## 4 Theoretical Analysis of DE-MPFSC Algorithm
### Validity Analysis of DE-MPFSC Algorithm  
The validity analysis of the DE-MPFSC algorithm is that whether the Markov model condition of the DE-MPFSC algorithm is true in the constrained feasible domain. It is necessary to judge whether the state transition of the population sequence is a Markov chain. So, we introduce the corresponding definitions and lemmas. **Definition 1** ([46,47]). : _Let \(\{\hat{X}_{n};n\geq 0\}\) be a random variable with a discrete value. The whole discrete values are recorded as \(S\), which is called the state space. If \(\forall n\geq 1,i_{k}\in S(k\leq n+1)\) and \(P\{\hat{X}_{n+1}=\frac{i_{n+1}}{\hat{X}_{n}}=i_{n},\cdots,\hat{X}_{0}=i_{0}\}= P\{\hat{X}_{n+1}=\frac{i_{n+1}}{\hat{X}_{n}}=i_{n}\}\), then \(\{\hat{X}_{n};n\geq 0\}\) is called the Markov chain._  
**Lemma 1** ([46,47]). : _The joint distribution of homogeneous Markov chain \(p_{ij}^{n}=P\{\hat{X}_{n+m}=\frac{j}{\hat{X}_{m}}=i\}\) is determined by the initial distribution \(P\{\hat{X}_{0}=i\}=p_{i},i\in S\) and the individual transition probability \(p_{ij}=P\{\hat{X}_{n}=\frac{j}{\hat{X}_{n-1}}=i\},(i,j\in S)\)._  
**Lemma 2** ([46,47]). : _For the homogeneous Markov chain \(p_{ij}^{n}=P\{\hat{X}_{n+m}=\frac{j}{\hat{X}_{m}}=i\}\) and \(\forall n,m\geq 0,i,j\in S\), there are \(p_{ij}^{n=m}=\sum_{k\in S}p_{ik}^{n}p\)._  
**Definition 2** ([46,47]). : _Let \(\{\hat{X}_{n};n\geq 0\}\) be the finite homogeneous Markov chain, \(p_{ij}^{n}\) is the \(n\)th iterative individual transition probability. If \(\exists n\geq 1\) makes \(p_{ij}^{n}\geq 0\), which is called a case in which status \(i\) can be transferred to state \(j\); otherwise, is called another case in which the status \(i\) cannot be transferred to state \(j\)._  
**Definition 3** ([46,47]). : _For Markov chain \(\{\hat{X}_{n};n\geq 0\}\), the greatest common divisor \(d_{i}\) of the state set \(\{i|n\geq 1,p_{ij}^{n}\geq 0\}\) is called the generalized period of the state set \(i\). If \(d_{i}>1\), then state set \(i\) is periodical. If \(d_{i}=1\), then state set \(i\) is nonperiodic. If \(d_{i}\) is a nonpositive real number, then \(i\) cannot be periodical._  
**Lemma 3** ([46,47]). : _If Markov chain \(\{\hat{X}_{n};n\geq 0\}\) is irreducible, that is, that all individual states are connected to each other, and \(\exists j\leq N\), and \(\exists j\leq N\) make \(p_{ij}>0\), then the chain is nonperiodic. Its transfer matrix is the primitive random matrix and there is a stationary distribution on the transfer matrix. Where there is a stationary distribution and a limited distribution \(\lim_{n\rightarrow\infty}P\{X_{n}=j\}=o_{j}\) on the transfer random matrix of the Markov chain being nonperiodic, irreducible and a finite state._  
Now, consider the [PERSON] processing model \(\{(x_{i}^{t},\delta_{i})|i=1,2,\cdots,NP\}\), the progeny \(x_{k}^{t+1}\) and iterative steps \(\delta_{i}\) of the DE-MPFSC algorithm are determined by the following formula. \[\delta_{i}^{t+1}=\left\{ \begin{array}{l}\gamma\delta_{i}^{t},lf\;\Psi(x_{k} ^{t}+\delta_{i}^{t}C^{t})<\Psi(x_{k}^{t}),\\ and\;g_{i}(x_{k}^{t}+\delta_{i}^{t}C^{t})\leq 0\\ \gamma^{-1}\delta_{i}^{t},\textit{otherwise}\end{array} \right. \tag{19}\]  
\[x_{k}^{t+1}=\left\{ \begin{array}{l}x_{k}^{t}+\delta_{i}^{t}C^{t},lf\;\Psi(x _{k}^{t}+\delta_{i}^{t}C^{t})<\Psi(x_{k}^{t}),\\ and\;g_{i}(x_{k}^{t}+\delta_{i}^{t}C^{t})\leq 0\\ x_{k}^{t},\textit{otherwise}\end{array} \right., \tag{20}\]  
where \(\gamma>1\) is a compilation parameter, \(\{C^{t},t\geq 0\}\) is a random vector and independent and identically distributed. Assume that the new individuals generated by the mutation operation of the DE algorithm are distributed on a circle centered on \(x_{k}^{t}\) with a radius of \(\delta_{i}^{t}\). When the mutation individual generated by the DE-MPFSC algorithm is valid, the step increases; otherwise, the step decreases. Where effective mutation individuals are superior than the parents. Then, each evolution can be performed simultaneously from the inside and outside of the feasible domain.