in Figure 5, the algorithm obtains three Pareto optimal solutions, H, I, and J, while the true Pareto front is formed by solutions A, B, C, D, E, F, and G. The minimum distances from the points on the true Pareto front to the points H, I, and J are \(d(A,H)\), \(d(B,H)\), \(d(C,I)\), \(d(D,I)\), \(d(E,J)\), \(d(F,J)\), and \(d(G,J)\), respectively. Hence, the IGD value is calculated as follows:

\[IGD =\frac{1}{\gamma}\big{(}d(A,H)+d(B,H)+d(C,I)\] \[\quad+d(D,I)+\cdots+d(G,J)\big{)} \tag{7}\]

When solving problems using multi-objective evolutionary algorithms, the diversity of the solutions is also an important consideration.