Ameisenalgorithmen

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Ameisenalgorithmen Ant Colony Optimization

• Lehrprobe zur Habilation,
• Barbara Hammer,
• AG LNM, Universität Osnabrück

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Optimization
General optimization problem given fX?R, find
xeX such that f(x) is minimum

• Given a graph with two specified vertices A and
  B, find a shortest path from A to B.
• Given a set of cities and pairwise distances,
  find a shortest tour.
• Given a sequence of amino acids of a protein,
  find the structure of the protein.
• Where is my manuscript for the talk, I put it on
  this pile of papers...

? shortest path problem, polynomial
? traveling salesperson problem, NP-hard
? protein structure prediction problem, NP-hard
? needle in a haystack, hopeless
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Ant colony
food
nest
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• Ant Colony Optimization (ACO)
• a heuristic optimization method for shortest path
  and other optimization problems which borrows
  ideas from biological ants

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Ant Colony Optimization

• Outline
• History ACO for shortest paths
• Traveling salesperson and ACO-metaheuristic
• Protein folding - state of the art ACO
• General comments - where is my manuscript?

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History ACO for shortest paths
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History ACO for shortest paths

• Goss et al. 1989, Deneuborg et al. 1990
• experiments with Argentine ants
• ants go from the nest to the food source and
  backwards
• after a while, the ants prefer the shortest path
  from the nest to the food source
• stigmercy
• the ants communicate indirectly laying pheromone
  trails and following trails with higher pheromone
• length gradient ? pheromone will accumulate on
  the shortest path
• Dorigo et al. 1991
• applications to shortest path problems

food
nest
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History ACO for shortest paths Idirected

• A first ACO for a simple shortest path problem
• directed acyclic graph (V0,...,N, Ei?j),
  ant hill 0, food source N

for all i pi0
/ant position init/
sihungry
/ant state init/ for all i?
j ti?jconst
/pheromone init/ repeat for all i
ant_step(i)
/ant step/ for all i? j
ti?j (1-?) ti?j /evaporate pheromone/
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History ACO for shortest paths Idirected
ant_step(i) if piN sisatisfied if pi0
sihungry
/collect food/deliver food/ if sihungry
choose j with pi?j with probability tpi?
j/Spi?jtpi?j /choose next step/
update ?tpi? j e pij
/update
pheromone/ if sisatisfied choose j with j?pi
with probability tj?pi/Sj?pitj?pi
update ?tj?pi e pij
/ reversed
directions/
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History ACO for shortest paths IIgeneral

• ...a more complex undirected cyclic graph ...

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History ACO for shortest paths IIgeneral
... Marc was not so happy with the result ...
449a
449a
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History ACO for shortest paths IIgeneral
for all i pi0
/ant position init/
si( ) /ant
brain is empty/ for all i-j ti-jconst
/pheromone
init/ repeat for all i construct_solution(i
) for all i
global_pheromone_update(i) for
all i-j ti-j (1-?) ti-j
/evaporate/
sihungry
minibrain
repeat for all i ant_step(i)
construct_solution(i) while pi?N
/no
solution/ choose j with pi-j with
probability tpi-j / Spi-jtpi-j pij
append j to si
/remember the trail/
minibrain
global_pheromone_update(i) for all j-j in si
?tj-j 1/length of the path stored in
si
update according to the quality
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History ACO for shortest paths IIgeneral
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History ACO for shortest paths
init pheromone ti-j repeat for all ants i
construct_solution(i) for all
ants i global_pheromone_update(i)
for all edges evaporate pheromone
construct_solution(i) init ant while not yet a
solution expand the solution by one edge
probabilistically according to the pheromone

global_pheromone_update(i) for all edges in the
solution increase the
pheromone according to the quality
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Traveling salesperson and ACO-metaheuristic
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Traveling salesperson
Traveling salesperson problem (TSP) given n
cities 1,...,N and distances dij 0 between the
cities, find a tour with shortest length, i.e. a
permutation p1,,N?1,,N such that the
length Sidp(i)p((i1)mod N) is
minimum classical NP-hard benchmark problem
?
?
A simple greedy heuristic start somewhere and
always add the closest not yet visited city to
the tour
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Traveling salesperson
A
init pheromone repeat for all ants i
construct_solution(i) for all
ants i global_pheromone_update(i)
for all edges evaporate pheromone
C
B
D
construct_solution(i) init ant while not yet a
solution expand the solution by one edge
probabilistically according to the pheromone

key observation a tour (A?C?D?B?A) decomposes
into edges A?C, C?D, D?B pheromone on the edges
global_pheromone_update(i) for all edge in the
solution increase the
pheromone according to the quality
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Traveling salesperson
init set tijconst for all cities i?j repeat
for all ants i construct_solution(i)
for all ants i global_pheromone_update(
i) for all edges i-j evaporate
pheromone
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Traveling salesperson
construct_solution(i) set ant to a randomly
chosen city while not yet a solution
jcurrent city, expand by j?k with
probability
only valid tours are constructed
a, ß gt0 control the mixture of the greedy
heuristic and the pheromone following
close cities are preferred
global_pheromone_update(i) for all j?k in the
solution ?tjk const
/ length of the constructed tour
short tours yield to most pheromone
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Traveling salesperson

• Results for a 30 cities instance (10 runs, one
  hour)
• Results for larger instances (25000 constructed
  tours, best tour documented)

best average std.deviation
ACO 420 420.4 1.3
Tabu-search 420 420.6 1.5
Sim. Annealing 422 459.8 25.1
ACO Gen.Alg. Evol.Prog. Sim.Ann.
50 cities 425 428 426 443
75 cities 535 545 542 580
100 cities 21282 21761
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ACO-metaheuristic

• Optimization problem for ACO
• over a set of basic components C c1,...,cn
• partial solutions are subsets s in C
• feasible (partial) solutions F in C
• solutions S in C
• cost function f for solutions
• Goal
• iteratively expand feasible partial solutions by
  components to reach a solution s with minimum
  f(s),
• pheromone attached to each component ci guides
  the search

edges i?j
partial tours
tours which visit each city at most once and in
consecutive order
valid tours
length of the tour
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ACO-metaheuristic
general ACO algorithm
init pheromone ticonst for each component
ci repeat for all ants i
construct_solution(i) for all
ants i global_pheromone_update(i)
for all pheromones i evaporate ti(1-?)ti
construct_solution(i) init s while s is
not a solution choose cj with probability
expand s by cj
? is a heuristic value, a,ß balance the
heuristic/pheromone
global_pheromone_update(i) for all cj in the
solution s increase
pheromone tjtj const / f(s)
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Protein folding - state of the art ACO
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Protein folding

• Protein folding
• given a sequence of amino acids s1sn
• where si in 1,0, i.e. hydrophobic/polar
• determine the structure of the protein
• i.e. coordinates in a 2D rectangular lattice,
  such that
• neighbored sequence entries are at neighbored
  positions
• each position is occupied at most once
• the number of 1-1 contacts in the 2D structure is
  maximized

in the 2D-HP-model (Dill)
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Protein folding
9 additional 1-1 contacts

• Dill, 1985 the HP model preserves important
  information of the biological conformation
• Crescenzi et al. and Berger/Leighton, 1998 the
  problem is NP hard
• Shmygelska/Hoss, 2003 ACO

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Protein folding
...
R
S
R

• start at the left end and iteratively fold one
  amino acid into a relative direction
• R,S,R,R,L,L,R,S,R,R,L,R,L,L,R,R,S,R in
  R,S,Llength-2
• basic components
• an element in i-R,i-S,i-L represents
  the local structural motif at position
  (i-1,i,i1)
• pheromone values ti-D , i 2.. length-1, D
  L,S,R

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Protein folding

• components local structural motifs i-D
• partial solutions subsets of local structural
  motifs
• feasible partial solutions sequences of
  consecutive structural motifs without overlap of
  the amino acids in the 2D lattice
• solutions final folds
• cost function to be maximized number of 1-1
  contacts in the 2D lattice

init pheromone ti-Dconst for each tuple
i-D repeat for all ants i
construct_solution(i) for the
best ants i optimize_solution(i)
for the best ants i global_pheromone_update(i)
for all pheromones i-D
evaporate ti-D(1-?)ti-D
daemon action local optimization
elitism
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Protein folding

• construct_solution(i)
• init s
• while s is not a solution position j
• choose a local structural motif j-D with
  probability proportional to
• 0 if the position is already occupied or the
  sequence gets trapped
• proportional to tj-Da?j-Dß
• expand s by the chosen motif

feasibility
? is related to the number of 1-1 contacts of
this motif
optimize_solution(i) perform a fixed number of
feasible and improving substitutions of local
structural motifs at random
global_pheromone_update(i) for all local
structural motifs in a solution tj-D tj-D
number of 1-1 contacts in the solution / const
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Protein folding

• Best reported results for different size
  instances

length GA EMC MSOE PERM ACO
20 9 9 9 9
24 9 9 9 9
25 8 8 8 8
36 14 14 14 14
48 23 23 23 23
50 21 21 21 21
60 34 35 36 36
64 37 39 42 38 42
85 52 53 51
100 50 50 47
100 47 48 47
GA genetic algorithm EMC evolutionary algorithm
Monte Carlo methods MSOE Monte Carlo
including overlapping conformations PERM
iterated heuristic growing
method
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General comments - where is my manuscript
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General comments

• ACOs nice, powerful, and robust metaheuristic
  for NP hard, possibly non -static optimization
  problems, the solutions of which decompose into
  single components
• Applications for ( state of the art results
  for some settings)
• quadratic assignment problems
• vehicle routing
• sequential ordering
• shortest common supersequence
• scheduling
• graph coloring and partitioning
• telecommunication networks and routing
• ...
• more info on ACOs
• Swarm Intelligence, From Natural to Artificial
  Systems, E.Bonabeau, M.Dorigo, G.Theraulaz, Santa
  Fe, 1999
• http//iridia.ulb.ac.be/mdorigo/ACO/
• Duft der Daten, Der Spiegel, November 13, 2000
• Swarm smarts, Scientific American, March 2000

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General comments

• Are ACOs better than other metaheuristics for
  general optimization problems?

No free lunch theorem (Macready/Wolpert) In the
mean, no optimization algorithm is to be
preferred! Precise
Assume A and B are
finite, B is totally ordered, F is a set of
functions from A to B which is closed under
permutation, H is a (randomized) search
heuristic. Then the expected time to reach the
first optimum is independent of H.
... so it might take a while until the ants find
my manuscript, but theyll find it.
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Rettet die Bildung!
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(No Transcript)
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ACO-metaheuristic

• ACO-metaheuristic
• applicable to general optimization problems as
  stated above
• robust and tolerant to changes, e.g. in
  non-static problems
• yields state of the art solvers for some problems
  
• with modifications
• daemon actions improve the found solutions using
  local search (e.g. k-opt for TSP)
• elitism update pheromone only for the (local or
  global) best ants
• ...

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Protein folding state of the art ACO

• optimize_solution(i)
• perform a fixed number of feasible and improving
  search moves randomly chosen from the following
  possibilities
• substitution of a single motif substitute one
  motif i-D by a different one i-D
• substitution of a sequence of motifs substitute
  all motifs within randomly chosen positions by
  different motifs
• long-range moves substitute one local motif and
  refold the two ends to feasible settings, whereby
  the respective original motifs are preserved if
  possible

LRSSRLS
LRSLRLS
LRSSRLS
LSRLRLS
LRSSRLS
LRSLRLS
SLSLRLR