# A Population-Based Search Approach to Solve Continuous Distributed Constraint Optimization Problems
## 1 Introduction  
The _Distributed Constraint Optimization Problems_ (DCOPs) [1, 2, 3] are a powerful framework for multi-agent modeling. In DCOPs, multiple agents communicate directly or indirectly and coordinate their value assignments to optimize the aggregate constraint utility. DCOPs have been widely applied in practical problems such as meeting scheduling [4, 5, 6], smart home [7, 8], sensor networks [9, 10], multi-robot coordination [11, 12], and path coordination [13]. Simultaneously, some important improvements and extensions have been proposed for DCOPs, such as the improving advanced methods [14, 15, 16, 17, 18], and the extensions of Asymmetric DCOPs [19, 20, 21, 22] and Dynamic DCOPs [23, 24]. The DCOP algorithms are simply classified as complete and incomplete algorithms. The complete algorithms aim to provide a global optimum solution, but the computationaland memory overheads are expensive, such as _Synchronous Branch-and-Bound_ (SyncBB) [25], _Asynchronous Distributed OPTtimization_ (ADOPT) [2], _Distributed Pseudo-tree Optimization Procedure_ (DPOP) [3; 26], and _Pseudo-trees-Forward Bounding_ (PT-FB) [27]. Conversely, the incomplete algorithms obtain an approximate solution by reducing computational and memory overheads, such as _Distributed Stochastic Algorithm_ (DSA) [28], _Max-Sum_ (MS) [9; 16], _Maximum Gain Message_ (MGM) [29], ACO_DCOP [30], and LSGA [31]. Although the DCOP framework plays an important role in applications of discrete variables (e.g., action sets of machines in the smart home), they make less sense in applications of continuous variables (e.g., positions of robots in the mobile network). Therefore, [PERSON] et al. [32] proposed the _Continuous DCOP_ (C-DCOP) framework to extend DCOPs whose variables are continuous and the constraint utilities are functional forms. The C-DOCP algorithms include _Continuous MS_ (CMS) [32], _Hybrid CMS_ (HCMS) [33], _Bayesian DPOP_ (B-DPOP) [34], _Particle Swarm Based Continuous DCOP_ (PCD) [35], _Exact Continuous DPOP_ (EC-DCOP) [36] and extensions (_Approximate Continuous DPOP_ (AC-DPOP), _Clustered AC-DPOP_ (CAC-DPOP), _Continuous DSA_ (C-DSA)), _Continuous Cooperative Constraint Approximation_ (C-CoCoA) [37], and _Particle Swarm with Local Decision Based Continuous DCOP_ (PCD-LD) [38]. Firstly, CMS and HCMS extend discrete MS, where CMS approximates utility functions to piecewise linear functions and HCMS introduces a continuous nonlinear optimization method. However, they are both limited by the forms of utility functions. Secondly, B-DPOP combines the _Bayesian optimization_ with DPOP, but it does not guarantee an intermediate solution quality before convergence. Further, EC-DPOP provides the global optimum solution. AC-DPOP and CAC-DPOP provide approximate solutions and reduce communication overheads by limiting the size of messages. Nevertheless, they both incur exponential memory and computational overheads. Then, PCD introduces _Particle Swarm Optimization_ (PSO) in C-DCOPs to reduce the computational and memory overhead. PCD-LD extends PCD to balance solution quality with memory overhead. Finally, C-CoCoA and C-DSA provide two low-cost methods to solve C-DCOPs, but none of the two algorithms have anytime properties. Based on the above analysis, we propose two population-based approaches by using basic mathematical operators such as addition and multiplication only to solve C-DCOPs, _Population-based Local Search Algorithm_ (PLSA), and the extension, _Population-based Global Search Algorithm_ (PGSA). The proposed PLSA and PGSA can solve problems with any form of utility functions and outperform gradient-based methods in terms of computational and memory overhead [35]. In the previous paper [39], the idea of population-based local search is applied to centralized optimization problems and exhibits excellent performance. Specifically, the algorithm (i) uses some elite solutions and useful information to guide local search; and (ii) systematically updates the population.