Spherical evolution for solving continuous optimization problems

Highlights

• Search pattern and search style are proposed by a mathematical model.

• Spherical search style is proposed.

• Spherical evolution algorithm is proposed.

• Data clustering problem is achieved by the proposed algorithm.

Abstract

In these years, more and more nature-inspired meta-heuristic algorithms have been proposed; search operators have been their core problem. The common characteristics or mechanism of search operators in different algorithms have not been represented by a standard format. In this paper, we first propose the concept of a search pattern and a search style represented by a mathematical model. Second, we propose a new search style, namely a spherical search style, inspired by the traditional hypercube search style. Furthermore, a spherical evolution algorithm is proposed based on the search pattern and spherical search style. At the end, 30 benchmark functions of CEC2017 and a real-world optimization problem are tested. Experimental results and analysis demonstrate that the proposed method consistently outperforms other state-of-the-art algorithms.

Introduction

Optimization has been an active area of research to provide satisfactory solutions for complex real-world problems over the past few decades. These problems have been more complex with the association of differentiability, multi-modality, increasing dimensionality, rotation characteristics and lack of knowledge of their mathematical expressions. This promotes researchers to develop accurate, fast and computationally efficient optimization algorithms.

More and more nature-inspired meta-heuristic approaches have been proposed and have achieved their goals successfully for solving some problems. The Genetic Algorithm (GA) [1] was an Evolutionary Algorithm (EA) based on Darwin’s theory of natural selection. The Particle Swarm Optimization (PSO) [2] was a well-known optimization algorithm mimicking the social behavior of birds flocking or fish schooling. The Differential Evolution (DE) [3] was an evolutionary algorithm, which was achieved by the mutation operator, crossover operator, and selection operator. The Ant Colony Optimization (ACO) [4] imitated the foraging behavior of an ant colony. The Artificial Bee Colony Algorithm (ABC) [5] was inspired by the foraging behavior of a bee colony, including onlookers, employed bees, and scouts. The Biogeography-Based Optimization Algorithm (BBO) [6] was achieved by a mathematical model of biogeography, which described how species migrate from one island to another, how new species arise, and how species become extinct. The Teaching-Learning-Based Optimization (TLBO) [7] was inspired by the study process between teachers and students. The Gravitational Search Algorithm (GSA) [8] was based on the law of gravity and mass interactions. The searcher agents were a collection of masses that interact with each other based on the Newtonian gravity theory and laws of motion. The Artificial Algae Algorithm (AAA) [9] was inspired by the living behaviors of a microalgae, photosynthetic species. The algorithm was based on its evolutionary process, adaptation process, and the movement of microalgae. The Thermal Exchange Optimization (TEO) [10] was based on Newton’s law of cooling. Each agent was considered as a cooling object, and by associating another agent as environment, a heat transference and thermal exchange happens between them. The Ant Lion Optimizer (ALO) [11] mimics the hunting mechanism of ant lions in nature. The five main steps of hunting prey (the random walk of ants, building traps, entrapment of ants in traps, catching preys, and re-building traps) were implemented. The Whale Optimization Algorithm (WOA) [12] mimics the social behavior of humpback whales. The algorithm was inspired by the bubble-net hunting strategy. The Grasshopper Optimization Algorithm (GOA) [13] mathematically modeled and mimicked the behavior of grasshopper swarms in nature. The Gray Wolf Optimizer [14] mimicked the leadership hierarchy and hunting mechanism of gray wolves in nature. Four types of gray wolves, namely, alpha, beta, delta, and omega, were employed for simulating the leadership hierarchy. The Sine Cosine Algorithm (SCA) [15] created multiple initial random candidate solutions and required them to fluctuate outwards or towards the best solution using a mathematical model based on sine and cosine functions. The Firefly Algorithm [16] was based on the behavior of flashing lights of fireflies. A less bright firefly gets attracted to the nearest bright firefly within its visual range, and the brightness of a firefly was determined by the objective function. The Vortex Search Algorithm (VS) [17] was inspired from the vortex pattern created by the vortical flow of the stirred fluids. To provide a good balance between the explorative and exploitative behavior of a search, the proposed method modeled its search behavior as a vortex pattern by using an adaptive step size adjustment scheme. The Harmony Search (HS) [18] emulated the process of playing different musical instruments based on an analogy to physical phenomena. The Charge System Search (CSS) [19] was based on electrostatic and Newtonian mechanics laws. The Mine Blast Algorithm (MBA) [20] was inspired by the mine bomb explosion. The Water Cycle Algorithm [21] was inspired from nature and based on the observation of the water cycle process. Streams flow to rivers or directly flow to the sea, which was modeled to solve optimization problems. The Water Wave Optimization (WWO) [22] was inspired by the shallow water wave theory. The phenomena of water waves, such as propagation, refraction, and breaking, were used to derive effective mechanisms for searching in a high-dimensional solution space. The Symbiotic Organisms Search (SOS) [23] was modeled based on three fundamental relationship structures, namely mutualism, commensalism and parasitism. These natural phenomena can be associated with the specific problem objective function to be optimized or solved. The Kidney-inspired Algorithm (KA) [24] was inspired by the kidney process in the human body. In this algorithm the solutions were filtered at a rate that was calculated based on the mean of objective functions of all solutions in the current population of each iteration. The Krill Herd Algorithm (KH) [25] was based on the simulation of the herding behavior of krill individuals. The minimum distances of each individual krill from food and from highest density of the herd were considered as the objective function for the krill movement. The Bird Mating Optimizer (BMO) [26] was inspired by mating strategies of bird species during mating season. BMO imitated the behavior of bird species metaphorically to breed broods with superior genes for designing optimum searching techniques. The Salp Swarm Algorithm (SSA) [27] was inspired by the swarming behavior of salps when navigating and foraging in oceans. The Invasive Tumor Growth Optimization (ITGO) [28] was inspired by the mechanism of tumor growth. The Across Neighborhood Search (ANS) [29] was motivated by a neighborhood search strategy. An individual directly searches across the neighborhoods of multiple superior solutions with the guidance of a Gaussian distribution. The Black Hole (BH) [30] was inspired by the black hole phenomenon. At each iteration of the Black Hole Algorithm, the best candidate was selected to be the black hole, which then started pulling other candidates around it, called stars. The Lightning Attachment Procedure Optimization (LAPO) [31] mimics the lightning attachment procedure including the downward leader movement, the upward leader propagation, the unpredictable trajectory of lightning downward leader, and the branch fading feature of lightning. The Forest Optimization Algorithm (FOA) [32] was inspired by the few trees in the forests that can survive for several decades, while other trees live only for a limited period. In FOA, the seeding procedure of the trees was simulated so that some seeds fall just under the trees, while others were distributed in wide areas by natural forces and the animals that feed on the seeds or fruits. Spotted hyena optimizer [33] was inspired by the behavior of spotted hyenas. The main concept behind this algorithm was the social relationship between spotted hyenas and their collaborative behavior. The three basic steps of SHO was searching for prey, encircling, and attacking prey; all three were mathematically modeled and implemented. Evolution strategies [34] belong to the general field of genetic algorithms. Evolutionary programming [35] was inspired by the inherent relationship and behavior between parents and offspring, which simulated the evolution at species level without a crossover operator. The Covariance Matrix Adaptation Evolution Strategy (CMA-ES) [36] was an improved version of Evolution Strategies, in which a covariance matrix adaptation was used.

By researching the mechanisms of nature-inspired meta-heuristic approaches (NMH), we can discover that the core problems of NMH algorithm’s design are focused on two aspects. The first issue is the connection between the search operator and the search pattern and search style. The search operators of an NMH algorithm can determine that if a solution of an individual in population can find a better solution. In fact, there are different operators for the different NMH algorithms. Are there some common characteristics or principles of NMH algorithms such as search pattern or search style? To date, we have not found the research to verify this. The second issue is the individual selection method for the search operators. Many individual selection methods, such as roulette wheel selection [37], tournament selection [38], and stochastic universal sampling method [39], have been proposed. More individual selection methods are inspired by the characteristics of individuals in nature. In this paper, we focus on research of these issues and attempt to propose the search pattern and search style. Moreover, we design a spherical evolution algorithm based on the search pattern and search style.

The rest of the paper is structured as follows. Section 2 introduces reviews of the related literatures. Section 3 proposes search pattern and search style. Section 4 presents the spherical evolution algorithm based on search pattern and search style. Section 5 introduces the experimental results and analysis. Section 6 concludes this study and gives directions for future research.