# A Phase Angle-Modulated Bat Algorithm with Application to Antenna Topology Optimization
## 1 Introduction  
With the rapid development of modern society, big data have become a research hotspot and a large number of data sets need to be optimized. High-dimensional optimization problems exist in many application scenarios, such as complex neural network weight optimization [1], text mining and analysis [2], sensor networking [3], vehicle routing [4], and antenna topology optimization [5]. To solve such optimization problems, meta-heuristic algorithms have been proposed as a reliable solution [6]. As an improvement of heuristic algorithms, meta-heuristic algorithms are the combination of random algorithms and local search algorithms. Meta-heuristic algorithms can find the optimal solution to the problem under limited time and space conditions. A large number of meta-heuristic algorithms have been proposed including the genetic algorithm (GA) [7], particle swarm optimization (PSO) [8], differential evolution (DE) [9], artificial bee colony (ABC) [10], the dragonfly algorithm (DA) [11], the fireworks algorithm (FA) [12], brain storm optimization (BSO) [13], harmony search (HS) [14], moth search (MS) [15], and the bat algorithm (BA) [16]. The BA, proposed by [PERSON] in 2010, is a meta-heuristic algorithm developed according to the principle of bat echolocation [16]. Compared with other meta-heuristic algorithms, BA has two characteristics: (i) frequency tuning can facilitate the control of the optimization direction of the algorithm, and (ii) dynamic conversion of global search and local search can effectively avoid premature convergence of the algorithm. The advantages of BA are its simplicity, fewer parameters, robustness, and ease of implementation. Due to its superiority, BA has been applied in various fields, such as wireless sensor network deployment in the 3D environment [17], data clustering [18], the low-carbon job shop scheduling problem [19], optimization of reactive power dispatch [20], and the blood vessel segmentation problem [21]. BA was first proposed to address the optimization problem of the continuousreal search space. In 2014, [PERSON] proposed a binary bat algorithm (BBA) [22] for discrete optimization problems. BBA maps continuous variables to discrete variables by introducing a transfer function. Similar operations are used in the binary particle swarm optimization algorithm (BPSO) [23] and the binary dragonfly algorithm (BDA) [11]. However, as the problem dimension increases, the BBA exploration capacity may become poor, and the global search capability will decrease [24]. Some researchers have also made improvements for solving the problem of insufficient optimization performance of traditional binary optimization algorithms. [PERSON] et al. proposed a binary fireworks algorithm (BFWA) [25] by introducing firework explosion, mutation explosion, and the selection of locations and showed its better convergence compared with GA, BPSO, and the binary cuckoo search algorithm (BCS). In [26], a binary multi-scale quantum harmonic oscillator algorithm (BMQHOA) with a genetic operator was proposed. The framework of BMQHOA consists of three nested phases including energy level stabilization, energy level decline, and scale adjustment, and it is superior to the BBA, BDA, and BPSO on computational accuracy, convergence capability, and stability. In 2015, [PERSON] et al. proposed an angle-modulated bat algorithm called AMBA [27]. The introduction of the angle modulation technology can achieve the rapid optimization and convergence by reducing the dimensions of variables. Furthermore, to improve the optimization ability of AMBA, some researchers proposed the amplitude angle-modulated bat algorithm (A-AMBA) [28] by adding one more parameter to control the amplitude of the sine wave in the generating function. Recently, meta-heuristic algorithms have found their applications in antenna designs because of their good global optimization ability.