List of metaphor-based metaheuristics

This is a chronologically ordered list of metaphor-based metaheuristics and swarm intelligence algorithms, sorted by decade of proposal.

Simulated annealing is a probabilistic algorithm inspired by annealing, a heat treatment method in metallurgy. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For problems where finding the precise global optimum is less important than finding an acceptable local optimum in a fixed amount of time, simulated annealing may be preferable to alternatives such as gradient descent.

The analogue of the slow cooling of annealing is a slow decrease in the probability of simulated annealing accepting worse solutions as it explores the solution space. Accepting worse solutions is a fundamental property of metaheuristics because it allows for a more extensive search for the optimal solution.

The ant colony optimization algorithm is a probabilistic technique for solving computational problems that can be reduced to finding good paths through graphs. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm aimed to search for an optimal path in a graph based on the behavior of ants seeking a path between their colony and a source of food. The original idea has since diversified to solve a wider class of numerical problems, and as a result, several problems^{[example needed]} have emerged, drawing on various aspects of the behavior of ants. From a broader perspective, ACO performs a model-based search and shares some similarities with the estimation of distribution algorithms.

Particle swarm optimization is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. It solves a problem by having a population of candidate solutions, dubbed particles, and moving these particles around in the search space according to simple mathematical formulae^[which?] over the particle's position and velocity. Each particle's movement is influenced by its local best known position but is also guided toward the best-known positions in the search space, which are updated as better positions are found by other particles. This is expected to move the swarm toward the best solutions.

PSO is originally attributed to Kennedy, Eberhart and Shi and was first intended for simulating social behaviour as a stylized representation of the movement of organisms in a bird flock or fish school. The algorithm was simplified, and it was observed to be performing optimization. The book by Kennedy and Eberhart describes many philosophical aspects of PSO and swarm intelligence. An extensive survey of PSO applications is made by Poli. A comprehensive review of theoretical and experimental works on PSO has been published by Bonyadi and Michalewicz.

Harmony search is a phenomenon-mimicking metaheuristic introduced in 2001 by Zong Woo Geem, Joong Hoon Kim, and G. V. Loganathan and is inspired by the improvization process of jazz musicians. In the HS algorithm, a set of possible solutions is randomly generated (called Harmony memory). A new solution is generated by using all the solutions in the Harmony memory (rather than just two as used in GA) and if this new solution is better than the worst solution in Harmony memory, the worst solution gets replaced by this new solution. The effectiveness and advantages of HS have been demonstrated in various applications like design of municipal water distribution networks, structural design, load dispatch problem in electrical engineering, multi-objective optimization, rostering problems, clustering, and classification and feature selection. A detailed survey on applications of HS can be found. and applications of HS in data mining can be found in.

Dennis (2015) claimed that harmony search is a special case of the evolution strategies algorithm. However, Saka et al. (2016) argues that the structure of evolution strategies is different from that of harmony search.

Artificial bee colony algorithm is a metaheuristic introduced by Karaboga in 2005 which simulates the foraging behaviour of honey bees. The ABC algorithm has three phases: employed bee, onlooker bee and scout bee. In the employed bee and the onlooker bee phases, bees exploit the sources by local searches in the neighbourhood of the solutions selected based on deterministic selection in the employed bee phase and the probabilistic selection in the onlooker bee phase. In the scout bee phase, which is analogous to bees abandoning exhausted food sources in the foraging process, solutions that are not beneficial anymore for search progress are abandoned, and new solutions are inserted instead to explore new regions in the search space. The algorithm has a well-balanced^{[weasel words]} exploration and exploitation ability.^{[clarification needed]}

The bees algorithm was formulated by Pham and his co-workers in 2005 and further refined in 2009. Modelled on the foraging behaviour of honey bees, the algorithm combines global explorative search with local exploitative search. A small number of artificial bees (scouts) explores randomly the solution space (environment) for solutions of high fitness (highly profitable food sources), whilst the bulk of the population search (harvest) the neighbourhood of the fittest solutions looking for the fitness optimum. A deterministic recruitment procedure which simulates the waggle dance of biological bees is used to communicate the scouts' findings to the foragers and distribute the foragers depending on the fitness of the neighbourhoods selected for local search. Once the search in the neighbourhood of a solution stagnates, the local fitness optimum is considered to be found, and the site is abandoned.

The imperialist competitive algorithm (ICA), like most of the methods in the area of evolutionary computation, does not need the gradient of the function in its optimization process. From a specific point of view, ICA can be thought of as the social counterpart of genetic algorithms (GAs). ICA is the mathematical model and the computer simulation of human social evolution, while GAs is based on the biological evolution of species.

This algorithm starts by generating a set of random candidate solutions in the search space of the optimization problem. The generated random points are called the initial Countries. Countries in this algorithm are the counterpart of Chromosomes in GAs and Particles in Particle Swarm Optimization and it is an array of values of a candidate solution of optimization problem. The cost function of the optimization problem determines the power of each country. Based on their power, some of the best initial countries (the countries with the least cost function value), become Imperialists and start taking control of other countries (called colonies) and form the initial Empires.

Two main operators of this algorithm are Assimilation and Revolution. Assimilation makes the colonies of each empire get closer to the imperialist state in the space of socio-political characteristics (optimization search space). Revolution brings about sudden random changes in the position of some of the countries in the search space. During assimilation and revolution, a colony might reach a better position and then have a chance to take the control of the entire empire and replace the current imperialist state of the empire.

Imperialistic Competition is another part of this algorithm. All the empires try to win this game and take possession of colonies of other empires. In each step of the algorithm, based on their power, all the empires have a chance to take control of one or more of the colonies of the weakest empire.

The algorithm continues with the mentioned steps (Assimilation, Revolution, Competition) until a stop condition is satisfied.

The above steps can be summarized as the below pseudocode:

0) Define objective function: ${\displaystyle f(\mathbf {x} ),\quad \mathbf {x} =(x_{1},x_{2},\dots ,x_{d});\,}$
1) Initialization of the algorithm. Generate some random solution in the search space and create initial empires.
    2) Assimilation: Colonies move towards imperialist states in different in directions.
    3) Revolution: Random changes occur in the characteristics of some countries.
    4) Position exchange between a colony and Imperialist. A colony with a better position than the imperialist,
       has the chance to take the control of empire by replacing the existing imperialist.
    5) Imperialistic competition: All imperialists compete to take possession of colonies of each other.
    6) Eliminate the powerless empires. Weak empires lose their power gradually and they will finally be eliminated.
    7) If the stop condition is satisfied, stop, if not go to 2.
8) End

River formation dynamics is based on imitating how water forms rivers by eroding the ground and depositing sediments (the drops act as the swarm). After drops transform the landscape by increasing/decreasing the altitude of places, solutions are given in the form of paths of decreasing altitudes. Decreasing gradients are constructed, and these gradients are followed by subsequent drops to compose new gradients and reinforce the best ones. This heuristic optimization method was proposed in 2007 by Rabanal et al. The applicability of RFD to other NP-complete problems has been studied, and the algorithm has been applied to fields such as routing and robot navigation. The main applications of RFD can be found at the survey Rabanal et al. (2017).

The gravitational search algorithm is based on the law of gravity and the notion of mass interactions. The GSA algorithm uses the theory of Newtonian physics and its searcher agents are the collection of masses. In GSA, there is an isolated system of masses. Using the gravitational force, every mass in the system can see the situation of other masses. The gravitational force is therefore a way of transferring information between different masses. In GSA, agents are considered as objects and their performance is measured by their masses. All these objects attract each other by a gravity force, and this force causes movement of all objects towards the objects with heavier masses. Heavier masses correspond to better solutions of the problem. The position of the agent corresponds to a solution of the problem, and its mass is determined using a fitness function. By lapse of time, masses are attracted by the heaviest mass, which would ideally present an optimum solution in the search space. The GSA could be considered as a small artificial world of masses obeying the Newtonian laws of gravitation and motion. A multi-objective variant of GSA, called MOGSA, was proposed by Hassanzadeh et al. in 2010.

Bat algorithm is a swarm-intelligence-based algorithm, inspired by the echolocation behavior of microbats. BA automatically balances exploration (long-range jumps around the global search space to avoid getting stuck around one local maximum) with exploitation (searching in more detail around known good solutions to find local maxima) by controlling loudness and pulse emission rates of simulated bats in the multi-dimensional search space.

The spiral optimization algorithm, inspired by spiral phenomena in nature, is a multipoint search algorithm that has no objective function gradient. It uses multiple spiral models that can be described as deterministic dynamical systems. As search points follow logarithmic spiral trajectories towards the common center, defined as the current best point, better solutions can be found, and the common center can be updated.

Artificial swarm intelligence is a real-time closed-loop system of human users connected over the internet and structured in a framework modeled after natural swarms such that it evokes the group's collective wisdom as a unified emergent intelligence. In this way, human swarms can answer questions, make predictions, reach decisions, and solve problems by collectively exploring a diverse set of options and converging on preferred solutions in synchrony. Invented by Dr. Louis Rosenberg in 2014, the ASI methodology has been noted for its ability to make accurate collective predictions that outperform the individual members of the swarm. In 2016, an Artificial Swarm Intelligence group from Unanimous A.I. was challenged by a reporter to predict the winners of the Kentucky Derby; it successfully picked the first four horses, in order, beating 540 to 1 odds.

Self-tuning metaheuristics have emerged as a significant advancement in the field of optimization algorithms in recent years, since fine tuning can be a very long and difficult process. These algorithms differentiate themselves by their ability to autonomously adjust their parameters in response to the problem at hand, enhancing efficiency and solution quality. This self-tuning capability is particularly important in complex optimization scenarios where traditional methods may struggle due to rigid parameter settings.

While individual metaphor-inspired metaheuristics have produced remarkably effective solutions to specific problems, metaphor-inspired metaheuristics in general have attracted criticism among researchers for hiding their lack of effectiveness or novelty behind elaborate metaphors. Kenneth Sörensen noted:

Sörensen and Glover stated:

In response, Springer's Journal of Heuristics has updated their editorial policy to state:

The policy of Springer's journal 4OR - A Quarterly Journal of Operations Research states:

The ACM Transactions on Evolutionary Learning Optimization and the Evolutionary Computation journal also include similar statements.

• Conceptual metaphor
• Scientific community metaphor
• Table of metaheuristics

• Sörensen, Kenneth; Sevaux, Marc; Glover, Fred (2017-01-16). "A History of Metaheuristics" (PDF). In Martí, Rafael; Panos, Pardalos; Resende, Mauricio (eds.). Handbook of Heuristics. Springer. ISBN 978-3-319-07123-7.
• Sörensen, Kenneth (2015). "Metaheuristics-the metaphor exposed". International Transactions in Operational Research. 22: 3–18. CiteSeerX 10.1.1.470.3422. doi:10.1111/itor.12001. S2CID 14042315.
• Lones, Michael A. (2014). "Metaheuristics in nature-inspired algorithms". Proceedings of the 2014 conference companion on Genetic and evolutionary computation companion - GECCO Comp '14. pp. 1419–22. CiteSeerX 10.1.1.699.1825. doi:10.1145/2598394.2609841. ISBN 9781450328814. S2CID 14997975.
• Fister, Iztok; Yang, Xin-She; Fister, Iztok; Brest, Janez; Fister, Dušan (2013). "A Brief Review of Nature-Inspired Algorithms for Optimization". Elektrotehniški Vestnik. arXiv:1307.4186.

• Evolutionary Computation Bestiary – a tongue-in-cheek account of all the weird metaphor-based metaheuristics out there in the wide world of academic publishing
• The Science Matrix's List of Metaheuristic^{[dead link]} – a complete list of metaheuristic algorithms filterable by name, author or year, providing a link to main publication of each algorithm