aco-03-04-2013

Transcript aco-03-04-2013

Ant Colony Optimization: an introduction

Daniel Chivilikhin 03.04.2013

Outline

1. Biological inspiration of ACO 2. Solving NP-hard combinatorial problems 3. The ACO metaheuristic 4. ACO for the Traveling Salesman Problem 2

Outline

1. Biological inspiration of ACO

2. Solving NP-hard combinatorial problems 3. The ACO metaheuristic 4. ACO for the Traveling Salesman Problem 3

Biological inspiration: from real to artificial ants

Ant colonies

• Distributed systems of social insects • Consist of

simple

individuals • Colony intelligence >> Individual intelligence 5

Ant Cooperation

•

Stigmergy

– indirect communication between individuals (ants) • Driven by environment modifications 6

Denebourg’s double bridge experiments

• Studied Argentine ants

I. humilis

• Double bridge from ants to food source 7

Double bridge experiments: equal lengths (1)

Double bridge experiments: equal lengths (2)

• Run for a number of trials • Ants choose each branch ~ same number of trials 9

Double bridge experiments: different lengths (2)

• The majority of ants follow the short path 11

Outline

1. Biological inspiration of ACO

2. Solving NP-hard combinatorial problems

3. The ACO metaheuristic 4. ACO for the Traveling Salesman Problem 12

Solving NP-hard combinatorial problems

Combinatorial optimization

• Find values of discrete variables • Optimizing a given

objective function

Combinatorial optimization

• • Π = (

, Ω) – problem instance

– set of candidate solutions

f –

objective function • • Ω – set of constraints ~



– set of feasible solutions (with respect to Ω) • Find globally optimal feasible solution

NP-hard combinatorial problems

• • Cannot be

exactly

solved in polynomial time

Approximate

methods – generate near optimal solutions in reasonable time • No formal theoretical guarantees • Approximate methods =

heuristics

Approximate methods

• Constructive algorithms • Local search 17

Constructive algorithms

• Add components to solution incrementally • Example – greedy heuristics:

add solution component with best heuristic estimate

Local search

• Explore neighborhoods of complete solutions • Improve current solution by local changes – first improvement – best improvement 19

What is a metaheuristic?

• A set of algorithmic concepts • Can be used to define heuristic methods • Applicable to a wide set of problems 20

Examples of metaheuristics

• Simulated annealing • Tabu search • Iterated local search • Evolutionary computation •

Ant colony optimization

• Particle swarm optimization • etc.

Outline

1. Biological inspiration of ACO 2. Solving NP-hard combinatorial problems

3. The ACO metaheuristic

4. ACO for the Traveling Salesman Problem 22

The ACO metaheuristic

ACO metaheuristic

• A colony of artificial ants cooperate in finding good solutions • Each ant – simple agent • Ants communicate indirectly using

stigmergy

Combinatorial optimization problem mapping (1)

• Combinatorial problem (

, Ω(t)) • Ω(t) – time-dependent constraints  Example – dynamic problems • Goal – find globally optimal feasible solution

* • Minimization problem • Mapped on another problem… 25

Combinatorial optimization problem mapping (2)

• •

= {

1 ,

2 , …,

c Nc

} – finite set of

components States

of the problem:

= {

= <

c i

c j

, …,

c h

, …>, |

| <

< + ∞} • Set of

candidate

solutions:



Combinatorial optimization problem mapping (3)

• Set of feasible states: ~



satisfying Ω(t) • Non-empty set of optimal solutions:

*  ~



Combinatorial optimization problem mapping (4)

• X – states • S – candidate solutions • ~

– feasible states •

* – optimal solutions

S X S*

Combinatorial optimization problem mapping (5)

• Cost

(

) for each • In most cases –



S g

(

) ≡

(

) 

 ~

•

G C =

(

) – completely connected graph • C – set of

components

• • L – edges fully connecting the components (

connections

)

G C

–

construction graph

Combinatorial optimization problem mapping (last



)

• Artificial ants build solutions by performing randomized walks on

G C

(

) 30

Construction graph

• Each component

c i

associated:   heuristic information pheromone trail or connection

l ij

have 31

Heuristic information

•

A priori

information about the problem • Does not depend on the ants • On components

c i

• On connections

l ij

–

η – η ij i

• Meaning: cost of adding a component to the current solution 32

Pheromone trail

• Long-term memory about the entire search process • On components

c i

• On connections

l ij

–

τ – τ ij i

• Updated by the ants 33

Artificial ant (1)

• Stochastic constructive procedure • Builds solutions by moving on

G C

• Has finite memory for: – Implementing constraints Ω(

) – Evaluating solutions – Memorizing its path 34

Artificial ant (2)

• Has a start state x • Has termination conditions

e k

• From state

x r

moves to a node from the neighborhood –

N k

(

x r

) • Stops if some

e k

are satisfied 35

Artificial ant (3)

• Selects a move with a probabilistic rule depending on:    Pheromone trails and heuristic information of neighbor components and connections Memory Constraints Ω 36

Artificial ant (4)

• Can update pheromone on visited components (nodes) • and connections (edges) • Ants act:  Concurrently  Independently 37

The ACO metaheuristic

While not doStop(): ConstructAntSolutions() UpdatePheromones() DaemonActions() 38

ConstructAntSolutions

• A colony of ants build a set of solutions • Solutions are evaluated using the objective function 39

UpdatePheromones

• Two opposite mechanisms:  Pheromone deposit  Pheromone evaporation 40

UpdatePheromones: pheromone deposit

• Ants increase pheromone values on visited components and/or connections • Increases probability to select visited components later 41

UpdatePheromones: pheromone evaporation

• Decrease pheromone trails on all components/connections by a same value • Forgetting – avoid rapid convergence to suboptimal solutions 42

DaemonActions

• Optional centralized actions, e.g.:  Local optimization  Ant elitism (details later) 43

ACO applications

• Traveling salesman • Quadratic assignment • Graph coloring • Multiple knapsack • Set covering • Maximum clique • Bin packing • … 44

Outline

1. Biological inspiration of ACO 2. Solving NP-hard combinatorial problems 3. The ACO metaheuristic

4. ACO for the Traveling Salesman Problem

ACO for the Traveling Salesman Problem

Traveling salesman problem

•

– set of nodes (cities), |

| =

• A – set of arcs, fully connecting

• Weighted graph

= (

) • Each arc has a weight

d ij

• Problem:

–

distance

Find minimum length Hamiltonian circuit

TSP: construction graph

• • Identical to the problem graph

•

• states = set of all possible partial tours 48

TSP: constraints

• All cities have to be visited • Each city – only once • Enforcing – allow ants only to go to new nodes 49

TSP: pheromone trails

• Desirability of visiting city

directly after

TSP: heuristic information

•

η ij =

1 /

d ij

• Used in most ACO for TSP 51

TSP: solution construction

• Select random start city • Add unvisited cities iteratively • Until a complete tour is built 52

ACO algorithms for TSP

• Ant System • Elitist Ant System • Rank-based Ant System • Ant Colony System • MAX-MIN Ant System 53

Ant System: Pheromone initialization

• Pheromone initialization

Τ ij = m

C nn , where:

 

m – number of ants C nn – path length of nearest-neighbor algorithm

Ant System: Tour construction

• Ant

is located in city

•

N i k

is the neighborhood of city

• Probability to go to city



N i k

p ij k

  

k l



N ij





  

  

 55

Tour construction: comprehension

p ij k

 



N k ij

  

il ij

  

il i

•

= 0 – greedy algorithm • β = 0 – only pheromone is at work  quickly leads to stagnation 56

Ant System: update pheromone trails – evaporation

Evaporation for all connections ∀ (

) ∈

τ ij

← (1 –

)

τ ij , ρ

∈ [0, 1] – evaporation rate

Prevents convergence to suboptimal solutions

• • •

Ant System: update pheromone trails – deposit

T k

– path of ant

k C k – length of path T k

Ants deposit pheromone on visited arcs: 

 



k m

  1  

ij k

,    

 

ij k

  1   0 , /

C k

, 

otherwise T k

Elitist Ant System

• Best-so-far ant deposits pheromone on each iteration: 

 



k m

  1  

ij k



 

ij best

 

ij best

  1   0 , /

C best

otherwise



T best

Rank-based Ant System

• Rank all ants • Each ant deposits amounts of pheromone proportional to its rank 60

MAX-MIN Ant System

1. Only iteration-best or best-so-far ant deposits pheromone 2. Pheromone trails are limited to the interval [

min,

max ] 61

Ant Colony System

• Differs from Ant System in three points:  More aggressive tour construction rule  Only best ant evaporates and deposits pheromone  Local pheromone update 62

Ant Colony System

1.Tour Construction

 arg 

like

max

Ant l



N i k

 

il System

if q



otherwise q

0 2. Local pheromone update:

τ ij

← (1 –

)

τ ij + ξτ 0 ,

Comparing Ant System variants

State of the art in TSP

• CONCORDE http://www.tsp.gatech.edu/concorde.html

• Solved an instance of 85900 cities • Computation took 286-2719 CPU days!

Current ACO research activity

• New applications • Theoretical proofs 66

Next time…

• Some proofs of ACO convergence 68

Thank you!

Any questions?

This presentation is available at: http://rain.ifmo.ru/~chivdan/presentations/ Daniel Chivilikhin [mailto: [email protected]] 69

Used resources

• • • • • • • • http://teamaltman.com/wp-content/uploads/2011/06/Uncertainty-Ant-Apple 1024x1024.jpg

http://myrealestatecoach.files.wordpress.com/2012/04/ant.jpg

http://ars.els-cdn.com/content/image/1-s2.0-S1568494613000264-gr3.jpg

http://www.theorie.physik.uni-goettingen.de/forschung/ha/talks/stuetzle.pdf

http://www.buyingandsellingwebsites.com/wp-content/uploads/2011/12/Ant-150.jpg

http://moodle2.gilbertschools.net/moodle/file.php/1040/Event 100_Days/Ant_Hormiga.gif

http://4.bp.blogspot.com/_SPAe2p8Y-kg/TK3hQ8BUMtI/AAAAAAAAAHA/p75K GcT_oo/s1600/ant+vision.GIF

http://1.bp.blogspot.com/ tXKJQ4nSqOY/UQZ3vabiy_I/AAAAAAAAAjA/IK8jtlhElqk/s1600/16590492-illustration of-an-ant-on-a-white-background.jpg

aco-03-04-2013

Transcript aco-03-04-2013

Ant Colony Optimization: an introduction

Outline

Outline

Ant colonies

Ant Cooperation

Outline

Combinatorial optimization

Combinatorial optimization

Approximate methods

Constructive algorithms

Local search

What is a metaheuristic?

Examples of metaheuristics

Outline

ACO metaheuristic

Construction graph

Heuristic information

Pheromone trail

Artificial ant (1)

Artificial ant (2)

Artificial ant (3)

Artificial ant (4)

The ACO metaheuristic

ConstructAntSolutions

UpdatePheromones

DaemonActions

ACO applications

Outline

Traveling salesman problem

TSP: construction graph

TSP: constraints

TSP: pheromone trails

TSP: heuristic information

TSP: solution construction

ACO algorithms for TSP

Elitist Ant System

Rank-based Ant System

MAX-MIN Ant System

Ant Colony System

Ant Colony System

State of the art in TSP

Current ACO research activity

Further reading

Next time…

Used resources

Directory