## 2 Methodology
### Problem Formulation  
The ReCon framework may be regarded as nonlinear programming (NLP), due to the nonlinear nature of satellite coverage and launch mass. Equation (1) is a general NLP formulation where the fitness function \(F\) or the constraint \(h\) is nonlinear. Both the constraint and the objective are nonlinear in the ReCon framework, as will be discussed in more detail in Sections 2.2.4 and 2.2.5. Multiple constraints are expressed as a vector \(h\), and the objective \(F\) is scalar. The design vector \(x\) (Section 2.2.1) is varied throughout the optimization process such that \(x_{\text{LB}}\leq x\leq x_{\text{UB}}\), but the operating parameter \(p\) remains unchanged (Section 2.2.3). Intermediate variables (Section 2.2.2) are not included in Equation (1), but they play a role as data inputs and output to relay internal states among subroutines. \[\text{Min}/(\mathbf{x},\mathbf{p})\ s.t.\ \mathbf{h}(\mathbf{x},\mathbf{p})\leq\mathbf{0}\enspace, \tag{1}\]  
Inequality constraint may be enforced by adding penalization to J where that larger values for penalty vector elements will more strongly discourage constraint violation. In Equation (2), each constraint violation is weight-accumulated only if \(h_{i}\) is positive as shown by indicator function \(\mathbf{1}_{\text{h>0}}\). \[\text{Min}J(\mathbf{x},\mathbf{p})+g\sum_{i}\{w_{h_{i}}h_{i}(\mathbf{x},\mathbf{p})\mathbf{1}_{h_{ i}>0}\}\enspace, \tag{2}\]  
Finally, non-penalized \(J\) is also a weighted sum of several figures of merit, as shown in Equation (3). Definitions for constraints and figures of merit are summarized in Table 1.  
\[\text{Min}\sum_{i}\{w_{F_{i}}F_{i}(\mathbf{x},\mathbf{p})\}+g\sum_{i}\{w_{h_{i}}h_{i}( \mathbf{x},\mathbf{p})\mathbf{1}_{h_{i}>0}\}\enspace. \tag{3}\]  
It would be noteworthy to mention that these figures of merit exhibit highly nonlinear behaviors and lack closed forms most of the time. Among four figures of merit, only reconfiguration time, as will be discussed in Section 2.3.3, has a closed form solution which is inversely proportional to the product of (Earth radius + altitude) and \(\Delta\)(altitude). Constellation mass is obtained through numerical iterations described in Section 2.3.4 to solve the rocket equation with exponential terms. The remaining two figures of merit are temporal coverage in GOM and revisit time in ROM. A potential region of interest (latitude band in this case) should be sufficiently observed by maximizing the temporal \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Figure of merit** & **Definition** & **Constraint** & **Definition** \\ \hline \(F_{1}\) & \((-1)\)\(\times\) GOM coverage (\%) & \(h_{1}\) & Minimum altitude (km) \\ \(F_{2}\) & ROM revisit time (s) & \(h_{2}\) & Maximum altitude (km) \\ \(F_{3}\) & Constellation mass (kg) & \(h_{3}\) & Maximum aperture (m) \\ \(F_{4}\) & Reconfiguration time (day) & \(h_{4}\) & Max propellant mass \\  & & & fraction \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Figures of merit and constraints in ReCon optimization. coverage in GOM, and an outstanding target within that region, once identified, must be visited as frequently as possible by minimizing the revisit time in ROM. There have been efforts to develop non-numerical algorithms for calculating the temporal coverage and revisit time of ground targets at the equator [38], but considerable errors would occur if targets are located at high latitudes where orbit groundtracks are cluttered (Figure 1). These methods also apply only to a single satellite, yet to be extended up to a constellation level. Numerical orbit simulation is therefore used in this work to compute coverage and revisit time. While coverage and revisit time are enhanced as the number of satellites increases, the larger launch mass (proxy of cost) must be accompanied. Combined with the inherent nonlinearity of each figure of merit, tensions among figures of merit lead to complex objective space of this problem, discussed again in Section 6.1 and exemplified by the poor performance of gradient-based approaches.