# Some Algorithms to Solve a Bi-Objectives Problem for Team Selection
## 2 DC Programming and DCA for MDSB
### DC Decomposition and DCA for MDSB-PMDSB  
As introduced in [2], the \(x_{i}=[0,1],i=1\ldots k\) are non-convex constraints. [PERSON] et al. [22] introduced the exact penalty technique to solve the difficulty caused by these constraints. The idea of this technique is that each constraint \(x_{i}=[0,1]\) is first equivalently formulated as:  
\[x_{i}\in[0,1]\text{ and }x_{i}(1-x_{i})\leq 0.\]After that, the constraint \(x_{i}(1-x_{i})\leq 0\) is penalized to the objective function while still keeping \(x_{i}\in[0,1]\) as the constraint. As a consequence, the penalized problem with continuous relaxation constraints corresponding to MDSB is given by  
\[PMDSB:\min\sum_{j=1}^{m}\left(E_{j}-\sum_{i=1}^{k}R_{ij}x_{i}\right)^{2}+\tau \sum_{i=1}^{k}x_{i}(1-x_{i}),\]  
subject to:  
\[x_{i}\in[0,1]\ \forall\ i=1\ldots k\]  
\[\sum_{i=1}^{k}x_{i}=h\]  
\[\sum_{i=1}^{k}(x_{i}*R_{i,j})\ \geq z_{j}\ \forall j=1..m\]  
\[\sum_{i=1}^{k}x_{i}*c_{i}\ \leq C,\]  
where \(\tau\) is a penalty parameter.  
It was shown in [2] that both the problems PMDSB and MDSB are equivalent if the penalty parameter \(\tau\) is appropriately chosen and sufficiently large enough. Thus, we focus on solving the penalized problem PMDSB. It is clear that this problem is not convex since the first term of the objective function \(\sum_{j=1}^{m}\left(E_{j}-\sum_{i=1}^{k}R_{ij}x_{i}\right)^{2}\) is convex, while the second term \(\sum_{i=1}^{k}x_{i}(1-x_{i})\) is a concave function. However, it can easily be formed as a DC program: \(PMDSB=\min\{g(X)-h(X)\}\), where both \(g(X)=\sum_{j=1}^{m}\left(E_{j}-\sum_{i=1}^{k}R_{ij}x_{i}\right)^{2}\) and \(h(X)=\sum_{i=1}^{k}x_{i}(x_{i}-1)\) are convex functions on \(R^{n}\). As a result, a DCA scheme can be developed to address the problem PMDSB.  
We propose the method to solve this optimization based on a general DC Algorithm:  
1. Randomly select \(X^{0}=\{x_{i}^{0}\}0\leq x_{i}^{0}\leq 1\forall i=1\ldots k\}\), and \(\epsilon\) is small enough and a suitable value of \(\tau\).
2. Calculate the approximation of \(h(X)\) at \(X^{l}\).
3. Compute \(X^{l+1}\) by solving the sub-problem: \[\min\ \sum_{j=1}^{m}\left(E_{j}-\sum_{i=1}^{k}R_{ij}x_{i}\right)^{2}-\tau \sum_{i=1}^{k}\left(2x_{i}^{l}-1\right)x_{i},\] subject to: \[x_{i}\in[0,1]\forall i=1\ldots k\] \[\sum_{i=1}^{k}x_{i}=h\] \[\sum_{i=1}^{k}(x_{i}*R_{i,j})\ \geq z_{j}\forall j=1..m\] \[\sum_{i=1}^{k}x_{i}*c_{i}\leq C.\] It is possible to solve the problem with several solvers such as CPLEX.
4. If \(\big{\|}f(X^{l+1})-f(X^{l})\big{\|}/\left(1+\big{|}f(X^{l})\big{|}\right)>\epsilon\), come back to step 2.  
Although we can guarantee convergence in the infinite limit of \(l\), complete convergence may take a long time in large-scale data, so \(\epsilon\) was used as a predetermined bound for obtaining an optimal solution. To make it more convenient for the implementation, we rewrote the sub-problem as a matrix form:  
\[\min\frac{1}{2}X^{T}AX-b^{T}X,\]  
There are several researchers that have designed their own genetic algorithm for solving the mixed binary-integer optimization such as [PERSON] [6], [PERSON] et al. [9], [PERSON] et al. [24], [PERSON] and [PERSON] [7] and [PERSON] et al. [25]. The genetic algorithm is a philosophy, not just a typical algorithm. Therefore, each study has a specific design scheme.