## 4 Optimal Mapping In Dima
### Solution of Multi-Objective Optimization
#### 4.2.3 Lexicographic Optimization  
Lexicographic optimization [63] is an iterative method to handle multiple objectives. It defines a fixed order of the objectives to be optimized in an iterative manner. A priority is assigned to each objective such that the set of objectives form a lexicographic order. It iteratively optimizes each objective while preserving the optimal values from previous iterations. Hence, an objective with a higher priority explores a larger solution space and makes it closer to its optimal value. [PERSON] et al. [64] proposed a physical and function layer modeling for the hardware and software mapping and addressed the DIMA optimization problem using lexicographic optimization. A binary program is presented to formulate the mapping, with the consideration of the slot resource, mass resource, cooling resource and segregation. The objective is to minimize the CPM burden, mass and SSC by optimally finding the installation mapping of the CPMs and RDCs. The CPM burden is defined as resource consumption of all tasks in the CPM device, which reflects the reliability. Besides the conventional measures on the costs and weights of DIMA devices, the reliability is particularly given with higher priority while the cost are given lower priority, including mass, SSC, OIC and IPC. The authors proposed an improved lexicographic optimization to manage the trade-off between the different quality measures. Unlike the traditional lexicographic optimization, the authors applied a relaxed solution space of the preceding optimization. The optimal value of preceding step is relaxed and an offset is added to relax constraints. By using that method, good trade-off between measures can be achieved, while higher priority of reliability is guaranteed. In addition, performance comparison between DIMA and IMA is conducted in the research, showing that DIMA can achieve higher reliability than IMA, while having more SSC and mass for CPMs and RDCs but less SSC and mass for wires. This is due to the distributed architecture of DIMA where all the peripherals are distributively connected to the RDCs. A list of previous work on DIMA optimization that we have discussed throughout this section are summarized in Tables 1 and 2. Table 1 summarizes the hardware and software mapping for DIMA optimization from different perspectives of assignment. Furthermore, Table 2 presents the solutions and approaches with more detail. It should be noted that the approximate algorithms are not adopted in the previous work. Most of the work relies on the exact algorithms using standard solvers while there are a few works using metaheuristic or heuristic algorithms that may not be able to guarantee the quality of solutions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Reference**} & **Task Assignment** & **Device Assignment** & **Link Assignment** & **Signal Assignment** \\ \hline
[34] & ✓ & ✓ & & \\ \hline
[48] & ✓ & & & \\ \hline
[28] & ✓ & & & \\ \hline
[35] & ✓ & ✓ & & \\ \hline
[49] & & ✓ & & \\ \hline
[54] & & & & ✓ & \\ \hline
[33] & ✓ & ✓ & & \\ \hline
[9] & ✓ & ✓ & & \\ \hline
[61] & & & ✓ & ✓ \\ \hline
[62] & & ✓ & & \\ \hline
[64] & & ✓ & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DIMA optimization under different mappings.