## 3 Proposed Method
### Individually Guided Multi-Task Evolutionary Optimization  
Putting GA, PSO, DE, and ABC into IMTO leads to the corresponding algorithms named IMGA, IMPSO, IMDE, and IMABC. To better show the designed framework, we give the IMGA implementation as an example. Its pseudocode is described in Algorithm 1, where its module performing horizontal transfer is realized in Algorithm 2. It is worth noting that individuals in MFO evolve together in each iteration, and the random mating probability (rmp) value decides the evolution directions of each individual. Different from MFO, vertical and horizontal evolutions are independent in IMTO and each task group of IMTO has its vertical evolution. The additional horizontal evolution is triggered when the convergence rate is less than zero, which is used to improve optimization performance. We use skill membership values to represent individuals' solving ability on different tasks, which can be used for guiding the division of different task groups before the evolution. **Definition 1** (Skill membership). : _For minimized optimization problems, the skill membership value of individual \(p_{i}\) is defined as:_  
\[h_{i}^{j}=\min\left(\
ot{p}\right)/f_{i}^{j} \tag{2}\]  
_where \(f_{i}^{j}\) is an objective function value of individual \(p_{i}\) on task \(T_{j}\), and \(\min\left(\
ot{p}\right)\) is the obtained minimum objective function value. If \(\min\left(\
ot{p}\right)\) or \(f_{i}^{j}\) is equal to zero, it will be replaced by random values \(\xi_{1}\) and \(\xi_{2}\) that are close to zero. The skill membership value \(\boldsymbol{\mu}_{i}=\mu_{ij=1}^{jk}\) reflects the solving ability of individual \(p_{i}\) on component task \(T_{j}\). Next, we show how to divide the initial population into different task groups according to skill membership values. This paper records the rank of skill membership values in order from the highest to lowest, and the higher skill membership values mean higher solving ability._  
At the beginning of IMGA, \(N_{0}\) individuals are randomly initialized within the unified search space. Given \(K\) optimization tasks, IMGA uses a partition strategy to divide the initial population \(P_{0}\) into \(K\) task groups that focus on solving component tasks. Since simply choosing better-performing individuals can be easily trapped into local optima, IMGA randomly chooses some individuals according to the randomly chosen ratio \(\gamma\). Specifically, \(N_{0}\) individuals of the initial population \(P_{0}\) are ranked according to their skill membership values. \(P_{0}\) is divided into \(P_{01}^{j}\) and \(P_{02}^{j}\) for task \(T_{j}\). \(P_{01}^{j}\) contains \(N\)(1-\(\gamma\)) individuals with the highest skill membership values, and \(P_{02}^{j}\) is composed of the remaining individuals of \(P_{0}\). IMGA chooses all candidates of \(P_{01}^{j}\) and randomly chooses \(N\gamma\) candidates from \(P_{02}^{j}\) to form task group \(G_{j}\). The setting of parameter \(\gamma\) with a default value 0.2 is to be discussed later. ```
0:\(N_{0}\), initial population size; \(N\), task group size; \(K\), number of tasks; \(\Omega\): communication rate. 0:\(\{x_{1}^{*},\ x_{2}^{*},\ldots,\ x_{k}^{*}\}\) (the best solution found for each component task). 1:\(P_{0}\leftarrow\) Random Initialization (\(N_{0}\)). 2: Evaluate each individual of the initial population on \(K\) component tasks. 3: Calculate the skill membership value \(\mu_{i}^{j}\) of individual \(p_{i}\) on the \(j\)th task. 4:for task \(j=1\) to \(K\)do
5: Sort (\(\mu_{i}^{j}\)) for task \(T_{j}\). 6: Divide \(P_{0}\) into \(P_{01}^{j}\) and \(P_{02}^{j}\). 7: Choose individuals from \(P_{01}^{j}\) and \(P_{02}^{j}\) to form task group \(G_{i}\). 8:endfor
9:while (stopping criterion is not met) do
10:for task \(j=1\) to \(K\)do
11: Mutation and crossover. 12: Fitness evaluations. 13: Finish vertical reproduction operations and select \(N\) candidates to update task group \(G_{i}\).