## 4 Optimal Mapping In Dima
### Solution of Multi-Objective Optimization
#### 4.2.2 Pareto Optimization  
The key concept of Pareto optimization is Pareto front that reflects the trade-offs among various objectives for finding good solutions. Pareto front is a set of Pareto optimal points on the boundary of the feasible space and none of them is dominated by others. A point, \(\mathbf{x}^{*}\in\mathbb{X}\), is Pareto optimal if and only if there does not exist another point, \(\mathbf{x}\in\mathbb{X}\), such that \(\mathbf{f}(\mathbf{x})\leq\mathbf{f}(\mathbf{x}^{*})\)[59]. Specifically, Pareto sampling calculates a set of solutions to consider the trade-off among different objectives [60]. [PERSON] et al. [9; 33; 61] presented approaches to calculate all solutions of the Pareto front to represent trade-off decisions for linear cost functions in DIMA optimization. The contributions of their works are summarized with details as follows. [PERSON] et al. [33] presented binary program formulations and extended the work [34] by optimizing multiple objectives. Mass, SSC, OIC and IPC are considered as cost functions, which results in a multi-objective optimization problem. The multi-objective optimization is solved by Pareto optimization, with Pareto front calculated. Results identified by Pareto front demonstrate that significant reduction of OIC can be achieved by increasing little mass for hardware mapping; for software mapping, on the other hand, IPC and OIC are less opposing and the objective values are primarily reduced by using less number of devices. [PERSON] et al. [9] also presented binary program formulations for the optimal mapping and allocation of resources in order to minimize mass and OIC, which are two contradicting objectives. Resource consumption and safety constraints (i.e., segregation with restricted installation locations) are considered in the formulation. Unlike in Reference [33], mass and OIC are modeled as linear and quadratic functions. The extension of using quadratic objective functions, however, renders the optimization problem to be more complicated; it seems to be applicable for very small problems. Two approaches were then investigated to tackle the multiple objectives, that is, Pareto front sampling with branch-and-cut and a genetic algorithm. Experimental results show that the genetic algorithm is not competitive to Pareto front sampling. It is also found that around 10% increase of mass can lead to around 50% decrease of OIC for dealing with the tradeoff between the mass the OIC. [PERSON] et al. [61] investigated the network topology optimization in DIMA using ILP formulation, by calculating the best trade-off solutions for multiple objectives. The objectives in the formulation are mass and OIC. Different from References [9; 33; 34], it considers the switch capacity of the signal and link for the node usage and edge usage for optimally finding ADCN topologies. Algorithms are derived to determine an optimal network topology for switches, switch locations, and interconnections, which reduces the economic cost while respecting bandwidth and safety limitations. Pareto front sampling is used to solve a series of single objective ILPs via the standard solver, for example, CPLEX. In addition, the authors further proposed a variant of the algorithm, which only considers pre-calculated paths for ignoring the paths that are unlikely to be chosen. As the number of variables is reduced, such a heuristic algorithm can reduce execution time for a large-scale system. [PERSON] et al. [62] considered a device assignment problem for allocating DIMA devices to installation locations of the aircraft, which is a hardware mapping only. Two objectives were considered for the optimization subject to the resource and safety constraints. A heuristic algorithm, proposed as a two-phase multiobjective local search, was developed to solve the optimization problem efficiently. First, a weighted sum approach was leveraged to approximate a good Pareto front. Second, Pareto local search was conducted on the solutions generated from the first phase to enable further approximation of Pareto front.