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 Ccost (or energy) function H on C

Find: minxC 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 T0 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 systemdescribed by H at arbitrarily low temperatures T, the system will be exclusively in the ground state in the limit T0.

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 T0,the temperature has to be decreased very, very slowly …

A simple combinatorial optimization problem:The (directed) polymer model:

Collection of optimal directed polymers

pathi ieE

]1,0[ieRandom bond energies:

Total energy:

Find ground state – i.e. optimal pathFrom top node to a bottom nodes:

Non-directed directed

Dijkstras algorithm for shortest paths in general graphs

algorithm Dijkstrabegin 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; endend

Start node: sMinimal distance (energy) from s to j: d(j)Predecessor of j: pred(j)

Performance O(N2), with heap reshuffling O(N log(N))

Strong disorder: Vrand >> ; Vint short ranged, hard core

ibondi

ineH )(

1,0in

]1,0[ie

From one line to many lines

Ground state ofN-line problem:Minimum Cost Flow problem

N

i

H

ijjiirand

i zrzrVzzrVdz

drdzH

1 0 )(int

2

)]()([]),([2

Continuum model for N interacting elastic lines in a random potential

Example conf. for 9 lines:

From one line to many lines

3

1

3

3

1

1

1

1

3 -1

3

3-1

-1 -1

-1 3

1

3

3

1

1

1

1

t

s

(with hard-core interactions)

Successive shortest path algorithm

Successive shortest path 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); endend

Find successively shortestpaths from s to t in theresidual network Gr(n)(r = residual capacities):Use node potentials (i)reduced energies e

ij(n) all positive Use Dijkstra’s algorithm

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

jii

ij SSJH 1,0 iij SJ

Si= +1 Si= -1

Find for given random bonds Jij

the ground stateconfiguration {Si}with fixed +/- b.c.

Find interface (cut) with minimum energy

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

})|({})|({ 0Aijjij

Ajijji

else

tiforv

siforv

nn

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 sS, tS‘=V\S,(S,S‘) = {(i,j) A|i S,j S‘}; capacity of the s-t-cut

)',(),(

',SSji ijuSSv

Min-Cut-Max-Flow-Theorem: max{n}v = min[S,S‘]v[S,S‘] and rij*=0 along (S,S‘)

jiijijij nnur n* maximum flow no directed path st in G(n*)

residual network G(n) with residual capacities

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

Strategy:1) flood the network from source2) propagate the flooding toward the target3) push excess flow back towards source

Preflow-Push Algorithm

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

})|({})|({

)(Aijjij

Ajijji nnie

distance function d(i) (w.r.t. target)

excess flow

Preflow-Push-Algorithm (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!

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: Jij>0, Thick lines: Jij<0

Further applicationsof combinatorial optimization methods in Stat-Phys.

o Flux lines with hard core interactionso Vortex glass with strong screeningo Interfaces, elastic manifolds, periodic mediao Wetting phenomena in random systemso Random field Ising systemso Spin glasses (2d polynomial, d>2 NP complete)o Statistical physics of complexity (K-Sat, vertex cover)o Random bond Potts model at Tc in the limit q ∞o ...