lecture 09: swarm intelligenceqf-zhao/teaching/mh/lec09.pdf · what is swarm intelligence? •...

Lecture 09: Swarm Intelligence

An Introduction to Meta-Heuristics, Produced by Qiangfu Zhao (Since 2012), All rights reserved © Lec09/1

What is swarm intelligence?

• Swarm intelligence (SI) is also a group of population based optimization/search algorithms.

• Instead of trying to emulate the evolution process, SI tries to emulate the “collective behaviors” of bacterial, ants, bugs, birds, fish, animals, etc.

• In nature, different “agents” are often governed by some very simple rules. They often make decisions in a local and decentralized way, but as a whole, some “intelligent” behaviors often emerge.


Ant colony optimization (ACO)

• ACO was first studied by Marco Dorigo in his doctor thesis in 1991.

• It simulates the collective behavior of an ant colony.

• ACO can be used directly for solving problems related to finding the shortest paths to reach some goal, such as traveling salesman problem (TSP).


http://en.wikipedia.org/wiki/

//upload.wikimedia.org/wikipedia/commons/3/34/Safari_ants.jpg

Why ant colony?

• Properties of ants:

– Cannot see well (can only “feel” locally).

– Deliver a chemical trail called pheromone while wandering around and searching for food.

– The ants choose their paths according to the strength of the pheromone deployed by its family members.


?

Why ant colony?

• Properties of the pheromone:

– The pheromone is olfactive and volatile.

– The pheromone is stronger if more ants go along the same path (reinforced by number).

– The pheromone is stronger if the path from the nest to the food is shorter

(less evaporation).


After a while, the shortest path will be found automatically!

A general algorithm

• Step 1: Initialize the pheromone information;

• Step 2: For each ant, do the following:

– Find a solution (a path) based on the current pheromone trail;

– Reinforcement: Add pheromone;

– Evaporation: Reduce pheromone;

• Step 3: Stop if terminating condition satisfied; return to Step 2 otherwise.


ACO for TSP

• Step 1: Initialize the pheromone information.

• Step 2: For each ant, do the following:

– Specify the initial node (fixed, or select at random);

– Find a solution by repeating the following steps until all nodes are visited:

• i=current node;

• Select an un-visited child node (linked to i) j with a probability pij.

• Step 3: Update the pheromone trails:

– Evaporation of pheromone value on each edge;

– Reinforcement of pheromone value on each edge contained in the found routes.



Representation of pheromone

• For solving the TSP problem, the pheromone is represented in a matrix T=[tij].

• Each element tij is the amount of pheromone on the edge (i,j) of the graph.

• Initially, all elements take the same value.

• During search, pheromones associated with edges contained in shorter routes will become stronger and stronger.


Probability of path selection

• The probability pij should be

– proportional to the pheromone value tij and

– proportional to the heuristic value hij (inverse of distance) of the edge.

𝑝𝑖𝑗 =𝜏𝑖𝑗𝛼 × ℎ𝑖𝑗

𝛽

σ𝑘∈Ω 𝜏𝑖𝑘𝛼 × ℎ𝑖𝑘

𝛽, ∀𝑗 ∈ Ω

where W is the set of not yet visited nodes, a and b are parameters representing the relative influence of the pheromone values and the heuristic values.


Updating the pheromone:Evaporation

• After finding the routes for all ants, the pheromone values associated with all edges are updated as follows:

where n is the total number of nodes in the graph, and r in [0,1] is the reduction (evaporation) rate of the pheromone.


],1[, , njiijij rtt

Updating the pheromone:Reinforcement

• After finding the routes for all ants, the pheromone values associated with edges contained in the k-throute (found by the k-th ant) are updated as follows:

where Q>0 is a constant, and L(k) is the total length of the k-th route.

• The above reinforcement is conducted for all routes.


routeth ),(if ),(/ kjikLQijij tt

Parameters and Their Values

• a: Pheromone influence (is fixed to 1).

• b: Heuristic influence (takes value from [1,15]).

• r: Evaporation rate (takes value from [0,1]).

• m: Number of ants.

• n: Number of nodes.

• Q: Constant (amount of pheromone that can be delivered by each ant).

• dij: Distance between the i-th node and the j-th node (the heuristic value hij is the inverse of dij).


Basic concept of ACO

• In ACO, in each iteration (generation) each ant (agent) generates a solution based on the information collected by all ants so far.

• The pheromone trail actually plays the role of a “message board”. All ants post some message to the board, and each ant tries to generate a better solution next time based on the information on the message board.

• The message board itself can be considered as the “culture” of the ant colony. In this sense, ACO is also a memetic algorithm.


Coffee Break


http://www.youtube.com/watch?v=VPrBmQER8LE&feature=relmfu

http://www.youtube.com/watch?v=qJjeHLcbQJ0

http://www.youtube.com/watch?v=VPrBmQER8LE&feature=relmfu

http://www.youtube.com/watch?v=qJjeHLcbQJ0

Particle Swarm Optimization (PSO)

• PSO is also a population based search algorithm.

• Instead of posting information to the message board as in ACO, in PSO, each agent tries to learn something directly from its “leader”.

• The leader can be a local one, or a global one.

• Through this kind of direct learning, the agents can become “good” much faster (compared with conventional EAs).

• In this sense, PSO is a kind of non-Darwinian evolution.


Basic considerations

• In PSO, the population is called swarm, and each individual is called a particle.

• For each particle, we need to record

– The current position x;

– The best position pb found so far; and

– The velocity v.

• For the whole swarm, we need to record the best particle (the global leader pg) found so far.


The algorithm

• Step 1: Randomly initialize the swarm.

• Step 2: Evaluate all particles.

• Step 3: For each particle

– Update its velocity;

– Update its position;

– Evaluate the particle.

• Step 4: Update if necessary the leader of the swarm and the best position obtained by each particle.



Update the velocity

• The velocity of a particle in the k-th iteration is updated as follows:

𝑣𝑖 𝑘 + 1 = 𝑎𝑣𝑖 𝑘 + 𝑏𝑤1 𝑝𝑏 𝑘 − 𝑥 𝑘 + 𝑐𝑤2(𝑝𝑔 𝑘 − 𝑥 𝑘 )

• Where k is the iteration number, 𝑎 is the inertia weight, 𝑏 and 𝑐 are the learning factors called personal influence and social influence, respectively, and w1 and w2 are random numbers taken from [0,1].


Update the position

• Based on the new velocity, the new position is obtained as follows:


)1()()1( kvkxkx

x(k)

x(k+1)pg(k)

pb(k)

v(k)

Physical meaning of PSO

• Each particle tries to learn from the current leader as well as the best position found by itself.

• The amount of information (influence) obtained from the leader and the best position found so far depends on the learning factors b and c.

• Also depends on the random factors w1 and w2. Thus, we will obtain a different solution in a different run.


Parameters and Their Values

• n: Number of particles (population size).

• a: the inertia weight, usually increases linearly during learning from a0 (=0.4) to a1 (=0.9).

• b:Cognitive (personal) learning factor which controls the movement towards its own success (=2).

• c: Social learning factor which controls the movement towards the leader (=2).

• A neighborhood size will also be used to define local leaders for different regions.


Team Work II

• Select one of the following topics:

– Try to solve the TSP problem with 10 nodes using GA, and ACO, and compare their performance.

– Try to find the minimum points of the following functions, using GA, DE, and PSO, and compare their performance.


048.2047.2,)1()(100),(

12.511.5,),,(

2

1

2

2

2

1212

3

1

2

3211

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lecture 09: swarm intelligenceqf-zhao/teaching/mh/lec09.pdf · what is swarm intelligence? •...

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