Harris hawks optimization: Algorithm and applications

Highlights

• A mathematical model is proposed to simulate the hunting behavior of Harris’ Hawks.

• An optimization algorithm is proposed using the mathematical model.

• The proposed HHO algorithm is tested on several benchmarks.

• The performance of HHO is also examined on several engineering design problems.

• The results show the merits of the HHO algorithm as compared to the existing algorithms.

Abstract

In this paper, a novel population-based, nature-inspired optimization paradigm is proposed, which is called Harris Hawks Optimizer (HHO). The main inspiration of HHO is the cooperative behavior and chasing style of Harris’ hawks in nature called surprise pounce. In this intelligent strategy, several hawks cooperatively pounce a prey from different directions in an attempt to surprise it. Harris hawks can reveal a variety of chasing patterns based on the dynamic nature of scenarios and escaping patterns of the prey. This work mathematically mimics such dynamic patterns and behaviors to develop an optimization algorithm. The effectiveness of the proposed HHO optimizer is checked, through a comparison with other nature-inspired techniques, on 29 benchmark problems and several real-world engineering problems. The statistical results and comparisons show that the HHO algorithm provides very promising and occasionally competitive results compared to well-established metaheuristic techniques. Source codes of HHO are publicly available at http://www.alimirjalili.com/HHO.html and http://www.evo-ml.com/2019/03/02/hho.

Introduction

Many real-world problems in machine learning and artificial intelligence have generally a continuous, discrete, constrained or unconstrained nature [1], [2]. Due to these characteristics, it is hard to tackle some classes of problems using conventional mathematical programming approaches such as conjugate gradient, sequential quadratic programming, fast steepest, and quasi-Newton methods [3], [4]. Several types of research have verified that these methods are not efficient enough or always efficient in dealing with many larger-scale real-world multimodal, non-continuous, and non-differentiable problems [5]. Accordingly, metaheuristic algorithms have been designed and utilized for tackling many problems as competitive alternative solvers, which is because of their simplicity and easy implementation process. In addition, the core operations of these methods do not rely on gradient information of the objective landscape or its mathematical traits. However, the common shortcoming for the majority of metaheuristic algorithms is that they often show a delicate sensitivity to the tuning of user-defined parameters. Another drawback is that the metaheuristic algorithms may not always converge to the global optimum. [6]

In general, metaheuristic algorithms have two types [7]; single solution based (i.g. Simulated Annealing (SA) [8]) and population-based (i.g. Genetic Algorithm (GA) [9]). As the name indicates, in the former type, only one solution is processed during the optimization phase, while in the latter type, a set of solutions (i.e. population) are evolved in each iteration of the optimization process. Population-based techniques can often find an optimal or suboptimal solution that may be same with the exact optimum or located in its neighborhood. Population-based metaheuristic (P-metaheuristics) techniques mostly mimic natural phenomena [10], [11], [12], [13]. These algorithms start the optimization process by generating a set (population) of individuals, where each individual in the population represents a candidate solution to the optimization problem. The population will be evolved iteratively by replacing the current population with a newly generated population using some often stochastic operators [14], [15]. The optimization process is proceeded until satisfying a stopping criteria (i.e. maximum number of iterations) [16], [17].

Based on the inspiration, P-metaheuristics can be categorized in four main groups [18], [19] (see Fig. 1): Evolutionary Algorithms (EAs), Physics-based, Human-based, and Swarm Intelligence (SI) algorithms. EAs mimic the biological evolutionary behaviors such as recombination, mutation, and selection. The most popular EA is the GA that mimics the Darwinian theory of evolution [20]. Other popular examples of EAs are Differential Evolution (DE) [21], Genetic Programming (GP) [20], and Biogeography-Based Optimizer (BBO) [22]. Physics-based algorithms are inspired by the physical laws. Some examples of these algorithms are Big-Bang Big-Crunch (BBBC) [23], Central Force Optimization (CFO) [24], and Gravitational Search Algorithm (GSA) [25]. Salcedo-Sanz [26] has deeply reviewed several physic-based optimizers. The third category of P-metaheuristics includes the set of algorithms that mimic some human behaviors. Some examples of the human-based algorithms are Tabu Search (TS) [27], Socio Evolution and Learning Optimization (SELO) [28], and Teaching Learning Based Optimization (TLBO) [29]. As the last class of P-metaheuristics, SI algorithms mimic the social behaviors (e.g. decentralized, self-organized systems) of organisms living in swarms, flocks, or herds [30], [31]. For instance, the birds flocking behaviors is the main inspiration of the Particle Swarm Optimization (PSO) proposed by Eberhart and Kennedy [32]. In PSO, each particle in the swarm represents a candidate solution to the optimization problem. In the optimization process, each particle is updated with regard to the position of the global best particle and its own (local) best position. Ant Colony Optimization (ACO) [33], Cuckoo Search (CS) [34], and Artificial Bee Colony (ABC) are other examples of the SI techniques.

Regardless of the variety of these algorithms, there is a common feature: the searching steps have two phases: exploration (diversification) and exploitation (intensification) [26]. In the exploration phase, the algorithm should utilize and promote its randomized operators as much as possible to deeply explore various regions and sides of the feature space. Hence, the exploratory behaviors of a well-designed optimizer should have an enriched-enough random nature to efficiently allocate more randomly-generated solutions to different areas of the problem topography during early steps of the searching process [35]. The exploitation stage is normally performed after the exploration phase. In this phase, the optimizer tries to focus on the neighborhood of better-quality solutions located inside the feature space. It actually intensifies the searching process in a local region instead of all-inclusive regions of the landscape. A well-organized optimizer should be capable of making a reasonable, fine balance between the exploration and exploitation tendencies. Otherwise, the possibility of being trapped in local optima (LO) and immature convergence drawbacks increases.

We have witnessed a growing interest and awareness in the successful, inexpensive, efficient application of EAs and SI algorithms in recent years. However, referring to No Free Lunch (NFL) theorem [36], all optimization algorithms proposed so-far show an equivalent performance on average if we apply them to all possible optimization tasks. According to NFL theorem, we cannot theoretically consider an algorithm as a general-purpose universally-best optimizer. Hence, NFL theorem encourages searching for developing more efficient optimizers. As a result of NFL theorem, besides the widespread studies on the efficacy, performance aspects and results of traditional EAs and SI algorithms, new optimizers with specific global and local searching strategies are emerging in recent years to provide more variety of choices for researchers and experts in different fields.

In this paper, a new nature-inspired optimization technique is proposed to compete with other optimizers. The main idea behind the proposed optimizer is inspired from the cooperative behaviors of one of the most intelligent birds, Harris’ Hawks, in hunting escaping preys (rabbits in most cases) [37]. For this purpose, a new mathematical model is developed in this paper. Then, a stochastic metaheuristic is designed based on the proposed mathematical model to tackle various optimization problems. It should be noted that the name Harris’s hawk and a similar inspiration have been used in [38], in which the authors modified the mathematical model of Grey Wolf Optimizer (MOGWO and GWO) and used it to solve multi-objective optimization problems. There were no new mathematical equations and the algorithm was developed completely based on MOGWO and GWO. In this work, however, we proposed new mathematical models to mimic all the stages of hunts used by Harris’s hawks and solve single-objective optimization problems efficiently

The rest of this research is organized as follows. Section 2 represents the background inspiration and info about the cooperative life of Harris’ hawks. Section 3 represents the mathematical model and computational procedures of the HHO algorithm. The results of HHO in solving different benchmark and real-world case studies are presented in Section 4. Section 5 discusses the results. Finally, Section 6 concludes the work with some useful perspectives.