Fireworks Algorithm

This chapter presents the motivation of when, why, and how the fireworks algorithm (FWA), as a novel swarm intelligence optimization algorithm, came out. After a concise review on swarm intelligence domain, a brief introduction to FWA is presented with primary focuses on four aspects of theoretical analysis, algorithm study, problem solving, and applications. The characteristics and advantages of FWA are also described. Finally, overviews of FWA research are detailed with completed reference citations.

Inspired by fireworks explosions in the sky at night, the fireworks algorithm (FWA) was proposed by the author in 2010, through the observation of the fact that fireworks explosion is similar to the way an individual searches for optimal solution in swarm intelligence algorithms. Recently, FWA has received extensive concerns from many active researchers in the swarm intelligence community. This chapter presents the fundamental principle, main constitution, implementation, and performance of the FWA, aiming to elaborate the FWA systematically and completely. The main contents include the key components, realization, characteristic, and impact of operations of FWA, as well as comparisons with genetic algorithm and particle swarm optimization.

In order to describe the convergence analysis of FWA, a Markov stochastic process modeling Fireworks Algorithm has been defined and established, in the first part of this chapter, then to prove the global convergence of FWA and analyze the time complexity of FWA based on an absorbing Markov stochastic process of FWA. After that, the computation of the approximation region of expected convergence time of Fireworks Algorithm has also been given through a detailed derivation procedure. In the second part of this chapter, we will present 13 commonly used random number generators (RNGs) and also try to discuss the impact of the RNGs on the performance of FWA.

In this chapter, we present an empirical study on the influence of approximation approaches on accelerating the fireworks algorithm search by elite strategy [1]. We use three sampling data methods to approximate fitness landscape, i.e., the best fitness sampling method, the sampling distance near the best fitness individual sampling method, and the random sampling method. For each approximation method, we conduct a series of combinative evaluations with different sampling methods and sampling numbers for accelerating fireworks algorithm. The experimental evaluations on benchmark functions show that this elite strategy can enhance the fireworks algorithm search capability effectively. We also analyze and discuss the related issues on the influence of approximation model, sampling method, and sampling number on the fireworks algorithm acceleration performance.

Since FWA was proposed in 2010 [1], it has shown its significance and superiority in dealing with the optimization problems. However, calculation of the number of explosion sparks and the amplitude of firework explosion of FWA should dynamically control the exploration and exploitation of searching space with iteration. The mutation operator of FWA needs to generate the search diversity. This chapter provides a new method to calculate the number of explosion sparks and the amplitude of firework explosions. By designing a transfer function, the rank number of a firework is mapped to the scale of the calculation of scope and spark number of a firework explosion. A parameter is used to dynamically control the exploration and exploitation of FWA with iteration going on. In addition, this chapter uses a new random mutation operator to control the diversity of FWA search.

In this chapter, we present an improved version of the recently developed Fireworks Algorithm (FWA) based on several modifications, according to our previous work “Enhanced Fireworks Algorithm” [1]. A comprehensive study on the operators of conventional FWA revealed that the algorithm works surprisingly well on benchmark functions which have their optimum at the origin of the search space. However, when being applied on shifted functions, the quality of the results of conventional FWA deteriorates severely and worsens with increasing shift values, i.e., with increasing distance between function optimum and origin of the search space. Moreover, compared to other meta-heuristic optimization algorithms, FWA has high computational cost per iteration. In order to tackle these limitations, we present five major improvements on FWA: (i) a new minimal explosion amplitude check, (ii) a new operator for generating explosion sparks, (iii) a new mapping strategy for sparks which are out of the search space, (iv) a new operator for generating Gaussian sparks, and (v) a new operator for selecting the population for the next iteration.

In this chapter, we present a dynamic version of fireworks algorithm, which is an improved version of the recently developed enhanced fireworks algorithm (EFWA) based on an adaptive dynamic local search mechanism. In EFWA, the explosion amplitude of each firework is computed based on the quality of the firework’s current location. This explosion amplitude is limited by a lower bound which decreases with the number of iterations in order to avoid the explosion amplitude to be [close to] zero, and in order to enhance global search abilities at the beginning and local search abilities toward the later phase of the algorithm. As the explosion amplitude in EFWA depends solely on the fireworks’ fitness and the current number of iterations, this procedure does not allow for an adaptive optimization process. To deal with these limitations, a dynamic search fireworks algorithm (dynFWA) is proposed, which uses a dynamic explosion amplitude for the firework at the current best position. If the fitness of the best firework could be improved, the explosion amplitude will increase to speed up convergence. On the contrary, if the current position of the best firework could not be improved, the explosion amplitude will decrease in order to narrow the search area. In addition, we show that one of the EFWA operators, i.e., Gaussian mutation operator, can be removed in dynFWA without a loss in accuracy—this makes dynFWA computationally more efficient than EFWA.

The explosion amplitude in FWA is a key factor influencing the performance of fireworks algorithm, which needs to be controlled precisely. In this chapter, a new FWA algorithm called Adaptive Fireworks Algorithm is proposed by replacing the explosion amplitude operator in FWA with an adaptive method.

In the previous studies on FWAs, researchers ignored the cooperation and interaction between the individual fireworks in the swarm, which are the most important core for any swarm intelligence algorithm. By incorporating a probabilistically oriented explosion mechanism (POEM) into the conventional FWA, a novel Cooperative Fireworks Algorithm (CoFWA, for short) is proposed to enhance the interactions among the individual fireworks in the swarm. In the CoFWA, the POEM mechanisms of sparks generation and fireworks selection are well designed to strengthen the cooperative capability of the individual fireworks in the CoFWA. It turns out by many experiments that the CoFWA significantly outperforms two most recent variants of FWA (i.e., EFWA and dynFWA) and SPSO2011 and shows a competitive performance against the state-of-the-art swarm intelligence algorithms.

Fireworks algorithm has a broad research area and is suitable for combination with other algorithms to produce a new hybrid algorithm. This chapter focuses on hybrid fireworks algorithms, including Fireworks Algorithm with Differential Mutation (FWA-DM), Hybrid Fireworks Optimization Method with Differential Evolution Operators (FWA-DE), Culture Fireworks Algorithm (CFWA), and Hybrid Biogeography-Based Optimization and Fireworks Algorithm (BBO_FWA).

This chapter is to present some research works of FWA for multiobjective optimization, of which this is a successful instance like the multiobjective fireworks algorithm (MOFWA) proposed by Zheng et al. (2013) Applied Soft Computing 13(11):4253–4263, [1] for oil crop fertilization, which takes into consideration not only crop yield and quality but also energy consumption and environmental effects. The variable-rate fertilization (VRF) is a key aspect of prescription generation in precision agriculture, which typically involves multiple criteria and objectives. To solve the problem efficiently, a hybrid multiobjective fireworks optimization algorithm (MOFWA) is proposed to evolve a set of solutions to the Pareto optimal front by mimicking the explosion of fireworks. Especially, MOFWA uses the concept of Pareto dominance for individual evaluation and selection, and combines differential evolution (DE) operators to increase information sharing among the individuals. The proposed MOFWA outperforms some state-of-the-art methods on a set of real-world VRF problems.

This chapter is to present how to apply FWA to solving multi-objective optimization problems with the help of a hypervolume indicator such as S-metric, then proposes a S-metric multi-objective fireworks algorithm (S-MOFWA). The S-metric is a frequently used quality measure for solution sets comparison in evolutionary multi-objective optimization algorithms (EMOAs), which is also used to evaluate the contribution of a single solution among the solution sets. Traditional multi-objective optimization algorithms usually perform a \((\mu + 1)\) strategy and update the external archive one by one, while the proposed S-MOFWA performs a \((\mu + \mu )\) strategy, thus converging faster to a set of Pareto solutions by three steps: (1) Exploring the solution space by mimicking the explosion of fireworks; (2) Performing a simple selection strategy for choosing the next generation of fireworks according to their S-metric; (3) Utilizing an external archive to maintain the best solution set ever found, with a new archive definition and a novel updating strategy, which can update the archive with \(\mu \) solutions in a single process. The detailed comparison results with NSGA-II, SPEA2, and PESA2 demonstrate the efficiency of the proposed S-MOFWA.

Considering the excellent performance of FWA for real parameter optimization problems, a novel FWA variant was proposed for tackling discrete optimization problems, especially for Travelling Salesman Problem (TSP). We first give a brief introduction to TSP, followed by the detailed description of the discrete fireworks algorithm (DFWA).

In recent years, the graphics processing unit (GPU) has gained much popularity in general purpose computing, thanks to its low price and easy access. In this chapter, a very efficient FWA variant based on GPUs, so-called GPU–FWA for short, is introduced. GPU–FWA modifies the original FWA to suit the particular architecture of the GPU. It does not need special complicated data structure, thus making it easy to implement; meanwhile, it can make full use of the great computing power of GPUs. The key components of GPU–FWA are FWA search, attract-repulse mutation, and implementation which are elaborated in this chapter.

This chapter presents the use of swarm intelligence algorithms for non-negative matrix factorization (NMF) Janecek and Tan (2011) International Journal of Swarm Intelligence Research (IJSIR) 2(4):12–34, [1]. The NMF is a special low-rank approximation which allows for an additive parts-based and interpretable representation of the data. Here, we present our efforts to improve the convergence and approximation quality of NMF using five different meta-heuristics based on swarm intelligence. Several properties of the NMF objective function motivate the utilization of meta-heuristics: this function is non-convex, discontinuous, and may possess many local minima. The proposed optimization strategies are twofold: On one hand, we present a new initialization strategy for NMF in order to initialize the NMF factors prior to the factorization; on the other hand, we present an iterative update strategy which improves the accuracy per runtime for the multiplicative update NMF algorithm.

In this chapter, we will present the applications of fireworks algorithm for dealing with practical optimization problems including, document clustering, spam detection, image recognition, and seismic inversion problem [1]. The experimental results given herein suggest that fireworks algorithm is one of the most promising swarm intelligence algorithms in dealing with those practical problems.

Swarm robotics is an emerging research area combining swarm intelligence and robotics. Thanks to the recent achievements in optimization problem using swarm intelligence, searching problems in swarm robotics have attracted a large number of researchers. In searching problems, a swarm of robots searches for multiple targets in the environment without knowing any prior knowledge about the targets. This progress is quite similar with that of optimization problems in many aspects. Moreover, in most of the swarm robotics searching problems so far, some kinds of fitness functions are introduced for guiding the search of the swarm. This makes it a natural advantage to introduce swarm intelligence algorithms into swarm robotics. In this chapter, inspired by the fireworks algorithm, the group explosion strategy (GES) is proposed for searching multiple targets in swarm robotics. In the GES model, the whole swarm is divided into several groups. Robots in a group are spatially adjacent within the sensing range of each other. The swarm searches and collects targets in the environment without prior knowledge. Different groups do not intersect directly and their search for targets is parallel and independent. Through certain strategies, groups that run into each other will be re-arranged into new groups with possibly different members and search directions. In this way, inter-group cooperation can emerge in the swarm. The simulation results indicate that the proposed method with GES in this chapter shows great advantage against the comparison algorithm inspired from PSO.