## 3 Min-Max Approach to MCGDM Problems
### Efficient Algorithm for the Min-Max Models  
In general, it is not easy to find a solution for a constrained Min-Max problem. We now develop an efficient algorithm to solve Models (3) and (4).  
Define a function \(F:R^{m}\to R\). For any \(u^{k}\in R^{m}\),  
\[F(u^{k})=\max\limits_{1\leq i\leq n}\{f_{i}(u^{k})\},\]  
where \(f_{i}:R^{m}\to R\), given by:  
\[f_{i}(u^{k})=\sum\limits_{j=1}^{m}\left(x_{ij}^{k}-u_{j}^{k}\sum\limits_{j=1} ^{m}x_{ij}^{k}\right)^{2}. \tag{8}\]  
It is clear that \(f_{i}\), \(i=1,2,\cdots,n\), are smooth quadratic functions in \(u\in R^{m}\).  
Clearly, we can write Model (3) in a compact form.  
\[\begin{array}{ll}\min&F(u^{k})\\ \text{s.t.}&l(u^{k})\leq 0,\;u^{k}\in R^{m},\end{array} \tag{9}\]where \(l:R^{m}\to R^{2m+2}\), specified by:  
\[l(u^{k})=\left(\begin{array}{c}\sum\limits_{j=1}^{m}u_{j}^{k}-1\\ 1-\sum\limits_{j=1}^{m}u_{j}^{k}\\ -u_{1}^{k}\\ \vdots\\ -u_{m}^{k}\\ u_{1}^{k}-1\\ \vdots\\ u_{m}^{k}-1\end{array}\right). \tag{10}\]  
For Model (9), we define:  
\[F(u^{k},t)=\theta_{1t}\ln\sum\limits_{i=1}^{n}e^{f_{i}(u^{k})/(\theta_{1}t)}, \ \ G(u^{k},t)=\theta_{2t}\ln\sum\limits_{\kappa=1}^{2m+2}e^{l_{\kappa}(u^{k})/( \theta_{2}t)},\]  
where \(t>0\) is a given perturbation parameter. As \(t\to 0\), the solution of the following unconstrained optimization problem tends to that of Model (9):  
\[\min L(u^{k},t)=F(u^{k},t)-t\ln(-G(u^{k},t)). \tag{11}\]  
Similarly, we define a function \(H:R^{p}\to R\), specified by:  
\[H(\omega)=\max\limits_{1\leq k\leq p}\{h_{k}(\omega)\},\]  
where \(h_{k}:R^{p}\to R\), and for any \(\omega\in R^{p}\),  
\[h_{k}(\omega)=\sum\limits_{i=1}^{n}\left(a_{ik}-\sum\limits_{k=1}^{p}\omega_{ k}a_{ik}\right)^{2}. \tag{12}\]  
It is clear that \(h_{k}\), \(k=1,2,\cdots,p\), are smooth quadratic functions in \(\omega\in R^{p}\).  
We write Model (4) in a compact form.  
\[\begin{array}{rl}\min&H(\omega)\\ \mbox{s.t.}&\Phi(\omega)\leq 0,\ \omega\in R^{p},\end{array} \tag{13}\]  
where \(\Phi:R^{p}\to R^{2p+2}\), specified by:  
\[\Phi(\omega)=\left(\begin{array}{c}\sum\limits_{k=1}^{p}\omega_{k}-1\\ 1-\sum\limits_{k=1}^{p}\omega_{k}\\ -\omega_{1}\\ \vdots\\ -\omega_{p}\\ \omega_{1}-1\\ \vdots\\ \omega_{p}-1\end{array}\right). \tag{14}\]For Model (13), we define:  
\[F(\omega,t)=\theta_{1t}\ln\sum_{k=1}^{p}e^{h_{k}(\omega)/(\theta_{1}t)},\ \ \ G(\omega,t)=\theta_{2t}\ln\sum_{\kappa=1}^{2p+2}e^{\Phi_{\kappa}(\omega)/( \theta_{2}t)},\]  
As \(t\to 0\), we can obtain an approximate solution of Model (13) by solving the following unconstrained optimization problem:  
\[\min L(\omega,t)=F(\omega,t)-t\ln(-G(\omega,t)). \tag{15}\]  
With the above preparation, we are now in a position to state the following framework of the algorithm to solve the optimization Models (9) and (13).  
**Algorithm 1**.: _Step 0. Given an initial guess of solution \(x_{0}\), choose \(0<t<1\), \(0<\theta_{1}\leq 1\), \(0<\theta_{2}\leq 1\). Set \(\
u:=0\)._  
_Step 1. Solve Problem (11) (or Problem (15)) by the modified conjugate gradient algorithms in [22, 23, 24]. Its optimal solution is referred to as \(u^{k*}\) (or \(\omega^{*}\)), \(k=1,2,\cdots,p\)._  
_Step 2. If \(u^{k*}\) and \(\omega^{*}\) are the solutions of Models (9) and (13), then the algorithm stops. Otherwise, go to Step 3._  
_Step 3. Set \(\
u:=\
u+1\), \(t:=t^{\
u}\). Go to Step 1._  
**Remark 2.** In the practical implementation of Algorithm 1, if the optimal solution of Problem (11) at the \(\
u\)-th iteration, being referred to as \(u^{k*}(\
u)\), approximately satisfies the constraints in Model (9), and the difference of the optimal solutions corresponding to \(\
u\) and \(\
u-1\) satisfies \(\|u^{k*}(\
u)-u^{k*}(\
u-1)\|\leq 0.5\times 10^{-4}\), then Algorithm 1 stops.