1. Optimization Methods H. Rieger, Saarland University, Saarbrücken, Germany Summerschool on Computational Statistical Physics, 4.-11.8.2010 NCCU Taipei, Taiwan

2. Optimization Task:Given a (discrete) configuration space C cost (or energy) function H on C Find:min x  C H(x) E.g.: in statistical physics: Ground state of a classical system in business: shortest path, optimal tour (Traveling Salesman Problem – TSP), optimal work assignement, optimal network flow, …

3. Stochastic optimization / Simulated Annealing Idea: Cost function = Energy H(a), a = configurations Boltzmann distribution for T  0 is non-zero only for the state(s) with lowest energy E = E 0 (  E 1  E 2  …):  P(a)  0 for  if H(a)>E 0, and P(a)  1/M for  if H(a)=E 0, where M is the degeneracy of the ground state Hence if one can equilbrate a Monte-Carlo simulation for the system described by H at arbitrarily low temperatures T, the system will be exclusively in the ground state in the limit T  0.

4. Meta-code for Simulated Annealing: procedure Sim-Ann choose random initial configuration x set initial temperature T to T 0 while ( T>T final ) x‘ = x + random change if H(x‘) < H(x) x=x‘ else if (rand()<exp[ -(H(x)-H(x‘))/T ] ) x=x‘ T=nextT(T) output x The art lies in the definition of the temperature – or cooling - protocol nexT(T), Which is a function of the MC-time, the number of MC moves. Obviously, to successfully equilibrate the system for T  0, the temperature has to be decreased very, very slowly …

5. A simple combinatorial optimization problem: The (directed) polymer model: Collection of optimal directed polymers Random bond energies: Total energy: Find ground state – i.e. optimal path From top node to a bottom nodes: Non-directeddirected

6. Dijkstras algorithm for shortest paths in general graphs algorithm Dijkstra begin S:={s}, S‘=N\{s}; d(s):=0, pred(s):=0; while |S|<|N| do begin choose (i,j): d(j):=min k,m {d(k)+c km |k in S, m in S‘}; S‘=S‘\{j}; S=S+{j}; pred(j):=i; end Start node: s Minimal distance (energy) from s to j: d(j) Predecessor of j: pred(j) Performance O(N 2 ), with heap reshuffling O(N log(N))

7. Strong disorder: V rand >>  ; V int short ranged, hard core From one line to many lines Ground state of N-line problem: Minimum Cost Flow problem Continuum model for N interacting elastic lines in a random potential Example conf. for 9 lines:

8. From one line to many lines 3 1 3 3 1 1 1 1 3 3 3 3 1 3 3 1 1 1 1 t s (with hard-core interactions) Successive shortest path algorithm

9. begin n:=0;  (i)=0; G r (n):=G; for line-counter = 1 to N do compute reduced energies e  ij (n)=e ij +  (i)-  (j); find the shortest path distance d(i) from s to all other nodes i in the residual network G r (n) w.r. to the reduced energies e  ij (n); increase flow on the shortest path from s to t by one unit; compute  (i):=  (i)-d(i); end Find successively shortest paths from s to t in the residual network G r (n) (r = residual capacities): Use node potentials  (i)  reduced energies e  ij (n) all positive  Use Dijkstra’s algorithm

10. Successive shortest path algorithm (2)

11. Details For details and applications see http://xxx.lanl.gov/abs/cond-mat/9705010

12. Another combinatorial optimization problem: Interfaces in random bond Ising ferromagnets S i = +1 S i = -1 Find for given random bonds J ij the ground state configuration {S i } with fixed +/- b.c.  Find interface (cut) with minimum energy

13. Min-Cut-Max-Flow Problem network G(V,A), arcs (Bonds) (i,j)  A have capacity u ij >0, flow 0<=n ij <=u ij fulfills mass balance constraint Find the maximum flow n * with value v from s to t s-t cut [S,S‘] is a partition of V in two disjoint sets with s  S, t  S‘=V\S, (S,S‘) = {(i,j)  A|i  S,j  S‘}; capacity of the s-t-cut Min-Cut-Max-Flow-Theorem: max {n} v = min [S,S‘] v[S,S‘] and r ij * =0 along (S,S‘) n * maximum flow  no directed path s  t in G(n * ) residual network G(n) with residual capacities

14. Augmenting path algorithm for max-flow For a flow x define the residual capacities: And the residual network G(x), consisting of edges with r ij >0

15. Finding augmenting paths: Labeling algorithm

16. Strategy: 1) flood the network from source 2) propagate the flooding toward the target 3) push excess flow back towards source Preflow-Push Algorithm begin d(i) = exact distance from target, d(s)=|V| for all (s,j)  A: n sj =u sj ; choose i  V\{s,t} with e(i)>0 if there is (i,j)  A with d(i)=d(j)+1: push  =min{e(i),r ij } flow units from i to j else relabel d(i)  min j {d(j)+1|(i,j)  A, r ij >0}; end distance function d(i) (w.r.t. target) excess flow

17. Preflow-Push- Algorithm (2) Interprete distance function (or labels) as height of the nodes.

18. Problems that can be mapped on min-cut / max-flow Interfaces / wetting in random media Random field Ising model (in any dimension) Periodic media (flux lines, CDW, etc.) in disordered environments Elastic manifolds with periodic potential and disorder

19. 2d spin glass (with free b.c.) with J ij random, of arbitrary sign!  Set of unsatisfied edges is given by a set of paths from one frustrated plaquette to another (plus closed circles)  Finding the ground state is a „minimum weighted perfect matching“ problem For details see http://xxx.lanl.gov/abs/cond-mat/9705010 Thin lines: J ij >0, Thick lines: J ij <0

20. Further applications of combinatorial optimization methods in Stat-Phys. o Flux lines with hard core interactions o Vortex glass with strong screening o Interfaces, elastic manifolds, periodic media o Wetting phenomena in random systems o Random field Ising systems o Spin glasses (2d polynomial, d>2 NP complete) o Statistical physics of complexity (K-Sat, vertex cover) o Random bond Potts model at T c in the limit q  ∞ o...