The continuous-discrete PSO algorithm for shape formation problem of multiple agents in two and three dimensional space

Shape formation problem of agents in the two or three dimensional space is one of the most important and challenging topics in the fields of evolutionary computation and multi-agents system, etc. Firstly, the basic concepts and objective functions of shape formation problem are introduced to deeply understand the considered shape formation problem. Three theorems of shape formation problem with three agents are addressed by the Lagrangian multiplier method, however, the Lagrangian multiplier method difficultly solves optimal shape formation problem where the number of agents is strictly larger than 3 and the number of constraints is larger than 2. In order to tackle the continuous and discrete optimization problem, the continuous-discrete particle swarm optimization (CDPSO) algorithm is developed to search for the rotated angle of the desired shape and the matching pair between points in the initial shape and points in the desired shape. Additionally, the parameters in CDPSO algorithm are set by three theorems on convergence analysis of the random PSO algorithm. To demonstrate the effectiveness and the feasibility of the CDPSO algorithm on the shape formation problem, numerical results not only discuss the optimal virtual helicopters formation between two typical shapes in the three dimensional space, but also provide one searching and rescuing strategy of MH370 plane to minimize the whole moving distance of all virtual rescuing ships. Moreover, the shape conversion problem including multiple agents is also solved by the CDPSO algorithm when the number of agents is equal to 100, 200, 500 and 1000. Additionally, the optimization results and the computational time are compared among the Lagrange multiplier method, CDPSO, CDDE, CDGA, CDPSOI and CDPSOE algorithms.