1

Simulated Annealing

A Basic Introduction

Olivier de Weck, Ph.D.

Massachusetts Institute of Technology

Lecture 10

2

Outline

• Heuristics

• Background in Statistical Mechanics

– Atom Configuration Problem

– Metropolis Algorithm

• Simulated Annealing Algorithm

• Sample Problems and Applications

• Summary

3

Learning Objectives

• Review background in Statistical Mechanics:

configuration, ensemble, entropy, heat capacity

• Understand the basic assumptions and steps in

Simulated Annealing (SA)

• Be able to transform design problems into a

combinatorial optimization problem suitable to

SA

• Understand strengths and weaknesses of SA

4

Heuristics

5

What is a Heuristic?

• A Heuristic is simply a rule of thumb that hopefully will find a good

answer.

• Why use a Heuristic?

– Heuristics are typically used to solve complex (large, nonlinear, non-

convex (i.e. contain local minima)) multivariate combinatorial optimization

problems that are difficult to solve to optimality.

• Unlike gradient-based methods in a convex design space, heuristics

are NOT guaranteed to find the true global optimal solution in a single

objective problem, but should find many good solutions (the

mathematician's answer vs. the engineer’s answer)

• Heuristics are good at dealing with local optima without getting stuck

in them while searching for the global optimum.

6

Types of Heuristics

• Heuristics Often Incorporate Randomization

• Two Special Cases of Heuristics

– Construction Methods

• Must first find a feasible solution and then improve it.

– Improvement Methods

• Start with a feasible solution and just try to improve it.

• 3 Most Common Heuristic Techniques

– Simulated Annealing

– Genetic Algorithms

– Tabu Search

– New Methods: Particle Swarm Optimization, etc…

7

Origin of Simulated Annealing (SA)

• Definition: A heuristic technique that mathematically mirrors the

cooling of a set of atoms to a state of minimum energy.

• Origin: Applying the field of Statistical Mechanics to the field of

Combinatorial Optimization (1983)

• Draws an analogy between the cooling of a material (search for

minimum energy state) and the solving of an optimization problem.

• Original Paper Introducing the Concept

– Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., “Optimization by Simulated

Annealing,” Science, Volume 220, Number 4598, 13 May 1983, pp. 671-

680.

8

MATLAB® “peaks” function

• Difficult due to plateau at z=0, local maxima

• SAdemo1

• xo=[-2,-2]

• Optimum at

• x*=[ 0.012, 1.524]

• z*=8.0484

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

x

Peaks

y

Start Point

Maximum

9

“peaks” convergence• Initially ~ nearly random search

• Later ~ gradient search

0 50 100 150 200 250 300-10

-8

-6

-4

-2

0

2

Iteration Number

Syste

m E

nerg

y

SA convergence history

current configuration

new best configuration

10

Statistical Mechanics

11

The Analogy

• Statistical Mechanics: The behavior of systems with many

degrees of freedom in thermal equilibrium at a finite temperature.

• Combinatorial Optimization: Finding the minimum of a given

function depending on many variables.

• Analogy: If a liquid material cools and anneals too quickly, then the

material will solidify into a sub-optimal configuration. If the liquid

material cools slowly, the crystals within the material will solidify

optimally into a state of minimum energy (i.e. ground state).

– This ground state corresponds to the minimum of the cost function in an

optimization problem.

12

Sample Atom Configuration

-1 0 1 2 3 4 5 6 7-1

0

1

2

3

4

5

6

7

1

2

3

4

Atom Configuration - Sample Problem

E=133.67

-1 0 1 2 3 4 5 6 7-1

0

1

2

3

4

5

6

7

1

23

4

Atom Configuration - Sample Problem

Original Configuration Perturbed Configuration

E=109.04

Perturbing = move a random atom

to a new random (unoccupied) slotEnergy of original (configuration)

13

Configurations

• Mathematically describe a configuration

– Specify coordinates of each “atom”

– Specify a slot for each atom

1 2 3 4

1 3 4 5 with i=1,..,4 , , ,

1 2 4 3ir r r r r

with i=1,..,4 1 12 19 23ir R

14

Energy of a state

• Each state (configuration) has an energy

level associated with it

totH T V E

, , , ,i i

i

H q q t q p L q q t Hamiltonian

Energy of configuration

=

Objective function value of designkinetic potential

Energy

15

Energy sample problem

• Define energy function for “atom” sample

problem1

2 2 2

potentialenergy

kineticenergy

N

i i i j i j

j i

E mgy x x y y

1

N

i i

i

E R E r Absolute and relative position of

each atom contributes to Energy

16

Compute Energy of Config. A

• Energy of initial configuration

1

2

3

4

1 10 1 5 18 20 20.95

1 10 2 5 5 5 26.71

1 10 4 18 5 2 47.89

1 10 3 20 5 2 38.12

E

E

E

E

Total Energy Configuration A: E({rA})= 133.67

-1 0 1 2 3 4 5 6 7-1

0

1

2

3

4

5

6

7

1

2

3

4

Atom Configuration - Sample Problem

17

Boltzmann Probability

!

!R

PN

P N

Number of configurations

What is the likelihood that a particular configuration will

exist in a large ensemble of configurations?

P=# of slots=25

N=# of atoms =4NR=6,375,600

({ }){ } exp

B

E rP r

k T

Boltzmann probability

depends on energy

and temperature

18

Boltzmann Collapse at low T

0 20 40 60 80 100 120 140 160 180 2000

2000

4000

6000

8000

10000

12000

14000

Energy

Occure

nces

T=10 - Eavg

=58.9245

T=10

0 20 40 60 80 100 120 140 160 180 2000

2000

4000

6000

8000

10000

12000

14000

Energy

Occure

nces

T=1 - Eavg

=38.1017

T=1

0 20 40 60 80 100 120 140 160 180 2000

2000

4000

6000

8000

10000

12000

Energy

Occure

nces

T=100 - Eavg

=100.1184

Boltzmann Distribution collapses to the lowest energy

state(s) in the limit of low temperature

Basis of search by Simulated Annealing

T=100

T high T low

19

Partition Function Z

• Ensemble of configurations can be

described statistically

• Partition function, Z, generates the

ensemble average

1

Tr exp expRN

i i

iB B

E EZ

k T k T

Initially (at T>>0) equal to the number of possible configurations

20

Free Energy

ln ( )BF T k T Z E T TS

1

( )RN

avg i

i

E E T E T

1

expln

( )1

RNi

i

i B

avg

B

E TE T

k T d ZE E T

Z d k T

Relates average Energy at T with Entropy S

Average Energy of all

Configurations in an

ensemble

21

Specific Heat and Entropy

• Specific Heat

• Entropy

2222

1

2 2

1( )

( )( )

RN

i

iR

B B

E T E TE T E TNdE T

C TdT k T k T

( ) ( )dS T C T

dT T

1

1

( )( ) ( )

T

T

C TS T S T dT

T

Entropy is ~ equal to ln(# of unique configurations)

22

Low Temperature Statistics

100

102

104

100

105

Temperature

# o

f configura

tions,

NT

100

102

104

0

2

4

6

8

10

Temperature

Specific Heat C and Entropy S

100

102

104

-150

-100

-50

0

50

100

150

Temperature

Fre

e E

nerg

y,

F

100

102

104

-150

-100

-50

0

50

100

150

Temperature

Avera

ge E

nerg

y,

Eavg

Approaches

E* from

below

Approaches

E* from

above

Entropy S

decreases

with lower T

Ht. Capacity# configs

decreases

23

Minimum Energy Configurations

• Sample Atom Placement Problem

-1 0 1 2 3 4 5 6 7-1

0

1

2

3

4

5

6

7

1

23

4

Atom Configuration - Sample Problem

Low gravity g=0.1

E*=14.26

-1 0 1 2 3 4 5 6 7-1

0

1

2

3

4

5

6

7

12 34

Atom Configuration - Sample Problem

E*=60

Strong gravity g=10

Uniqueness?

24

Simulated Annealing

25

Dilemma

• Cannot compute energy of all configurations !

– Design space often too large

– Computation time for a single function evaluation can

be large

• Use Metropolis Algorithm, at successively lower

temperatures to find low energy states

– Metropolis: Simulate behavior of a set of atoms in

thermal equilibrium (1953)

– Probability of a configuration existing at T

Boltzmann Probability P(r,T)=exp(-E(r)/T)

26

The SA Algorithm• Terminology:

– X (or R or ) = Design Vector (i.e. Design, Architecture, Configuration)

– E = System Energy (i.e. Objective Function Value)

– T = System Temperature

– = Difference in System Energy Between Two Design Vectors

• The Simulated Annealing Algorithm

1) Choose a random Xi, select the initial system temperature, and specify the

cooling (i.e. annealing) schedule

2) Evaluate E(Xi) using a simulation model

3) Perturb Xi to obtain a neighboring Design Vector (Xi+1)

4) Evaluate E(Xi+1) using a simulation model

5) If E(Xi+1)< E(Xi), Xi+1 is the new current solution

6) If E(Xi+1)> E(Xi), then accept Xi+1 as the new current solution with a

probability e(- /T) where = E(Xi+1) - E(Xi).

7) Reduce the system temperature according to the cooling schedule.

8) Terminate the algorithm.

27

SA Block Diagram

n

Reached equilbrium at Tj ?

End Tmin

Start To

T<Tmin?

Define initialconfiguration Ro

Evaluate energy E(Ro)

Evaluate energy E(Ri+1)

Accept Ri+1 asnew configuration

Keep Ri as current configuration

Perturb configuration Ri

_> Ri+1

Compute energy difference E = E(Ri+1)-E(Ri)

Reduce temperature Ti+1 =Tj - T E < 0?

Create randomnumber v in [0,1]

exp - E > v?T

Metropolis step

SA Block Diagram

y

y

y

n

n

n

y

Image by MIT OpenCourseWare.

28

MATLAB Function: SA.m

xo

Initial Configuration

Simulated Annealing

Algorithm

evaluation

functionperturbation

function

file_eval.m file_perturb.m

Option Flags

Best Configuration

options

SA.m xbest Ebest

Historyxhist

29

Exponential Cooling

• Typically (T1/To)~0.7-0.91

1

k

k k

o

TT T

T

0 5 10 15 20 25 30 35 40 450

1

2

3

4

5

6

7

8

9

10

Temperature Step

Simulated Annealing Evolution

C-specific heat

S-entropy

T/10-temperature

Can also do linear

or step-wise cooling

…

But exponential cooling

often works best.

30

Key Ingredients for SA

1. A concise description of a configuration (architecture, design,

topology) of the system (Design Vector).

2. A random generator of rearrangements of the elements in a

configuration (Neighborhoods). This generator encapsulates rules

so as to generate only valid configurations. Perturbation function.

3. A quantitative objective function containing the trade-offs that

have to be made (Simulation Model and Output Metric(s)).

Surrogate for system energy.

4. An annealing schedule of the temperatures and/or the length of

times for which the system is to be evolved.

31

Sample Problems

32

Traveling Salesman Problem

• N cities arranged randomly on [-1,1]

• Choose N=15

• SAdemo2.m

• Minimize “cost” of route (length, time,…)

• Visit each city once, return to start city

2 22 2

1 1

1 1 1

visit N cities return home to first city

N

j i j i j j N

i j j

l R x R x R x R x R

33

TSP Problem (cont.)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Start1

2

3

45 6

7

8

9

10

11

12

13

14

15

Length of TSP Route: 17.4309

Initial (Random) Route

Length: 17.43

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Start

1

2

3

45 6

7

8

9

10

11

12

13

14

15

Length of TSP Route: 8.2384

Shortest Route

Final (Optimized) Route

Length: 8.24

Result with SA

34

Structural Optimization

• Define:

– Design Domain

– Boundary Conditions

– Loads

– Mass constraint

• Subdivide domain

– N x M design “cells”

– Cell density =1 or =0

• Where to put material to minimize compliance?

1

max

1

find 1,...,

min ( )

s.t.

s.t.

i

T

i

N

i i

i

i N

C f u

u K f

V m

35

Structural Topology Optimization (II)

1 2

1211

2 3

1312

3 4

1413

4 5

1514

5 6

1615

6 7

1716

7 8

1817

8 9

1918

9 10

201911 12

2221

12 13

2322

13 14

2423

14 15

2524

15 16

2625

16 17

2726

17 18

2827

18 19

2928

19 20

302921 22

3231

22 23

3332

23 24

3433

24 25

3534

25 26

3635

26 27

3736

27 28

3837

28 29

3938

29 30

403931 32

4241

32 33

4342

33 34

4443

34 35

4544

35 36

4645

36 37

4746

37 38

4847

38 39

4948

39 40

5049

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

28 29 30 31 32 33 34 35 36

1 2

1211

2 3

1312

3 4

1413

45

1514

56

1615

67

1716

78

1817

89

1918

910

2019

11 12

2221

12 13

2322

13 14

2423

1415

2524

1516

2625

1617

2726

1718

2827

1819

2928

1920

3029

21 22

3231

22 23

3332

23 24

3433

24 25

3534

2526

3635

2627

3736

2728

3837

2829

3938

2930

4039

31 32

4241

32 33

4342

33 34

4443

3435

4544

3536

4645

3637

4746

3738

4847

3839

4948

3940

5049

Deformation not drawn to scale

“Energy” = strain energy = compliance

Computed via Finite Element Analysis

Cantilever

Boundary

condition

Applied Load

F=-2000

36

Structural Toplogy Optimization

0 2 4 6 8 10-1

0

1

2

3

4

5

6

7

8

9

Initial Structural Configuration: xo

Initial Structural Toplogy “Final” Toplogy

Compliance C=593.0

Compliance C=236.68

“island”Structural Configuration

Not “optimal” – often

need post-processing

37

Structural Optimization –

Convergence Analysis

0 5 10 15 20 25 30 35 400

2

4

6

8

10

12

14

Temperature Step

Simulated Annealing Evolution

C-specific heat

S-entropy

ln(T)-temperature

200 400 600 800 1000 1200 1400 1600 1800 2000 22000

5

10

15

20

25

30

35

40

45

T=3e+002

Energy

Occure

nces

T=2.7e+002T=2.4e+002

T=2.2e+002T=1.9e+002

T=1.8e+002

T=1.6e+002

T=1.4e+002

T=1.3e+002

T=1.1e+002T=1e+002

T=93

T=84

T=75

T=68

T=61

T=55

T=49T=45

T=40

T=36

T=32

T=29

T=26

T=24

T=21

T=19

T=17

T=16

T=14

T=13T=11T=10

T=9.2T=8.2

T=7.4

T=6.7

T=6

Evolution of

- Entropy, Temperature, Specific Heat

Don’t terminate

when C is still

large !

Boltzmann Distributions

38

Premature termination

0 500 1000 1500 2000 2500 3000200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Iteration Number

Syste

m E

nerg

y

SA convergence history

current configuration

new best configuration

Indicator: Best Configuration found only shortly

before Simulated Annealing terminated.

39

Final Example: Telescope Array

Optimization

-200 -100 0 100 200-200

-100

0

100

200Cable Length = 1064.924

-100 -50 0 50 100-100

-50

0

50

100UV Density = 0.67236

• Place N=27 stations in xy within a 200 km radius

• Minimize UV density metric

• Ideally also minimize cable length

Initial

Configuration

40

Optimized Solution

-200 -100 0 100 200-200

-100

0

100

200Cable Length = 1349.609

-100 -50 0 50 100-100

-50

0

50

100UV Density = 0.33333

Simulated AnnealingImproved UV density from 0.67 to 0.33

• Simulated Annealing transforms the array:

– Hub-and-Spoke Circle-with-arms

Cohanim B. E., Hewitt J. N., and de Weck O.L., “The Design of Radio Telescope Array

Configurations using Multiobjective Optimization: Imaging Performance versus Cable

Length”, The Astrophysical Journal, Supplement Series, 154, 705-719, October 2004

41

Summary

42

Summary: Steps of SA

• The Simulated Annealing Algorithm

1) Choose a random Xi, select the initial system temperature, and outline the

cooling (ie. annealing) schedule

2) Evaluate E(Xi) using a simulation model

3) Perturb Xi to obtain a neighboring Design Vector (Xi+1)

4) Evaluate E(Xi+1) using a simulation model

5) If E(Xi+1)< E(Xi), Xi+1 is the new current solution

6) If E(Xi+1)> E(Xi), then accept Xi+1 as the new current solution with a

probability e(- /T) where = E(Xi+1) - E(Xi).

7) Reduce the system temperature according to the cooling schedule.

8) Terminate the algorithm.

43

Recent Research in SA

• Alternative Cooling Schedules and Termination

criteria

• Adaptive Simulated Annealing (ASA) –

determines its own cooling schedule

• Hybridization with other Heuristic Search

Methods (GA, Tabu Search …)

• Multiobjective Optimization with SA

44

References• Cerny, V., "Thermodynamical Approach to the Traveling Salesman Problem: An Efficient

Simulation Algorithm", J. Opt. Theory Appl., 45, 1, 41-51, 1985 .

• de Weck O. “System Optimization with Simulated Annealing (SA), Memorandum SA.pdf

• Cohanim B. E., Hewitt J. N., and de Weck O.L., “The Design of Radio Telescope Array

Configurations using Multiobjective Optimization: Imaging Performance versus Cable

Length”, The Astrophysical Journal, Supplement Series, 154, 705-719, October 2004

• Jilla, C.D., and Miller, D.W., “Assessing the Performance of a Heuristic Simulated

Annealing Algorithm for the Design of Distributed Satellite Systems,” Acta Astronautica,

Vol. 48, No. 5-12, 2001, pp. 529-543.

• Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., “Optimization by Simulated

Annealing,” Science, Volume 220, Number 4598, 13 May 1983, pp. 671-680.

• Metropolis,N., A. Rosenbluth, M. Rosenbluth, A. Teller, E. Teller, "Equation of State

Calculations by Fast Computing Machines", J. Chem. Phys.,21, 6, 1087-1092, 1953.

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