蚁群优化算法

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Ant behavior was the inspiration for the metaheuristic optimization technique

Ant behavior was the inspiration for the metaheuristic optimization technique

蚂蚁行为是元启发式优化技术的灵感来源

When a colony of ants is confronted with the choice of reaching their food via two different routes of which one is much shorter than the other, their choice is entirely random. However, those who use the shorter route reach the food faster and therefore go back and forth more often between the anthill and the food.

当一群蚂蚁面临通过两条不同的路径获取食物的选择时，其中一条路径比另一条路径短得多，它们的选择完全是随机的。然而，那些使用较短的路线到达食物更快，因此往返于蚁丘和食物之间的频率更高。 In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial Ants stand for multi-agent methods inspired by the behavior of real ants. The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of Artificial Ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.g., vehicle routing and internet routing. The burgeoning activity in this field has led to conferences dedicated solely to Artificial Ants, and to numerous commercial applications by specialized companies such as AntOptima. In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.

在蚁群算法中，对原有的蚁群算法进行了三方面的改进: (i)边缘选择偏向于挖掘(即边缘选择偏向于挖掘边缘)。(2)在建立解时，蚂蚁通过应用局部信息素更新规则来改变它们选择的边的信息素水平; (3)在每次迭代结束时，只允许最优的蚂蚁通过应用一个修正的全局信息素更新规则来更新路径

To optimize the form of antennas, ant colony algorithms can be used. As example can be considered antennas RFID-tags based on ant colony algorithms (ACO),

为了优化天线的形状，可以采用蚁群算法。作为例子可以考虑天线射频识别标签基于蚁群算法(ACO) ,

As an example, Ant colony optimization is a class of optimization algorithms modeled on the actions of an ant colony. Artificial 'ants' (e.g. simulation agents) locate optimal solutions by moving through a parameter space representing all possible solutions. Real ants lay down pheromones directing each other to resources while exploring their environment. The simulated 'ants' similarly record their positions and the quality of their solutions, so that in later simulation iterations more ants locate better solutions. One variation on this approach is the bees algorithm, which is more analogous to the foraging patterns of the honey bee, another social insect.

This algorithm is a member of the ant colony algorithms family, in swarm intelligence methods, and it constitutes some metaheuristic optimizations. Initially proposed by Marco Dorigo in 1992 in his PhD thesis, the first algorithm was aiming 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 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 estimation of distribution algorithms.

The ACO algorithm is used in image processing for image edge detection and edge linking.Randomly place [math]\displaystyle{ K }[/math] ants on the image [math]\displaystyle{ I_{M_1 M_2} }[/math] where [math]\displaystyle{ K= (M_1*M_2)^\tfrac{1}{2} }[/math] . Pheromone matrix [math]\displaystyle{ \tau_{(i,j)} }[/math] are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.

将蚁群算法应用于图像处理中，实现了图像的边缘检测和边缘连接。随机地点 < math > k </math > 蚂蚁在图像 < math > i _ { m _ 1 m _ 2} </math > < math > > k = (m _ 1 * m _ 2) ^ tfrac {1}{2} </math > 。信息素矩阵 < math > tau _ {(i，j)} </math > 由一个随机值初始化。在初始化过程中的主要挑战是确定启发式矩阵。

Overview

There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:

确定启发式矩阵的方法有多种。对于下面的例子，启发式矩阵是根据当地统计数据计算出来的:

the local statistics at the pixel position (i,j).

像素位置(i，j)的局部统计。

In the natural world, ants of some species (initially) wander randomly, and upon finding food return to their colony while laying down pheromone trails. If other ants find such a path, they are likely not to keep travelling at random, but instead to follow the trail, returning and reinforcing it if they eventually find food (see Ant communication).

[math]\displaystyle{ \eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)} }[/math]

< math > eta _ (i，j)} = tfrac {1}{ z } * Vc _ (i，j)} </math >

Over time, however, the pheromone trail starts to evaporate, thus reducing its attractive strength. The more time it takes for an ant to travel down the path and back again, the more time the pheromones have to evaporate. A short path, by comparison, gets marched over more frequently, and thus the pheromone density becomes higher on shorter paths than longer ones. Pheromone evaporation also has the advantage of avoiding the convergence to a locally optimal solution. If there were no evaporation at all, the paths chosen by the first ants would tend to be excessively attractive to the following ones. In that case, the exploration of the solution space would be constrained. The influence of pheromone evaporation in real ant systems is unclear, but it is very important in artificial systems.

Where [math]\displaystyle{ I }[/math] is the image of size [math]\displaystyle{ M_1*M_2 }[/math]

大小如下的图像在哪里

The overall result is that when one ant finds a good (i.e., short) path from the colony to a food source, other ants are more likely to follow that path, and positive feedback eventually leads to many ants following a single path. The idea of the ant colony algorithm is to mimic this behavior with "simulated ants" walking around the graph representing the problem to solve.

[math]\displaystyle{ Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j}) }[/math], which is a normalization factor

[ math ] z = sum { i = 1: m _ 1} sum { j = 1: m _ 2} Vc (i _ { i，j }) </math > ，这是一个归一化因子

Ambient networks of intelligent objects

[math]\displaystyle{ \begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\ \lt math \gt begin { align } Vc (i _ { i，j }) = & f left (left vert i _ {(i-2，j-1)}-i _ {(i + 2，j + 1)}右垂直 + 左垂直 i _ {(i-2，j + 1)}-i _ (i + 2，j-1)}右垂直。\\ New concepts are required since “intelligence” is no longer centralized but can be found throughout all minuscule objects. Anthropocentric concepts have been known to lead to the production of IT systems in which data processing, control units and calculating forces are centralized. These centralized units have continually increased their performance and can be compared to the human brain. The model of the brain has become the ultimate vision of computers. [[Ambient networks]] of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles. However, once those objects are interconnected they dispose of a form of intelligence that can be compared to a colony of ants or bees. In the case of certain problems, this type of intelligence can be superior to the reasoning of a centralized system similar to the brain.\lt ref name="Waldner 2008 214"\gt {{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = [[ISTE Ltd|ISTE]] John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 214}}\lt /ref\gt & +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\ & +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\ & +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\ & +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\ Nature offers several examples of how minuscule organisms, if they all follow the same basic rule, can create a form of [[collective intelligence]] on the macroscopic level. Colonies of social insects perfectly illustrate this model which greatly differs from human societies. This model is based on the co-operation of independent units with simple and unpredictable behavior.\lt ref\gt {{cite book|last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Inventer l'Ordinateur du XXIème Siècle |publisher = [[Hermes Science]] |place = London |year = 2007 | pages = 259–265 |isbn = 978-2-7462-1516-0}}\lt /ref\gt They move through their surrounding area to carry out certain tasks and only possess a very limited amount of information to do so. A colony of ants, for example, represents numerous qualities that can also be applied to a network of ambient objects. Colonies of ants have a very high capacity to adapt themselves to changes in the environment as well as an enormous strength in dealing with situations where one individual fails to carry out a given task. This kind of flexibility would also be very useful for mobile networks of objects which are perpetually developing. Parcels of information that move from a computer to a digital object behave in the same way as ants would do. They move through the network and pass from one knot to the next with the objective of arriving at their final destination as quickly as possible.\lt ref\gt {{cite book |last = Waldner |first = Jean-Baptiste |authorlink = Jean-Baptiste Waldner |title = Nanocomputers and Swarm Intelligence |publisher = ISTE John Wiley & Sons |place = London |year = 2008 |isbn = 978-1-84704-002-2 | page = 215}}\lt /ref\gt & + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align} }[/math]

左边。(i-1，j + 2)}-i _ {(i-1，j-2)}右 vert + 左 vert i _ {(i，j-1)}-i _ {(i，j + 1)}右 vert 右 vert { align } </math >

Artificial pheromone system

[math]\displaystyle{ f(\cdot) }[/math] can be calculated using the following functions:
[math]\displaystyle{ f(x) = \lambda x, \quad \text{for x ≥ 0; (1)} }[/math]
[math]\displaystyle{ f(x) = \lambda x^2, \quad \text{for x ≥ 0; (2)} }[/math]
[math]\displaystyle{ f(x) = \lt math \gt f (cdot) }[/math] 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) =

Pheromone-based communication is one of the most effective ways of communication which is widely observed in nature. Pheromone is used by social insects such as

\begin{cases}

开始{ cases }

bees, ants and termites; both for inter-agent and agent-swarm communications. Due to its feasibility, artificial pheromones have been adopted in multi-robot and swarm robotic systems. Pheromone-based communication was implemented by different means such as chemical or physical (RFID tags, light, sound) ways. However, those implementations were not able to replicate all the aspects of pheromones as seen in nature.

\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\

Sin (frac { pi x }{2 lambda }) ，& text { for 0≤ x ≤} lambda text { ; (3)}

0, & \text{else}

0，& text { else }

Using projected light was presented in an 2007 IEEE paper by Garnier, Simon, et al. as an experimental setup to study pheromone-based communication with micro autonomous robots. Another study that proposed a novel pheromone communication method, COSΦ, for a swarm robotic system is based on precise and fast visual localization.

\end{cases}</math>
[math]\displaystyle{ f(x) = 结束{ cases } }[/math] < br/> < math > f (x) =

The system allows simulation of a virtually unlimited number of different pheromones and provides the result of their interaction as a gray-scale image on a horizontal LCD screen that the robots move on. In order to demonstrate the pheromone communication method, Colias autonomous micro robot was deployed as the swarm robotic platform.

\begin{cases}

开始{ cases }

\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\

\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\

Algorithm and formulae

0, & \text{else}

0，& text { else }

In the ant colony optimization algorithms, an artificial ant is a simple computational agent that searches for good solutions to a given optimization problem. To apply an ant colony algorithm, the optimization problem needs to be converted into the problem of finding the shortest path on a weighted graph. In the first step of each iteration, each ant stochastically constructs a solution, i.e. the order in which the edges in the graph should be followed. In the second step, the paths found by the different ants are compared. The last step consists of updating the pheromone levels on each edge.

\end{cases}</math>
The parameter [math]\displaystyle{ \lambda }[/math] in each of above functions adjusts the functions’ respective shapes.
Step 2 Construction process:
The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation [math]\displaystyle{ P_{x,y} }[/math]
Step 3 and Step 5 Update process:
The pheromone matrix is updated twice. in step 3 the trail of the ant (given by [math]\displaystyle{ \tau_{(x,y)} }[/math] ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.
[math]\displaystyle{ 上面每个函数中的参数 \lt math \gt lambda }[/math] 调整函数各自的形状。< br/> 步骤2构建过程: < br/> 蚂蚁的移动是基于4个连接的像素或8个连接的像素。根据概率方程 < math > p _ { x，y } </math > < br/> 步骤3和步骤5更新过程给出蚂蚁移动的概率: < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁的踪迹(由 < math > tau _ {(x，y)} </math > 给出)被更新，在步骤5中，蚂蚁的踪迹蒸发率被更新，这个更新是由下面的方程给出的。[数学]

\tau_{new} \leftarrow

左塔罗

procedure ACO_MetaHeuristic is

(1-\psi)\tau_{old} + \psi \tau_{0}

(1-psi) tau _ { old } + psi tau _ {0}

    while not_termination do

</math>, where [math]\displaystyle{ \psi }[/math] is the pheromone decay coefficient [math]\displaystyle{ 0\lt \tau \lt 1 }[/math]

这里是信息素衰变系数

        generateSolutions()

        daemonActions()

Step 7 Decision Process:
Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on Otsu's method.

步骤7决策过程: 一旦 k 蚂蚁在 n 次迭代中移动了一个固定的距离 l，判断它是否是一个边缘是基于信息素矩阵 τ 上的阈值 t。下面例子的阈值是根据 Otsu 的方法计算的。

        pheromoneUpdate()

    repeat

Image Edge detected using ACO:
The images below are generated using different functions given by the equation (1) to (4). which is a very general framework in which ant colony algorithms fit.

使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。这是一个非常普遍的框架，蚁群算法适合。

end procedure

Edge selection

There is in practice a large number of algorithms claiming to be "ant colonies", without always sharing the general framework of optimization by canonical ant colonies.). A technical report extracted from the thesis and co-authored by V. Maniezzo and A. Colorni

在实践中，有大量的算法声称是“蚁群” ，但并不总是共享规范的蚁群优化的一般框架。).摘自论文的技术报告，由 v. Maniezzo 和 a. Colorni 合著

Each ant needs to construct a solution to move through the graph. To select the next edge in its tour, an ant will consider the length of each edge available from its current position, as well as the corresponding pheromone level. At each step of the algorithm, each ant moves from a state [math]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math], corresponding to a more complete intermediate solution. Thus, each ant [math]\displaystyle{ k }[/math] computes a set [math]\displaystyle{ A_k(x) }[/math] of feasible expansions to its current state in each iteration, and moves to one of these in probability. For ant [math]\displaystyle{ k }[/math], the probability [math]\displaystyle{ p_{xy}^k }[/math] of moving from state [math]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math] depends on the combination of two values, the attractiveness [math]\displaystyle{ \eta_{xy} }[/math] of the move, as computed by some heuristic indicating the a priori desirability of that move and the trail level [math]\displaystyle{ \tau_{xy} }[/math] of the move, indicating how proficient it has been in the past to make that particular move. The trail level represents a posteriori indication of the desirability of that move.

In general, the [math]\displaystyle{ k }[/math]th ant moves from state [math]\displaystyle{ x }[/math] to state [math]\displaystyle{ y }[/math] with probability

[math]\displaystyle{ p_{xy}^k = \frac { (\tau_{xy}^{\alpha}) (\eta_{xy}^{\beta}) } { \sum_{z\in \mathrm{allowed}_x} (\tau_{xz}^{\alpha}) (\eta_{xz}^{\beta}) } }[/math]

where

[math]\displaystyle{ \tau_{xy} }[/math] is the amount of pheromone deposited for transition from state [math]\displaystyle{ x }[/math] to [math]\displaystyle{ y }[/math], 0 ≤ [math]\displaystyle{ \alpha }[/math] is a parameter to control the influence of [math]\displaystyle{ \tau_{xy} }[/math], [math]\displaystyle{ \eta_{xy} }[/math] is the desirability of state transition [math]\displaystyle{ xy }[/math] (a priori knowledge, typically [math]\displaystyle{ 1/d_{xy} }[/math], where [math]\displaystyle{ d }[/math] is the distance) and [math]\displaystyle{ \beta }[/math] ≥ 1 is a parameter to control the influence of [math]\displaystyle{ \eta_{xy} }[/math]. [math]\displaystyle{ \tau_{xz} }[/math] and [math]\displaystyle{ \eta_{xz} }[/math] represent the trail level and attractiveness for the other possible state transitions.

Pheromone update

Trails are usually updated when all ants have completed their solution, increasing or decreasing the level of trails corresponding to moves that were part of "good" or "bad" solutions, respectively. An example of a global pheromone updating rule is

[math]\displaystyle{ \tau_{xy} \leftarrow (1-\rho)\tau_{xy} + \sum_{k}\Delta \tau^{k}_{xy} }[/math]

where [math]\displaystyle{ \tau_{xy} }[/math] is the amount of pheromone deposited for a state transition [math]\displaystyle{ xy }[/math], [math]\displaystyle{ \rho }[/math] is the pheromone evaporation coefficient and [math]\displaystyle{ \Delta \tau^{k}_{xy} }[/math] is the amount of pheromone deposited by [math]\displaystyle{ k }[/math]th ant, typically given for a TSP problem (with moves corresponding to arcs of the graph) by

[math]\displaystyle{ \Delta \tau^{k}_{xy} = \begin{cases} Q/L_k & \mbox{if ant }k\mbox{ uses curve }xy\mbox{ in its tour} \\ 0 & \mbox{otherwise} \end{cases} }[/math]

where [math]\displaystyle{ L_k }[/math] is the cost of the [math]\displaystyle{ k }[/math]th ant's tour (typically length) and [math]\displaystyle{ Q }[/math] is a constant.

Common extensions

Here are some of the most popular variations of ACO algorithms.

Ant System (AS)

The Ant System is the first ACO algorithm. This algorithm corresponds to the one presented above. It was developed by Dorigo.

Ant Colony System (ACS)

In the Ant Colony System algorithm, the original Ant System was modified in three aspects: (i) the edge selection is biased towards exploitation (i.e. favoring the probability of selecting the shortest edges with a large amount of pheromone); (ii) while building a solution, ants change the pheromone level of the edges they are selecting by applying a local pheromone updating rule; (iii) at the end of each iteration, only the best ant is allowed to update the trails by applying a modified global pheromone updating rule.

Category:Articles which contain graphical timelines

类别: 包含图形时间表的文章

Elitist Ant System

Category:Nature-inspired metaheuristics

类别: 自然启发的启发式元推理

In this algorithm, the global best solution deposits pheromone on its trail after every iteration (even if this trail has not been revisited), along with all the other ants.

Category:Optimization algorithms and methods

类别: 优化算法和方法

This page was moved from wikipedia:en:Ant colony optimization algorithms. Its edit history can be viewed at 蚁群优化算法/edithistory