## 4 Distribution Method for Observation Scheduling in the AEOS Constellation
### The Decentralized Framework of PI  
We start by introducing the \"_significance value_\" (respectively, the \"_marginal significance value_\") of a target with regard to the assigned (resp. non-assigned) satellite in the PI framework. (1) _Significance_: Assume \(g\in\mathbf{a}_{s}\), the significance value \(q_{si}(\mathbf{a}_{s}\ominus g)\) represents the variation of \(F(\mathbf{a}_{s})\) after removing \(g\) from \(\mathbf{a}_{s}\), it can be formulated as follows:  
\[q_{s}(\mathbf{a}_{s}\ominus g)=F(\mathbf{a}_{s})-F(\mathbf{a}_{s}\ominus g) \tag{16}\]  
where \(\mathbf{a}_{s}\ominus g\) is the sequence after removing \(g\) from \(\mathbf{a}_{s}\). For completeness, we set \(q_{s}(\mathbf{a}_{s}\ominus g)=0\) for \(g\
otin\mathbf{a}_{s}\). (2) _Marginal significance:_ Assume \(g\
otin\mathbf{a}_{s}\), the marginal significance \(q_{c}\)*(\(\mathbf{a}_{c}\oplus g\)) represents the maximal variation of \(F(\mathbf{a}_{s})\) and after adding \(g\) into \(\mathbf{a}_{s}\), it can be formulated as follows:  
\[q_{s}^{*}(\mathbf{a}_{s}\oplus g)=\max_{k\in\{1,2,\ldots,|\mathbf{a}_{s}|+1\}} \{F(\mathbf{a}_{s}\oplus_{k}g)-F(\mathbf{a}_{s})\} \tag{17}\]  
where \(\mathbf{a}_{s}\oplus_{k}g\) is the target sequence after inserting \(g\) into the \(k\)-th position of \(\mathbf{a}_{s}\). If \(g\in\mathbf{a}_{s}\), let \(q_{s}\)*(\(\mathbf{a}_{s}\oplus g\)) toward minus infinity, i.e., \(q_{s}\)*(\(\mathbf{a}_{c}\oplus g\)) \(\rightarrow-\infty\). The basic idea of PI framework is as follows [34]: each AEOS \(s\) transmits the significance value \(q_{s}(\mathbf{a}_{s}\ominus g)\) of target \(g\in\mathbf{a}_{s}\) to its neighbor satellite \(c\) through ISLs. Then, satellite compares the received \(q_{s}(\mathbf{a}_{c}\odot g)\) with the marginal significance value \(q_{c}\)*(\(\mathbf{a}_{c}\odot g\)) computed based on its sequence \(\mathbf{a}_{c}\). If the criterion in (18) is satisfied (see Proposition 1 later), target \(g\) is deleted from \(\mathbf{a}_{s}\) but added into some proper position of \(\mathbf{a}_{c}\) such that the global fitness value Y(\(\mathbf{a}\)) is increased. Similarly, the significance of target \(g\) is further transmitted to other satellites and this value is continuously updated. The above process is repeated until (18) is no longer satisfied or Y(\(\mathbf{a}\)) cannot be increased by exchanging targets between satellites. Note that the _convergence_ of PI algorithm is naturally guaranteed since the global fitness Y(\(\mathbf{a}\)) is increased whenever the target assignment is changed. To facilitate the understanding of the PI procedure, the following conclusion reveals how global fitness is increased by exchanging targets between satellites. Proposition 1: _Given the global fitness value Y(\(\mathbf{a}\)) of a solution \(\mathbf{a}\), two AEOSs \(s\) and \(c\) with \(H(s,c)=1\), and a target \(g\in\mathbf{a}_{s}\backslash\mathbf{a}_{c}\). Then, if (18) is satisfied, the global fitness value would be increased after removing \(g\) from \(\mathbf{a}_{s}\) but inserting it into the proper position of \(\mathbf{a}_{c}\)._  
\[q_{c}^{*}(\mathbf{a}_{c}\oplus g)>q_{s}(\mathbf{a}_{s}\odot g) \tag{18}\]  
Proof. : Let \(p_{c}=\operatorname*{argmax}_{k\in\{1,2,\ldots\mid\mathbf{a}_{c}\mid+1\}}[F( \mathbf{a}_{c}\odot g)-F(\mathbf{a}_{c})]\). Suppose we remove \(g\) from \(\mathbf{a}_{s}\) and add it into the \(p_{c}\)-th position of \(\mathbf{a}_{c}\), resulting in a new solution denoted as \(\mathbf{a}^{\prime}\). By (16) and (17), we can calculate the fitness of \(\mathbf{a}^{\prime}\) as Y(\(\mathbf{a}^{\prime}\)) = Y(\(\mathbf{a}\)) + \(q_{c}\)*(\(\mathbf{a}_{c}\odot g\)) - \(q_{s}(\mathbf{a}_{c}\odot g)\). Therefore, the inequality Y(\(\mathbf{a}^{\prime}\)) > Y(\(\mathbf{a}\)) holds and the global fitness value increases only when \(q_{c}\)*(\(c\oplus g\)) > \(q_{s}(\odot g\)). \(\square\)  
Specifically, we use \(F_{s}\) and \(F_{c}\) to denote the partial fitness value of \(s\) and \(c\), respectively.