## 1 Introduction  
Nature-inspired optimization algorithms have become of increasing interest to many researchers in optimization fields in recent years [1]. After half a century of development, nature-inspired optimization algorithms have formed a great family. It not only has a wide range of contact with biological, physical, and other basic science, but also involves many fields such as artificial intelligence, artificial life, and computer science. Due to the differences in natural phenomena, these optimization algorithms can be roughly divided into three types. These are algorithms based on biological evolution, algorithms based on swarm behavior, and algorithms based on physical phenomena. Typical biological evolutionary algorithms are Evolutionary Strategies (ES) [2], Evolutionary Programming (EP) [3], the Genetic Algorithm (GA) [4; 5; 6], Genetic Programming (GP) [7], Differential Evolution (DE) [8], the Backtracking Search Algorithm (BSA) [9], Biogeography-Based Optimization (BBO) [10; 11], and the Differential Search Algorithm (DSA) [12]. In the last decade, swarm intelligence, as a branch of intelligent computation models, has been gradually rising [13]. Swarm intelligence algorithms mainly simulate biological habits or behavior, including foraging behavior, search behavior and migratory behavior, brooding behavior, and mating behavior. Inspired by these phenomena, researchers have designed many intelligent algorithms, such as Ant Colony Optimization (ACO) [14], Particle Swarm Optimization (PSO) [15], Bacterial Foraging (BFA) [16], Artificial Bee Colony (ABC) [17; 18; 19], Group Search Optimization (GSO) [20],Cuckoo Search (CS) [21; 22], Seeker Optimization (SOA) [23], the Bat Algorithm (BA) [24], Bird Mating Optimization (BMO) [25], Brain Storm Optimization [26; 27] and the Grey Wolf Optimizer (GWO) [28]. In recent years, in addition to the above two kinds of algorithms, intelligent algorithms simulating physical phenomenon have also attracted a great deal of researchers' attention, such as the Gravitational Search Algorithm (GSA) [29], the Harmony Search Algorithm (HSA) [30], the Water Cycle Algorithm (WCA) [31], the Intelligent Water Drops Algorithm (IWDA) [32], Water Wave Optimization (WWO) [33], and States of Matter Search (SMS) [34]. The classical optimization algorithm about a physical phenomenon is simulated annealing, which is based on the annealing process of metal [35; 36]. Though these meta-heuristic algorithms that have many advantages over traditional algorithms, especially in NP hard problems, have been proposed to tackle many challenging complex optimization problems in science and industry, there is no single approach that is optimal for all optimization problems [37]. In other words, an approach may be suitable for solving these problems, but it is not suitable for solving those problems. This is especially true as global optimization problems have become more and more complex, from simple uni-modal functions to hybrid rotated shifted multimodal functions [38]. Hence, more innovative and effective optimization algorithms are always needed. A natural phenomenon provides a rich source of inspiration to researchers to develop a diverse range of optimization algorithms with different degrees of usefulness and practicability. In this paper, a new meta-heuristic optimization algorithm inspired by the phase transition phenomenon of elements in a natural system for continuous global optimization is proposed, which is termed phase transition-based optimization (PTBO). From a statistical mechanics point of view, a phase transition is a non-trivial macroscopic form of collective behavior in a system composed of a number of elements that follow simple microscopic laws [39]. Phase transitions play an important role in the probabilistic analysis of combinatorial optimization problems, and are successfully applied in Random Graph, Satisfiability, and Traveling Salesman Problems [40].