Optimization Methods
H. Rieger, Saarland University, Saarbrücken, Germany
Summerschool on Computational Statistical Physics, 4.-11.8.2010
NCCU Taipei, Taiwan
Optimization
Task:
Given a (discrete) configuration space C
cost (or energy) function H on C
Find:
minx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,
…
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 = E0 (  E1  E2  …):
 P(a)  0 for  if H(a)>E0, and
P(a) 1/M for  if H(a)=E0,
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.
Meta-code for Simulated Annealing:
procedure Sim-Ann
choose random initial configuration x
set initial temperature T to T0
while ( T>Tfinal )
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 …
A simple combinatorial optimization problem:
The (directed) polymer model:
Non-directed
Random bond energies:
Total energy:
ei [0,1]
E  i path ei
Find ground state – i.e. optimal path
From top node to a bottom nodes:
Collection of optimal directed polymers
directed
Dijkstras algorithm
for shortest paths in general graphs
Start node: s
Minimal distance (energy)
from s to j: d(j)
Predecessor of j: pred(j)
algorithm Dijkstra
begin
S:={s}, S‘=N\{s};
d(s):=0, pred(s):=0;
while |S|<|N| do
begin
choose (i,j):
d(j):=mink,m{d(k)+ckm|k in S, m in S‘};
S‘=S‘\{j}; S=S+{j};
pred(j):=i;
Performance O(N2),
end
with heap reshuffling O(N log(N))
end
From one line to many lines
Continuum model for N interacting elastic lines in
a random potential
2


   dri 

H    dz     Vrand [ri ( z ), z ]  Vint [ri ( z )  rj ( z )]


i 1 0
j ( i )
 2  dz 

N H
Strong disorder: Vrand >> ; Vint short ranged, hard core

H
e n
i i
i ( bond )
Ground state of
N-line problem:
Minimum Cost
Flow problem
ni  0,1
ei [0,1]
Example conf.
for 9 lines:
From one line to many lines
(with hard-core interactions)
s
3
1
3
3
-1
-1
1
3
-1
1
1
1
3
-1
3
3
1
-1
1
1
3
1
3
1
t
Successive shortest path algorithm
Successive shortest path algorithm
Find successively shortest
paths from s to t in the
residual network Gr(n)
(r = residual capacities):
Use node potentials (i)
reduced energies eij(n)
all positive 
Use Dijkstra’s algorithm
begin
n:=0; (i)=0; Gr(n):=G;
for line-counter = 1 to N do
compute reduced energies eij(n)=eij + (i)- (j);
find the shortest path distance d(i) from s to all other nodes i
in the residual network Gr(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
end
Successive shortest path algorithm (2)
Details
For details and applications see
http://xxx.lanl.gov/abs/cond-mat/9705010
Another combinatorial optimization problem:
Interfaces in random bond Ising ferromagnets
H   J ij Si S j
J ij  0, Si  1
i
Find for given
random bonds Jij
the ground state
configuration {Si}
with fixed +/- b.c.
Si= +1
 Find interface (cut)
with minimum
energy
Si= -1
Min-Cut-Max-Flow Problem
network G(V,A), arcs (Bonds) (i,j)A have capacity uij>0,
flow 0<=nij<=uij fulfills mass balance constraint
for i  s
v

n

n

 v for i  t


ji
ij
{ j|( ji )A}
{ j|(ij )A}
 0 else

Find the maximum flow n* with value v from s to t
residual network G(n) with residual capacities
rij u ij nij  n ji
n* maximum flow  no directed path st in G(n*)
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 vS , S '  (i , j )( S , S ') uij
Min-Cut-Max-Flow-Theorem: max{n}v = min[S,S‘]v[S,S‘] and rij*=0 along (S,S‘)
Augmenting path algorithm for max-flow
For a flow x define the residual capacities:
And the residual network G(x), consisting of edges with rij>0
Finding augmenting paths: Labeling algorithm
Preflow-Push
Algorithm
Strategy:
1) flood the network from source
2) propagate the flooding toward the target
3) push excess flow back towards source
excess flow e(i ) 
n
ji
{ j |( ji ) A}

n
ij
{ j |( ij ) A}
distance function d(i) (w.r.t. target)
begin
d(i) = exact distance from target, d(s)=|V|
for all (s,j)  A: nsj=usj;
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),rij} flow units from i to j
else
relabel d(i)minj{d(j)+1|(i,j)A, rij>0};
end
Preflow-PushAlgorithm (2)
Interprete
distance function (or labels) as
height of the nodes.
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
2d spin glass (with free b.c.)
with Jij random, of arbitrary sign!
Thin lines: Jij>0, Thick lines: Jij<0
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
Further applications
of combinatorial optimization methods in Stat-Phys.
o
o
o
o
o
o
o
o
o
Flux lines with hard core interactions
Vortex glass with strong screening
Interfaces, elastic manifolds, periodic media
Wetting phenomena in random systems
Random field Ising systems
Spin glasses (2d polynomial, d>2 NP complete)
Statistical physics of complexity (K-Sat, vertex cover)
Random bond Potts model at Tc in the limit q ∞
...