**5.2 Generalized Monte Carlo Gauss-Seidel iterative algorithm**

Based on the fixed-point type necessary condition, the generalized Monte Carlo Gauss-Seidel iterative algorithm is presented in Algorithm 2.

Remark 7: For any initial values *I* ð Þ 0 <sup>1</sup> , … , *I* ð Þ 0 *<sup>L</sup>* ; *<sup>F</sup>*0 0ð Þ � �, the Monte Carlo cost function *CMC I* ð Þ *n* <sup>1</sup> , … , *I* ð Þ *n <sup>j</sup>* , *I* ð Þ *n <sup>j</sup>*þ1, … , *<sup>I</sup>* ð Þ *n <sup>L</sup>* ; *F*0ð Þ *<sup>n</sup>* ; *Pce*0, *Pce*<sup>1</sup> , *N* � � must converge to a stationary point and Algorithm 2 terminates after a finite number of iterations. The proofs are similar to those of Lemma 3 and Theorem 1.2.

**Algorithm 2**: Simultaneous optimization of the sensor rules and the fusion rule.


$$I\_j^{(0)}(Y\_{\vec{\eta}}) = \mathbf{0}/\mathbf{1}, F^{0(0)}(\mathfrak{s}\_{k'}) = \mathbf{0}/\mathbf{1}.$$

• Step 3: Iteratively search for the *L* sensor rules and the fusion rule until a termination criterion in Step 4 is satisfied. The *n* þ 1th iteration is given as follows: for *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>* and *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> 1, … , 2*<sup>L</sup>*

$$\begin{split} I\_{1}^{(n+1)}(Y\_{i1}) &= I \left[ P\_{11} \left( I\_{2}^{(n)}(Y\_{i2}), I\_{3}^{(n)}(Y\_{i3}), \ldots, I\_{L}^{(n)}(Y\_{iL}); F^{0(n)}; \mathbf{P}^{x0}, \mathbf{P}^{x1} \right) \cdot \hat{L}(Y\_{i}) \right], \\ I\_{2}^{(n+1)}(Y\_{i2}) &= I \left[ P\_{21} \left( I\_{1}^{(n+1)}(Y\_{i1}), I\_{3}^{(n)}(Y\_{i3}), \ldots, I\_{L}^{(n)}(Y\_{iL}); F^{0(n)}; \mathbf{P}^{x0}, \mathbf{P}^{x1} \right) \cdot \hat{L}(Y\_{i}) \right], \\ &\cdots \\ I\_{L}^{(n+1)}(Y\_{iL}) &= I \left[ P\_{L1} \left( I\_{1}^{(n+1)}(Y\_{i1}), I\_{2}^{(n+1)}(Y\_{i2}), \ldots, I\_{L-1}^{(n+1)}(Y\_{i(L-1)}); F^{0(n)}; \mathbf{P}^{x0}, \mathbf{P}^{x1} \right) \cdot \hat{L}(Y\_{i}) \right], \\ &F^{0(n+1)}(s\_{k'}) &= I \left[ \sum\_{i=1}^{N} P \left( s\_{k'} \left| \left( I\_{1}^{(n+1)}(Y\_{i1}), I\_{2}^{(n+1)}(Y\_{i2}), \ldots, I\_{L}^{(n+1)}(Y\_{iL}) \right) \right) \frac{\hat{L}(Y\_{i})}{\mathbf{g}(Y\_{i})} \right) \right]. \end{split}$$

• Step 4: For *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>* and *<sup>k</sup>*<sup>0</sup> <sup>¼</sup> 1, 2, … , 2*<sup>L</sup>*, the termination criterion of the iteration process is

$$\begin{aligned} I\_1^{(n+1)}(Y\_{i1}) &= I\_1^{(n)}(Y\_{i1}), \\ I\_2^{(n+1)}(Y\_{i2}) &= I\_2^{(n)}(Y\_{i2}), \\ \cdots \\ I\_L^{(n+1)}(Y\_{iL}) &= I\_L^{(n)}(Y\_{iL}); \\ F^{0(n+1)}(s\_{k'}) &= F^{0(n)}(s\_{k'}). \end{aligned}$$
