# Multi-Objective PSO with Variable Number of Dimensions for Space Robot Path Optimization
## 1 Introduction
The VND problems have in common that their solution is built using more structurally similar segments and the optimal number of those segments is usually a priori unknown. The family of metameric problems contains a large variety of problems, starting from a wind farm turbine placement [14], composite laminate metamaterials design [15], and neural network architecture design [16], to biomedical inverse imaging [17]. A comprehensive review of VND problems can be found in [18]. Pure VND algorithms face two tasks during the optimization process: (1) they have to find the optimal dimension of the final solution (i.e., number of segments), and (2) they have to find the optimal values of the individual decision space variables (DSV) [19]. Clearly, the SRPP problem belongs to the multi-objective VND problems because the optimal number of turning points depends on the configuration of the obstacles and more than one objective should be considered during the path selection. Most of the initial works regarding VND optimization focused on modifications of genetic algorithms [20; 21] because of their natural ability to represent the grid-based problems with the binary DSVs. Then, the attention moved to less limited evolutionary optimization algorithms (EOA) such as differential evolution (DE) [22], or heuristic algorithms (HA) such as particle swarm optimization (PSO) [23; 24]. Single-objective algorithms based on PSO were proposed in [19; 25]. Although there are quite few studies dealing with a single-objective VNDPSO, e.g., [26; 27; 28; 29], studies dealing with multi-objective VNDPSO-based algorithms are very rare. The so-called social class MOPSO algorithm (SCMOPSO) was introduced in [30] to solve the VND MO problem of wireless sensor network design. In SCMOPSO, the particles are clustered according to their dimension to different classes. Then, the particles move to another class (change their dimension) after a certain number of iterations in which they do not improve. However, the exchange of information between particles is limited only to members of one class. Authors in [31] proposed the multi-objective variable length PSO (VLMOPSO). VLMOPSO uses an additional binary variable that enables or disables any DSV (the metameric segment to be more precise), which deteriorates the convergence of the algorithm since the decision space dimension is enlarged significantly. In [32], we proposed the variable number of dimensions generalized differential evolution (VNDGDE3)--a VND extension based on the multi-objectiveGDE3 algorithm [33]. In VNDGDE3, a large portion of the decision space variables for the crossover operation is selected randomly. This behavior results in the generation of infeasible DSVs (i.e., paths going through obstacles in the case of SRPP). Promising results of VNDPSO on single-objective problems such as printed circuit board decoupling [34], microwave imaging [17], or design of an antenna array [35] lead us to the idea of extending it to solve multi-objective problems as well. Therefore, the contributions of this paper can be summarized as follows. A novel variable number of dimension multi-objective optimization (VNDMOPSO) algorithm is proposed. Its convergence properties are assessed on a set of VND benchmark problems. The convergence of VNDMOPSO is compared with other available VND algorithms, namely VNDGDE3 [32] and VLOPOSO [31]. The VNDMOPSO is then applied to solve various instances of the multi-objective space robot pathfinding problem where both distance and number of turning points should be minimized. The rest of the paper is organized as follows. The space robot path planning problem is formulated as the VND two-objective problem in Section 2. Section 3 describes the proposed VNDMOPSO algorithm. Then, Section 4 contains the discussion of obtained results for the VNDMOPSO algorithm and compares them with results obtained by the reference algorithms. The influence of control parameters of VNDMOPSO is assessed here, also. Finally, Section 5 reviews the main conclusions of the paper.