## 4 Theoretical Analysis of DE-MPFSC Algorithm
### Convergence Analysis of DE-MPFSC Algorithm  
**Definition 4** ([48; 49; 50]). : \(\{\hat{X}_{n};n\geq 0\}\) _is the set that is strongly convergent to the global optimal solution in probability. If \(\lim_{n\to\infty}P\{[\hat{X}_{n}\subset M]\}=1\) is true, which is marked as \(\hat{X}_{n}\to M(P.S. )\)._  
**Definition 5** ([48; 49; 50]). : \(B\subset S\) _is a satisfactory solution set. If \(\forall X\in B,Y\
ot\in B\) make \(f(X)>f(Y)\)._  
**Theorem 2**. : _(Theorem 2 shows that the Markov chain of the DE-MPFSC algorithm is convergent in the distribution and does not depend on the choice of the initial population, which is expressed as \(o(Z)=\sum_{Y\in S}P\{X_{t}=\frac{Z}{X_{0}}=Y\}\)). (Limit Distribution Theorem) There is a limited distribution for the population sequence \(\{\hat{X}_{n};n\geq 0\}\) of the DE-MPFSC algorithm._  
Proof. : From Theorem 1, we know that the sequence \(\{\hat{X}_{t};n\geq 0\}\) of the DE-MPFSC algorithm is a finite, homogeneous, irreducible and nonperiodic Markov chain. Then, from lemma 3, we obtain  
\[\lim_{t\to\infty}P\{T(X_{t})=Y\}=\lim_{t\to\infty}\sum_{X_{0}\in S}P\{T(X_{t}) =\frac{Y}{X_{0}}\}P\{X_{0}\}=o(Y). \tag{27}\]  
Then, \(o(Y)\) is the distribution on state population \(S\) and  
\[\sum_{Y\in S}P\{X_{t}=\frac{Z}{X_{0}}=Y\}=o(Z),(Z\in S). \tag{28}\]  
**Theorem 3**. : _(The Strong Convergence Theorem in the Probability of the DE-MPFSC Algorithm) (Theorem 3 shows that the convergence of the Markov sequence \(\{X_{t}\}\), \(\{X_{t}\}\) on the state population \(S\) of the DE-MPFSC algorithm is related to parameters \(\{\alpha_{t}^{D}\}\), \(\{\beta_{t}^{D}\}\). \(\alpha_{t}^{D}\) is the probability that the population individuals in a Markov sequence \(\{X_{t}\}\) will leave the next satisfactory solution sets [50]. However, from Formulas (1), (2) of theorem 3, \(\alpha_{t}^{D}\) is a negative correlation with \(\beta_{t}^{D}\). So, for the Markov sequence \(\{X_{t}\}\), there is a nonconvergent state. In order to enhance the convergence analysis of the satisfactory solution to the DE-MPFSC algorithm, we need to perform strong convergence analysis around all feasible points that satisfy the limit distribution. The theorem theoretically explains the necessary conditions for the absolute convergence of the DE-MPFSC algorithm.). Let \(P\) be the probability distribution on the state population \(S\), \(\{X(t)\}\) is the Markov sequence on \(S\), \(M\) is the global optimal solution set, \(D\) is any satisfactory solution set. If \(\{\tilde{\alpha}_{t}^{D}\}\), \(\{\tilde{\beta}_{t}^{D}\}\) satisfies these following conditions: (1) \(\sum_{t=1}^{\infty}(1-\tilde{\beta}_{t}^{D})=\infty\); (2) \(\lim_{t\to\infty}\frac{\tilde{\alpha}_{t}^{D}}{1-\tilde{\beta}_{t}^{D}}=0\)._  
Then, \(X_{t}\) is strongly convergent in probability to satisfactory solution sets \(D\), which is to say \(\lim_{t\to\infty}P\{X_{t}\subset D\}=1\). Where \(\tilde{\alpha}^{D}=P\{X(t+1)\cap D^{c}\
eq\emptyset\}\), \(\beta_{t}^{D}=P\{X(t+1)\cap D^{c}\
eq\emptyset\}\). Proof. : Let \(P_{0}=P\{X_{t}\cap D=\emptyset\}\), from the _Bayes_ formula, we know  
\[P_{0}(t+1)=P\{X(t)\cap D=\emptyset\}\] \[=P\{X(t+1)\cap D\
eq\emptyset\}\cdot P\{X(t)\cap D\
eq\emptyset \}+P\{X(t+1)\cap D=\emptyset\}\}\cdot P\{X(t)\cap D=\emptyset\} \tag{29}\] \[\leq\tilde{\alpha}_{t}^{D}+\tilde{\beta}_{t}^{D}P_{0}(t).\]  
From Formula (4), we know \(\forall\varepsilon>0,\exists N_{1}\), when \(t\geq N_{1}\), \(\frac{\tilde{\alpha}_{t}^{D}}{1-\tilde{\beta}_{t}^{D}}\leq\frac{\varepsilon}{2}\); thus, \(\tilde{\alpha}_{t}^{D}\leq\frac{\varepsilon}{2}(1-\tilde{\beta}_{t}^{D})\).