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Ameisenalgorithmen – Ant Colony Optimization Lehrprobe zur Habilation, Barbara Hammer, AG LNM, Universität Osnabrück
Optimization ,[object Object],[object Object],[object Object],[object Object],General optimization problem: given f:X  ℝ, find x ε X such that f(x) is minimum    needle in a haystack, hopeless    traveling salesperson problem, NP-hard     shortest path problem, polynomial    protein structure prediction problem, NP-hard
Ant colony food nest
[object Object],[object Object]
Ant Colony Optimization ,[object Object],[object Object],[object Object],[object Object],[object Object]
History: ACO for shortest paths …
History: ACO for shortest paths ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],nest food
History: ACO for shortest paths I:directed ,[object Object],[object Object],for all i: p i :=0;  /*ant position init*/ s i :=hungry;  /*ant state init*/ for all i   j:  τ i  j :=const;  /*pheromone init*/ repeat  for all i:  ant_step (i);  /*ant step*/ for all i   j:  τ i  j  := (1- ρ )  τ i  j  ;  /*evaporate pheromone*/
History: ACO for shortest paths I:directed ant_step (i): if p i =N: s i :=satisfied;  if p i =0: s i :=hungry;  /*collect food/deliver food*/ if s i =hungry:  choose j with p i  j with probability  τ pi   j / Σ pi  j’ τ pi  j’  /*choose next step*/ update  Δτ pi   j  :=  ε ; p i :=j;  /*update pheromone*/ if s i =satisfied:  choose j with j  p i  with probability  τ j  pi / Σ j’  pi τ j’  pi update  Δτ j  pi :=  ε ; p i :=j;  /* reversed directions*/
History: ACO for shortest paths II:general ,[object Object],WC4 WC5 Barbara Marc 449a Anja Dagmar Espresso 322 339 WC3 Friedhelm Fachschaft WC2 Rechner Astrid Zeitschriften WC Bibo RZ-Sekretariat Toiletten Cafete RZ Getraenke-automat Mensa
History: ACO for shortest paths II:general 449a 449a ... Marc was not so happy with the result ...
History: ACO for shortest paths II:general for all i: p i :=0;  /*ant position init*/ s i :=( );   /*ant brain is empty*/ for all i-j:  τ i-j :=const;  /*pheromone init*/ repeat  for all i:  construct_solution (i);  for all i:  global_pheromone_update (i);  for all i-j:  τ i-j  := (1- ρ )  τ i-j ;  /*evaporate*/ construct_solution (i):  while p i ≠N   /*no solution*/ choose j with p i -j with probability  τ pi-j  /  Σ pi-j’ τ pi-j’ ; p i :=j; append j to s i ;  /*remember the trail*/   global_pheromone_update (i):  for all j-j’ in s i :  Δτ j-j’ := 1/length of the path stored in s i ;   minibrain update according  to the quality minibrain s i :=hungry repeat  for all i:  ant_step (i);
History: ACO for shortest paths II:general WC4 WC5 Barbara Marc 449a Anja Dagmar Espresso 322 339 WC3 Friedhelm Fachschaft WC2 Rechner Astrid Zeitschriften WC Bibo RZ-Sekretariat Toiletten Cafete RZ Getraenke Mensa
History: ACO for shortest paths   init pheromone t i-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 ;
Traveling salesperson and ACO-metaheuristic …
Traveling salesperson Traveling salesperson problem  (TSP): given n cities {1,...,N} and distances d ij  ≥0  between the cities, find a tour with shortest length, i.e. a permutation  π :{1,…,N}  {1,…,N} such that the length =  Σ i d π (i) π ((i+1)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
Traveling salesperson init pheromone; 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 edge in the solution:  increase the pheromone according to the quality ;   A B C D key observation : a tour (A  C  D  B  A) decomposes into edges A  C, C  D, D  B pheromone  on the edges
Traveling salesperson init: set  τ ij :=const 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;
Traveling salesperson global_pheromone_update (i);  for all j  k in the solution:  Δτ jk  := const / length of the constructed tour short tours yield to most pheromone construct_solution (i):  set ant to a randomly chosen city; while not yet a solution:  j=current city, expand by j  k with probability = only valid tours are constructed close cities are preferred α ,  β  >0 control the mixture  of the greedy heuristic  and the pheromone following
Traveling salesperson ,[object Object],[object Object],25.1 459.8 422 Sim. Annealing 1.5 420.6 420 Tabu-search 1.3 420.4 420 ACO std.deviation average best  21761 21282 100 cities 580 542 545 535 75 cities 443 426 428 425 50 cities Sim.Ann. Evol.Prog. Gen.Alg. ACO
ACO-metaheuristic ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],edges i  j partial tours tours which visit each city at most once and in consecutive order length of the tour valid tours
ACO-metaheuristic init pheromone  τ i =const for each component c i ; repeat  for all ants i:  construct_solution (i);  for all ants i:  global_pheromone_update (i);  for all pheromones i:  evaporate:  τ i =(1- ρ ) ∙τ i ; construct_solution (i);  init s={ }; while s is not a solution: choose c j  with probability =   expand s by c j ;  global_pheromone_update (i);  for all c j  in the solution s:  increase pheromone:  τ j = τ j + const / f(s);   η  is a heuristic value, α , β  balance the  heuristic/pheromone general ACO algorithm
Protein folding - state of the art ACO …
Protein folding ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],in the 2D-HP-model (Dill)
Protein folding ,[object Object],[object Object],[object Object],9 additional 1-1 contacts
Protein folding ,[object Object],[object Object],[object Object],[object Object],[object Object],R S R ... R L S
Protein folding ,[object Object],[object Object],[object Object],[object Object],[object Object],init pheromone  τ i-D =const 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:  τ i-D =(1- ρ ) ∙τ i-D ; daemon action: local optimization elitism
Protein folding ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],η  is related to the number of 1-1 contacts of this motif feasibility 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: τ j-D  :=  τ j-D  + number of 1-1 contacts in the solution / const;
Protein folding ,[object Object],GA: genetic algorithm EMC: evolutionary algorithm  + Monte Carlo methods MSOE: Monte Carlo including  overlapping conformations PERM: iterated heuristic growing method 47 48 47 100 47 50 50 100 51 53 52 85 42 38 42 39 37 64 36 36 35 34 60 21 21 21 21 50 23 23 23 23 48 14 14 14 14 36 8 8 8 8 25 9 9 9 9 24 9 9 9 9 20 ACO PERM MSOE EMC GA length
General comments -  where is my manuscript …
General comments ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
General comments ,[object Object],No free lunch theorem  (Macready/Wolpert)  I n 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 they’ll find it.
Rettet die Bildung! Jawoll! Gegen Stellenk ürzungen im Hochschulbereich!
 
ACO-metaheuristic ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
Protein folding – state of the art ACO ,[object Object],[object Object],[object Object],[object Object],[object Object],LRS S RLS LRS L RLS L RSS RLS L SRL RLS LRS S RLS LRS L RLS SL S L RL R

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Aco

  • 1. Ameisenalgorithmen – Ant Colony Optimization Lehrprobe zur Habilation, Barbara Hammer, AG LNM, Universität Osnabrück
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  • 6. History: ACO for shortest paths …
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  • 9. History: ACO for shortest paths I:directed ant_step (i): if p i =N: s i :=satisfied; if p i =0: s i :=hungry; /*collect food/deliver food*/ if s i =hungry: choose j with p i  j with probability τ pi  j / Σ pi  j’ τ pi  j’ /*choose next step*/ update Δτ pi  j := ε ; p i :=j; /*update pheromone*/ if s i =satisfied: choose j with j  p i with probability τ j  pi / Σ j’  pi τ j’  pi update Δτ j  pi := ε ; p i :=j; /* reversed directions*/
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  • 11. History: ACO for shortest paths II:general 449a 449a ... Marc was not so happy with the result ...
  • 12. History: ACO for shortest paths II:general for all i: p i :=0; /*ant position init*/ s i :=( ); /*ant brain is empty*/ for all i-j: τ i-j :=const; /*pheromone init*/ repeat for all i: construct_solution (i); for all i: global_pheromone_update (i); for all i-j: τ i-j := (1- ρ ) τ i-j ; /*evaporate*/ construct_solution (i): while p i ≠N /*no solution*/ choose j with p i -j with probability τ pi-j / Σ pi-j’ τ pi-j’ ; p i :=j; append j to s i ; /*remember the trail*/ global_pheromone_update (i): for all j-j’ in s i : Δτ j-j’ := 1/length of the path stored in s i ; minibrain update according to the quality minibrain s i :=hungry repeat for all i: ant_step (i);
  • 13. History: ACO for shortest paths II:general WC4 WC5 Barbara Marc 449a Anja Dagmar Espresso 322 339 WC3 Friedhelm Fachschaft WC2 Rechner Astrid Zeitschriften WC Bibo RZ-Sekretariat Toiletten Cafete RZ Getraenke Mensa
  • 14. History: ACO for shortest paths init pheromone t i-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 ;
  • 15. Traveling salesperson and ACO-metaheuristic …
  • 16. Traveling salesperson Traveling salesperson problem (TSP): given n cities {1,...,N} and distances d ij ≥0 between the cities, find a tour with shortest length, i.e. a permutation π :{1,…,N}  {1,…,N} such that the length = Σ i d π (i) π ((i+1)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
  • 17. Traveling salesperson init pheromone; 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 edge in the solution: increase the pheromone according to the quality ; A B C D key observation : a tour (A  C  D  B  A) decomposes into edges A  C, C  D, D  B pheromone on the edges
  • 18. Traveling salesperson init: set τ ij :=const 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;
  • 19. Traveling salesperson global_pheromone_update (i); for all j  k in the solution: Δτ jk := const / length of the constructed tour short tours yield to most pheromone construct_solution (i): set ant to a randomly chosen city; while not yet a solution: j=current city, expand by j  k with probability = only valid tours are constructed close cities are preferred α , β >0 control the mixture of the greedy heuristic and the pheromone following
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  • 22. ACO-metaheuristic init pheromone τ i =const for each component c i ; repeat for all ants i: construct_solution (i); for all ants i: global_pheromone_update (i); for all pheromones i: evaporate: τ i =(1- ρ ) ∙τ i ; construct_solution (i); init s={ }; while s is not a solution: choose c j with probability = expand s by c j ; global_pheromone_update (i); for all c j in the solution s: increase pheromone: τ j = τ j + const / f(s); η is a heuristic value, α , β balance the heuristic/pheromone general ACO algorithm
  • 23. Protein folding - state of the art ACO …
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  • 30. General comments - where is my manuscript …
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  • 33. Rettet die Bildung! Jawoll! Gegen Stellenk ürzungen im Hochschulbereich!
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