# A Novel Method for Building Displacement Based on Multipopulation Genetic Algorithm
## 2 Related Work
### Optimal Approaches
#### 2.2.2 Functional Optimization Approaches  
Functional optimization approaches introduce models and ideas from other fields, such as the mechanics of materials, physics and engineering, and construct displacement mathematical equation systems (e.g., least squares adjustment method, energy minimization method, finite element method (FEM)). Compared with combinatorial optimization approaches, functional optimization approaches are more complex, the key of which is how to transform related cartographic constraints into mathematical equations. [PERSON] [29; 30; 31] proposed the constraint method for solving spatial conflicts, which expressed the constraints as linear equations and used a least squares adjustment to find the optimal solution. Since the constraints are contradictory and all constraints cannot be completely satisfied, the constraints were set to different weights according to the constraint type to obtain the optimal compromise scheme. Again, [PERSON] [32] adopted the same idea and used least squares adjustment for the displacement and shape simplification of building groundplans. [PERSON] [33] modelled the displacement as the deformation of an elastic body and proposed the FEM to handle conflicts during the generalization of maps. The FEM first discretized the whole map space though constrained Delaunay triangulation and then introduced boundary constraints to control the shape and size of objects. Because of the holistic propagation of displacement, the FEM can maintain spatial relationships and patterns. However, the FEM cannot resolve all spatial conflicts and will cause large deformations of objects. For the linear feature displacement problem, [PERSON] and [PERSON] [34] proposed the snake algorithm to resolve spatial conflicts. In the snake model, the deformation of linear objects generates internal energy, and the conflicts between linear objects and other objects generate external energy. When the total energy is the minimum, the linear objects obtain the optimal position and shape. The snake model is very suitable for the displacement of linear objects because it considers the propagation of the displacement, which can maintain the shape of linear objects during the displacement. Later, [PERSON] et al. [35] combined the snake model into a multi-agent process to resolve multiple types of spatial conflicts and obtained good results. To solve the area feature displacement problem, [PERSON] et al. [36; 37] adopted the idea of the snake model and proposed the Beams algorithm. The Beams model first adopts an enhanced minimum spanning tree as an elastic truss to capture the important spatial patterns of building clusters. Then, the Beams algorithm iteratively deforms the truss to minimize energy until a user-defined threshold value is achieved. [PERSON] et al. [38] proposed a fully automated building displacement algorithm by combining the constrained Delaunay triangulation skeleton and elastic beams, which is an improvement on the elastic beams algorithm. Through the literature summarized above, we can conclude that (1) although the sequential approaches are relatively simple to resolve the spatial conflicts by one displacement, they may generate secondary conflicts; (2) the functional optimization approaches use mathematical ideas to model the spatial relationships and constraints between objects, which can well maintain the spatial relations between objects in the process of displacement, but the modelling process of this kind of algorithm is complex and cannot solve all possible spatial conflicts; and (3) the combinatorial optimization method uses a heuristic search strategy to obtain the optimal solution, which meets the current goal of the intelligent and automatic displacement methods. However, the current combinatorial optimization algorithms, such as GA and IGA, cannot resolve all possible spatial conflicts and even generate secondary conflicts.