Msdo 2015 Lecture 9 Sa

7/21/2019 Msdo 2015 Lecture 9 Sa

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IN THE NAME OF A

THE MOST BENEFTHE MOST MER



[email protected] ; 0321-9

_________________PhD, FLIGHT VEHICLE DESIGNBEIJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS, BUAA, P.R.CHINA, 2009

MS, FLIGHT VEHICLE DESIGNBEIJING UNIVERSITY OF AERONAUTICS AND ASTRONAUTICS, BUAA, P.R.CHINA, 2006

BE, MECHANICAL ENGINEERINGNATIONAL UNIVERSITY OF SCIENCE AND TECHNOLOGY, NUST, PAKISTAN, 2000

EMAIL: [email protected]

[email protected]

TEL: +92-320-9595510

WEB:

www.ist.edu.pk/qasim-zeeshan LINKEDIN: pk.linkedin.com/pub/qasim-zeeshan/67/554/ba7

Dr Qasim Zeeshan

http://pk.linkedin.com/pub/qasim-zeeshan/67/554/ba7

mailto:[email protected]


http://www.ist.edu.pk/qasim-zeeshan














MULTIDISCIPLI

SY

DOPTIMIZA

LECTURE # 9



STATUS

PHASE-I

Introduction to Multidisciplinary System Design Optimizatio

Terminology and Problem Statement

Introduction to Optimization

Classification of Optimization Problems

Numerical/ Classical Optimization

MSDO Architectures

Practical Applications: Structure, Aero etc



STATUS

PHASE-II WEEK 8: Genetic Algorithm

WEEK 9: Particle Swarm Optimization

WEEK 10: Simulated Annealing

WEEK 11: MID TERM

WEEK 12:

Ant Colony Optimization, Tabu Search, Pattern Search

WEEK 13:

LAB, Practical Applications

00

20

40

60

-0.5

0

0.5

1



STATUS

PHASE-III WEEK 14: Design of Experiments, Meta-modeling, and Ro

WEEK 15: Multi-objective Optimization

Hybrid Optimization & Hyper Heuristic Optimiz

WEEK 16: Post Optimality Analysis/ Revision & Discussion

WEEK 17: END TERM/ Paper Presentations ?



Previous Lecture : GA

HAVE U TRIED GA at home?

SOME QUESTIONS ?

Is GA a LOCAL SEARCH or a GLOBAL SEARCH algorithm? Or BOTH

What will happen if you change different parameters of GA?

POPULATION SIZE/ TYPE

SELECTION CROSSOVER

MUTATION

What will you do if you have a very short time and you want a reaso

Will GA give the same result if RUN on different Machines with same

Will GA give the same result every time you Run the algorithm with sa



M

________________SIMULATED ANNE

Dr. Qasim ZeeshanLECTURE # 9



SIMULATED ANNEA

_______________________GLOBAL OPT



Difficulty in Searching Global Op

Local search techniques, such as steepest

descend method, are very good infinding local optima.

However, difficulties arise when theglobal optima is different from the localoptima.

Since all the immediate neighboringpoints around a local optima is worse

than it in the performance value, localsearch can not proceed once trapped ina local optima point.

We need some mechanism that can helpus escape the trap of local optima.

And the simulated annealing is one ofsuch methods.

startingpoint

descenddirection

local minima

gl

ba



ANNEA



What is Annealing?

The name and inspiration come from annealing inmetallurgy, a technique involving heating andcontrolled cooling of a material to increase thesize of its crystals and reduce their defects.

The heat causes the atoms to become unstuck from

their initial positions (a local minimum of theinternal energy) and wander randomly throughstates of higher energy

The slow cooling gives them more chances offinding configurations with lower internal energy

than the initial one.



What is Annealing?



What is Annealing?

Annealing, in metallurgy and materials science, is a heat treatment wherein a mater

altered, causing changes in its properties such as strength and hardness. It is a procthat produces conditions by heating to above the recrystallization temperature,maintaining a suitable temperature, and then cooling.

Annealing is used to induce ductility, soften material, relieve internal stresses, refinestructure by making it homogeneous, and improve cold working properties.

In the cases of copper, steel, silver, and brass, this process is performed by substantheating the material (generally until glowing) for a while and allowing it to cool.

Unlike ferrous metals — which must be cooled slowly to anneal — copper, silver and bcan be cooled slowly in air or quickly by quenching in water. In this fashion the metasoftened and prepared for further work such as shaping, stamping, or forming



What is Annealing?

Full annealing is the process of slowly raising thetemperature about 50 ºC (90 ºF) above the Austenitictemperature line A3 or line ACM in the case ofHypoeutectoid steels (steels with < 0.77% Carbon)and 50 ºC (90 ºF) into the Austenite-Cementite regionin the case of Hypereutectoid steels (steels with >0.77% Carbon).

It is held at this temperature for sufficient time for all

the material to transform into Austenite or Austenite-Cementite as the case may be. It is then slowly cooledat the rate of about 20 ºC/hr (36 ºF/hr) in a furnaceto about 50 ºC (90 ºF) into the Ferrite-Cementiterange. At this point, it can be cooled in roomtemperature air with natural convection.

The grain structure has coarse Pearlite with ferrite orCementite (depending on whether hypo or hyper

eutectoid). The steel becomes soft and ductile.



What is Annealing?

The benefits of annealing are: Improved ductility

Removal of residual stresses that result from cold-workingor machining

Improved machinability

Grain refinement

Full annealing consists of

(1) recovery (stress-relief )

(2) recrystallization

(3) grain growth stages.

Annealing reduces the hardness, yield strength and

tensile strength of the steel.



Annealing of solids

Annealing: the physical process of heating up a solid and then cdown slowly until it crystallizes.

The atoms in the material have high energies at high temperatures and havfreedom to arrange themselves. As the temperature is reduced, the atomic decrease.

A crystal with regular structure is obtained at the state where the system haenergy.

If the cooling is carried out very quickly, which is known as rapid quenchingirregularities and defects are seen in the crystal structure.

The system does not reach the minimum energy state and ends in a polycrywhich has a higher energy.



The study of statistical mechanics shows that, at a given atom remaining at the state of r satisfies Boltzmann’s pr

distribution (Boltzmann’s law)

E (r) denotes the energy at state r，k＞0 is the Boltzmann’s cons

random variable representing energy, Z(T ) is the standardizatio

probability distribution

1 ( )( ( )) exp( )

( )

E r P E E r

Z T kT

( )( ) exp( )

s D

E s Z T

kT

Annealing of Solids



Given two energies E1 < E2 ，at the same temperature T :

The probability of atom remaining at low energy state is gprobability remaining at high energy state.

When temperature is very high, the probabilities of each sbasically the same, close to the average value of 1 / |D|number of state in the state space D).

The lower the temperature (T 0), the higher the probabilit

lower energy state.

0)()( 21 E E P E E P

Annealing of Solids



0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5

P r o b a b i l i t y

When temperature is higher (T = 20), the

difference between the probabilities remaining at

four energies is relatively small. However the

probability at the lowest energy state, x =1, is

0.269, which exceeds the average of 0.25. This

can be seen as the random move of atoms.

With temperature drops (T = 5), the probabilityat state x = 4 becomes relatively small.

At T = 0.5, the probability at state x = 1 is

0.865, while the probabilities at other three

states are very small.

Annealing of Solids

x=1

x=4

x=3

x=2

Analogy between



Analogy betweenOptimization and Annealing

Analogy the states of the solid represent feasible solutions of the optimization pro

the energies of the states correspond to the values of the objective funct

the minimum energy state corresponds to the optimal solution to the prob

rapid quenching can be viewed as local optimization

The optimal solution of an optimization problem can be

to the minimum energy state in an annealing process, i.e

with the greatest probability at the lowest temperature.



SIMULA

ANNEAL



What is Simulated Annealing?

Simulated annealing (SA) is a generic probabilistic metaheuristic fooptimization problem of locating a good approximation to the glogiven function in a large search space.

It is often used when the search space is discrete (e.g., all tours thaof cities).

For certain problems, simulated annealing may be more effective tenumeration — provided that the goal is merely to find an acceptasolution in a fixed amount of time, rather than the best possible solu



What is Simulated Annealing?

The name and inspiration come from annealing

in metallurgy, a technique involving heatingand controlled cooling of a material toincrease the size of its crystals and reduce theirdefects.

The heat causes the atoms to become unstuck

from their initial positions (a local minimum ofthe internal energy) and wander randomlythrough states of higher energy; the slowcooling gives them more chances of findingconfigurations with lower internal energy thanthe initial one.



What is Simulated Annealing?36

In annealing, a material is heated to high energy where there are frchanges (disordered). It is then gradually (slowly) cooled to a low ewhere state changes are rare (approximately thermodynamic equili

“frozen state”).

For example, large crystals can be grown by very slow cooling, butcooling or quenching is employed the crystal will contain a number imperfections (glass).



MINIMUM ENERGY CONFIGURATIO37



Simulated Annealing

Simulated Annealing (SA) is a generalization of a Monte Carlo methodthe equations of state and frozen states of n-body systems Metropolis

The concept is based on the manner in which liquids freeze or metals rthe process of annealing. In an annealing process a melt, initially at higand disordered, is slowly cooled so that the system at any time is apprthermodynamic equilibrium.

As cooling proceeds, the system becomes more ordered and approachground state at T=0. Hence the process can be thought of as an adiabto the lowest energy state.

If the initial temperature of the system is too low or cooling is done insslowly the system may become quenched forming defects or freezing ometastable states (ie. trapped in a local minimum energy state).



Simulated Annealing

The original Metropolis scheme was that an initial state of a thermosystem was chosen at energy E and temperature T, holding T constaconfiguration is perturbed and the change in energy dE is compute

If the change in energy is negative the new configuration is accepte

If the change in energy is positive it is accepted with a probability

Boltzmann factor exp -(dE/T).

This processes is then repeated sufficient times to give good samplithe current temperature, and then the temperature is decremented process repeated until a frozen state is achieved at T=0.



Simulated Annealing

The current state of the thermodynamic system is analogous to the current scombinatorial problem, the energy equation for the thermodynamic systemat the objective function, and ground state is analogous to the global minim

The major difficulty (art) in implementation of the algorithm is that there isanalogy for the temperature T with respect to a free parameter in the comproblem.

Furthermore, avoidance of entrainment in local minima (quenching) is depe

annealing schedule

choice of initial temperature

how many iterations are performed at each temperature

and how much the temperature is decremented at each step as cooling proceed



Simulated Annealing

SA is one of the most flexible techniques available for solving hard coproblems.

The main advantage of SA is that it can be applied to large problemsthe conditions of differentiability, continuity, and convexity that are noin conventional optimization methods.

The algorithm starts with an initial design. New designs are then rando

in the neighborhood of the current design.

The change of objective function value, ( ΔE), between the new and theis calculated as a measure of the energy change of the system.

At the end of the search, when the temperature is low the probability worse designs is very low. Thus, the search converges to an optimal sol



Simulated Annealing

The Metropolis algorithm generates a sequence of states of a solid as follows: givin

with energy Ei, the next state Sj is generated by a transition mechanism that consists perturbation with respect to the original state, obtained by moving one of the partiby the Monte Carlo method.

Let the energy of the resulting state, which also is found probabilistically, be Ej; if thless than or equal to zero, the new state Sj is accepted. Otherwise, in case the diffezero, the new state is accepted with probability

Where T is the temperature of the solid and kB is the Boltzmann constant. This accepknown as Metropolis criterion and the algorithm summarized above is the Metropolitemperature is assumed to have a rate of variation such that thermodynamic equilibthe current temperature level, before moving to the next level. This normally requirestate transitions of the Metropolis algorithm.

T k

E E

P B

ji

exp



Adaptive Simulated Annealing

Adaptive simulated annealing (ASA) is a variant of simulated annealing (SA) algo

algorithm parameters that control temperature schedule and random step selection adjusted according to algorithm progress.

This makes the algorithm more efficient and less sensitive to user defined parameter

These are in the standard variant often selected on the basis of experience and expoptimal values are problem dependent), which represents a significant deficiency in

The algorithm works by representing the parameters of the function to be optimized

numbers, and as dimensions of a hypercube (N dimensional space). Some SA algorithms apply Gaussian moves to the state, while others have distributio

temperature schedules.

Imagine the state as a point in a box and the moves as a rugby-ball shaped cloud a

The temperature and the step size are adjusted so that all of the search space is saresolution in the early stages, whilst the state is directed to favorable areas in the la

http://en.wikipedia.org/wiki/Simulated_annealing

http://en.wikipedia.org/wiki/Gaussian

http://en.wikipedia.org/wiki/Gaussian




SIMULATED ANNEA

_______________________VOCABU

TERMINO

Analogy between combinatorial



gyoptimization and annealing

Analogy the states of the solid represent feasible solutions of the optimizat

the energies of the states correspond to the values of the objectiv

the minimum energy state (Frozen State) corresponds to the optim

the problem

rapid quenching can be viewed as local optimization

The optimal solution of a combinatorial optimization problemanalogous to the minimum energy state in an annealing procestate with the greatest probability at the lowest temperature



TERMINOLOGY47

Simulated AnAnnealing Feasible solutionsSystem states

Cost (Objective)Energy

Neighboring solution (Change of state

Control parameterTemperature

Final solutionFrozen state



TERMINOLOGY

Annealing Schedule

The annealing schedule is the rate by which the tempera

decreased as the algorithm proceeds.

The slower the rate of decrease, the better the chances

finding an optimal solution, but the longer the run time.

TERMINOLOGY



TERMINOLOGY

Reannealing

Annealing is the technique of closely controlling the temp

cooling a material to ensure that it is brought to an optim

Reannealing raises the temperature after a certain num

points have been accepted, and starts the search againtemperature.

Reannealing avoids getting caught at local minima.



Basic principle of SA

According to the analogy between the annealing of solids andof solving combinatorial optimization problems, in 1983, Kirkp

al．proposed the SA algorithm based on Metropolis’s criterion

In 1953, Metropolis et al. presented the so-called Metropolis’s

their study of how to simulate the annealing process of solids. They produced an algorithm to provide efficient simulation of a collec

equilibrium at a given temperature.

This algorithm is also called Metropolis’s algorithm, which follows Boltz

B i i i l f SA



A solution and its objective function of a combinatorial optimizaproblem are equivalent to a state and its energy of a solid res

Set a control parameter t , whose value decreases in the coursealgorithm. Let t play the role of temperature T in a solid annea

For each value taken by t , the algorithm repeats an iterative p

“produce a new solution - judge - accept /reject”. This correspprocess tending to a thermal equilibrium of solids.

The action course of “produce a new solution - judge - accept /rejecttime of implementing Metropolis’s algorithm.


B i i i l f SA



At a given temperature, the iteration number of impMetropolis’s algorithm is called the length of Markov

SA starts from an initial solution and an initial tempe

Reduce the value of control parameter T gradually,

implementing Metropolis’s algorithm.

Finally when T tends to 0, the optimal solution of anoptimization problem can be obtained.


B i i i l f SA



Solid annealing must be "slowly" cooled down, so as

a thermal equilibriums at every temperature, and ult

tends to the minimum energy state.

Similarly, the control parameter t must also decreaseensure that SA eventually converge to the global op

solution of an optimization problem.





SA produces a sequence of solutions by using Metropolis’s algorithmcorresponding transfer probability Pt based on Metropolis’s criterio

Pt determines whether or not to transfer from the current solution i tsolution j.

At the beginning, t takes a greater value, after enough transfers, slowly redof t .

Repeat the process, until some stopping criterion is met.

1, ( ) ( )

( ) ( ) ( )exp( ) ( ) ( )

t

f j f i

P i j f i f j f j f i

t

i f

i f





The SA algorithm consists of a sequence of iterations

Each iteration consists of randomly changing the current solution to

solution in the neighborhood of the current solution.

Once a new solution is created, the corresponding change in the c

computed to decide whether the newly produced solution can be a

the current solution.

If the change in the cost function is negative, the newly produced s

directly taken as the current solution.

Otherwise, it is accepted according to Metropolis’s criterion based

probability.


METROPOLIS CRITERION



If the difference f ji=f (j)-f (i) between the cost function vacurrent and the newly produced solution is equal to or la

zero, a random number in [0, l] is generated from a un

distribution and if

then the newly produced solution j is accepted as the curr

If not, the current solution i keeps unchanged.

[ / ] ji k f t

e

METROPOLIS CRITERION

Metropolis’s criterion



- f ji

exp(- f ji /t k )

0

1

Low t k High t k

Metropolis’s criterion

FLOWCHART



Initial solution

Final solution

Evaluate solution

Update the current solution

Decrease temperature

Generate a

new solution

Accepted ?

Change

temperature ?

Terminate

the search ?

No

Yes

Yes

No

No

Yes

FLOWCHART



SIMULATED ANNEA

_______________________

PROCED

Procedure of a basic SA



Begin

Initialize (i 0, t0, L0);k: =0; i : =i 0;

Repeat

For l : =1 to Lk do

Begin

Generate ( j from N(i ));

If f ( j ) < f (i ) then i := j ;

ElseIf exp (-(f ( j )-f (i )/tk)>random [0,1] then i : = j ;

End;

k: =k+1;Calculate Length (Lk);

Calculate Control (tk);

Until stop criterion

End;

Procedure of a basic SAI

Fin

Evalu

Update

Decre

Generate a

new solution

A

tem

t

No

No




STEP 1

Generate an initial solution i 0; Set i :=i 0; k:=0; t0:=tmax (initial temperature).

STEP 2 If the stopping condition for internal iteration is met, then go to

STEP 3; Otherwise select a solution j from neighborhood N(i ) randomly,

compute f ji =f ( j )-f (i ); if f ji <0 set i := j ;

else if exp(- f ji /tk)> random(0, 1) set i := j repeat STEP 2.

STEP 3 Set tk+1:=g(tk); k:=k+1; If the stopping condition for external iteration is met, then

terminate the algorithm; otherwise return to STEP 2.


Initial s

Final solu

Evaluate so

Update the cur

Decrease te

Generate a

new solution

Accepted

Chang

temperatu

Termin

the sear

No

No




SA includes an internal cycle and an

external cycle The internal cycle is STEP 2. It indicates that, at

the same temperature t k, some solutions are

explored randomly.

The external cycle is STEP 3, including drops in

the temperature t k+1

: = g (t k), increasing in the

number of iterative steps k: = k +1, and

stopping conditions.


Initial s

Final solu

Evaluate so

Update the cur

Decrease te

Generate a

new solution

Accepted

Chang

temperatu

Termin

the sear

No

No




The internal cycle simulates the process of the

thermal equilibrium at a constant temperature. It is an important component that ensuring SA

can converge to the global optimal solution.

In each iteration of the internal cycle,

Metropolis’s algorithm is implemented once.

The iteration number in internal cycle is calledthe length of Markov chain, or the length of

time.


Initial s

Final solu

Evaluate so

Update the cur

Decrease te

Generate anew solution

Accepted

Changtemperatu

Termin

the sear

No

No




SA makes a tradeoff between

concentralization and diversification

strategies

For the next solution, if it is better than the

current solution, then select it as new current

solution. (concentralization strategy)

Otherwise, select it as new current solution with

a probability. (diversification strategy)


Initial s

Final solu

Evaluate so

Update the cur

Decrease te

Generate a

new solution

Accepted

Chang

temperatu

Termin

the sear

No

No



SIMULATED ANNEA

_______________________

BASIC ELEM

Basic elements of SA



In order to implement the SA algorithm for a problem, fichoices must be made:

Representation of solutions

Definition of the cost function

Definition of the generation mechanism for the neighbors

Setting stopping criteria

Designing a cooling schedule

Basic elements of SA



SIMULATED ANNEA _______________________

BASIC ELEMREPRESENTATION OF SOLU




The representation of solutions for SA is similar to

including binary coding, real number coding, and

others.

For complex optimization problems, the real numbis usually used.




SIMULATED ANNEA _______________________

BASIC ELEMDEFINITION OF COST FUN




The cost function can be defined like the fitness definGAs.

Or just take the objective function f ( x) (or – f ( x) ) as t

function SAs do not require the cost function must be non-neg


SIMULATED ANNEA



SIMULATED ANNEA

_______________________BASIC ELEM

GENERATION MECHANISNEIGH

Generation mechanism for neigh



Generation mechanism for neigh

Various generation mechanisms could be developedcould be borrowed from GAs, for example, mu

inversion.

A neighbor can be generated by changing the value of

variable or several decision variables with a small magnitu

For instance, X =(2.17,3.45, 6.32, 9.88)

Neighbors X' =(2.17, 3.35, 6.32,9.88)

X' =(2.17, 3.35, 6.42,9.88)



SIMULATED ANNEA _______________________

BASIC ELEMCOOLING SCH

Cooling schedule



g

In designing the cooling schedule for a SA algorithmparameters must be specified:

an initial temperature, t 0

a temperature update rule, g(t k)

a final temperature, t f

the number of iterations to be performed at each temperat

the length of Markov chain, Lk



HISTORICAL PERSPECTIVE



79

After the World War II he returned to the faculty of the Universityof Chicago as an Assistant Professor. He came back to Los Alamos in

1948 to lead the group in the Theoretical (T) Division that designedand built the MANIAC Icomputer in 1952 and MANIAC II in 1957.

He chose the name MANIAC in the hope of stopping the rash ofsuch acronyms for machine names, but may have, instead, onlyfurther stimulated such use)

The MANIAC ( Mathematical Analyzer, Numerical I ntegrator, and Computer or Mathematical Analyzer, Numerator, I ntegrator, and Computer)was an early computer built under the direction of Nicholas

Metropolis at the Los Alamos Scientific Laboratory. It was based onthe von Neumann architecture of the IAS, developed by John vonNeumann. As with all computers of its era, it was a one of a kindmachine that could not exchange programs with other computers(even other IAS machines)

From 1957 to 1965 he was Professor of Physics at the University ofChicago and was the founding Director of its Institute for ComputerResearch. In 1965 he returned to Los Alamos where he was made aLaboratory Senior Fellow in 1980


http://en.wikipedia.org/wiki/World_War_II

http://en.wikipedia.org/wiki/MANIAC_I

http://en.wikipedia.org/wiki/MANIAC_II

http://en.wikipedia.org/wiki/Nicholas_Metropolis


http://en.wikipedia.org/wiki/Los_Alamos_National_Laboratory

http://en.wikipedia.org/wiki/Von_Neumann_architecture

http://en.wikipedia.org/wiki/IAS_machine

http://en.wikipedia.org/wiki/John_von_Neumann


http://en.wikipedia.org/w/index.php?title=Institute_for_Computer_Research&action=edit&redlink=1






http://en.wikipedia.org/wiki/IAS_machine

http://en.wikipedia.org/wiki/Von_Neumann_architecture

http://en.wikipedia.org/wiki/Los_Alamos_National_Laboratory



http://en.wikipedia.org/wiki/MANIAC_II



http://en.wikipedia.org/wiki/World_War_II



METROPOLIS (1915-1999)

In the 1950s, a group of researchers led by Metropolis developedthe Monte Carlo method. Generally speaking, the Monte Carlomethod is a statistical approach to solve deterministic many-bodyproblems. In 1953 Metropolis co-authored the first paper on atechnique that was central to the method now known as simulatedannealing.

This landmark paper showed the first numerical simulations ofa liquid . Although credit for this innovation has historically beengiven to Metropolis, the entire theoretical development in factcame from Marshall Rosenbluth, who later went on to distinguishhimself as the most dominant figure in plasma physics during thelatter half of the 20th century.


http://en.wikipedia.org/wiki/Monte_Carlo_method



http://en.wikipedia.org/wiki/Marshall_Rosenbluth

http://en.wikipedia.org/wiki/Marshall_Rosenbluth



http://en.wikipedia.org/wiki/Monte_Carlo_method




The algorithm — the Metropolis algorithm or Metropolis-Hastingsalgorithm -- for generating samples from the Boltzmanndistribution was later generalized by W.K. Hastings.

He is credited as part of the team that came up with the nameMonte Carlo method in reference to a colleague's relative's lovefor the Casinos of Monte Carlo.

Monte Carlo methods are a class of computational algorithms thatrely on repeated random sampling to compute their results.


http://en.wikipedia.org/wiki/Metropolis-Hastings_algorithm


http://en.wikipedia.org/wiki/Boltzmann_distribution


http://en.wikipedia.org/wiki/Monte_Carlo_methods

http://en.wikipedia.org/wiki/Monte_Carlo_methods










In statistical mechanics applications prior to the introduction of theMetropolis algorithm, the method consisted of generating a largenumber of random configurations of the system, computing theproperties of interest (such as energy or density) for each configuration,and then producing a weighted average where the weight of eachconfiguration is its Boltzmann factor, e − E / kT , where E is the energy, T isthe temperature, and k is the Boltzmann constant

The key contribution of the Metropolis paper was the idea that

Instead of choosing configurations randomly, then weighting them with

exp( −E/kT), we choose configurations with a probability exp( −E/kT) and

weight them evenly.

– Metropolis et al .

Historical Perspective

http://en.wikipedia.org/wiki/Weighted_average

http://en.wikipedia.org/wiki/Boltzmann_factor

http://en.wikipedia.org/wiki/Energy

http://en.wikipedia.org/wiki/Temperature

http://en.wikipedia.org/wiki/Boltzmann_constant

http://en.wikipedia.org/wiki/Boltzmann_constant



http://en.wikipedia.org/wiki/Boltzmann_factor

http://en.wikipedia.org/wiki/Weighted_average



The Boltzmann constant (k or kB) is the physicalconstant relating energy at the individual particle level

with temperature observed at the collective or bulk level.It is the gas constant R divided by the Avogadroconstant NA:

It has the same units as entropy. It is named afterthe Austrian physicist Ludwig Boltzmann.

Ludwig Eduard Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist famous forhis founding contributions in the fields of statisticalmechanics and statistical thermodynamics. He was one ofthe most important advocates for atomic theory at a timewhen that scientific model was still highly controversial.

BOLTZMAN CONSTANT



http://en.wikipedia.org/wiki/Gas_constant

http://en.wikipedia.org/wiki/Avogadro_constant




http://en.wikipedia.org/wiki/Gas_constant





The logarithmic connection between entropy and probability was firststated by L. Boltzmann in his kinetic theory of gases. This famousformula for entropy S

where k = 1.3806505(24) × 10−23 J K−1 is Boltzmann's constant,and the logarithm is taken to the natural base e. W is theWahrscheinlichkeit, the frequency of occurrence of a macrostate

On September 5, 1906, while on a summer vacation in Duino, nearTrieste, Boltzmann hanged himself during an attack of depression. Heis buried in the Viennese Zentralfriedhof; his tombstone bears theinscription



Simulated Annealing in Practi



method proposed in 1983 by IBM researchers for solving VLSI layout prob(Kirkpatrick et al, Science, 220:671-680, 1983).

Very-large-scale integration (VLSI) is the process of creating integrated circuits bythousands of transistors into a single chip. VLSI began in the 1970s when complexand communication technologies were being developed. The microprocessor is a V

theoretically will always find the global optimum (the best solution)

useful for some problems, but can be very slow slowness comes about because T must be decreased very gradually to retain optimality

In practice how do we decide the rate at which to decrease T? (this is a practical method)



SIMULATED ANNEA

_______________________

MONTE CA

MONTE CARLO



Monte Carlo methods (or Monte Carlo experiments) are a classof computational algorithms that rely on repeated random sampling to

compute their results. Monte Carlo methods are often usedin simulating physical and mathematical systems. These methods are mostsuited to calculation by a computer and tend to be used when it isinfeasible to compute an exact result with a deterministic algorithm.Thismethod is also used to complement the theoretical derivations.

Monte Carlo methods are especially useful for simulating systems withmany coupled degrees of freedom, such as fluids, disordered materials,strongly coupled solids, and cellular structures (see cellular Potts model).

They are used to model phenomena with significant uncertainty in inputs,such as the calculation of risk in business.

They are widely used in mathematics, for example to evaluatemultidimensional definite integrals with complicated boundary conditions.When Monte Carlo simulations have been applied in space explorationand oil exploration, their predictions of failures, cost overruns and scheduleoverruns are routinely better than human intuition or alternative "soft"methods.

MONTE CARLO



The Monte Carlo method was coined in the 1940s by John vonNeumann and Stanislaw Ulam, while they were working on nuclear weapon

projects in the Los Alamos National Laboratory. It was named in homageto Monte Carlo casino, a famous casino, where Ulam's uncle would oftengamble away his money

John von Neumann (December 28, 1903 – February 8, 1957) as aHungarian-American mathematician who made major contributions to a vastrange of fields, including set theory, functional analysis, quantum mechanics,ergodic theory, continuous geometry, economics and game theory, computerscience, numerical analysis, hydrodynamics, and statistics, as well as many

other mathematical fields. He is generally regarded as one of the greatestmathematicians in modern history

Stanisław Marcin Ulam (April 13, 1909 – May 13, 1984) was arenowned American mathematician of Polish-Jewish origin, whoparticipated in the Manhattan Project and originated the Teller – Ulamdesign of thermonuclear weapons. He also proposed the idea of nuclearpulse propulsion and developed a number of mathematical methods innumber theory, set theory, ergodic theory and algebraic topology



SIMULATED ANNEA

_______________________METROPOLIS –HASTINGS ALGO

Metropolis –Hastings algorithm



Marshall Nicholas Rosenbluth (5 February, 1927 – 28 September,2003) was an American plasma physicist and member of the

National Academy of Sciences. In 1997 he was awarded the National Medal of Science for

discoveries in controlled thermonuclear fusion, contributions toplasma physics and work in computational statistical mechanics. Hewas also a recipient of the E.O. Lawrence Prize (1964), the AlbertEinstein Award (1967), the James Clerk Maxwell Prize in Plasma

Physics (1976), and the Enrico Fermi Award (1985). In 1953, Rosenbluth derived the Metropolis algorithm; cited in

Computing in Science and Engineering (Jan. 2000) as being amongthe top 10 algorithms having the "greatest influence on thedevelopment and practice of science and engineering in the 20thcentury."

Metropolis –Hastings algorithm



In mathematics and physics, the Metropolis –

Hastings algorithm is a Markov chain MonteCarlo method for obtaining a sequence

of random samples from a probability

distribution for which direct sampling is

difficult.

This sequence can be used to approximate the

distribution (i.e., to generate a histogram), or

to compute an integral (such as an expected

value).



SIMULATED ANNEA

______________________MARKOV CHAIN MONTE C

Markov Chain Monte Carlo



Markov chain Monte Carlo (MCMC) methods (which include random walmethods) are a class of algorithms for sampling from probability distribuon constructing a Markov chain that has the desired distribution as its eqdistribution.

The state of the chain after a large number of steps is then used as a sadesired distribution. The quality of the sample improves as a function of of steps.

Usually it is not hard to construct a Markov chain with the desired propemore difficult problem is to determine how many steps are needed to costationary distribution within an acceptable error.

A good chain will have rapid mixing — the stationary distribution is reachstarting from an arbitrary positio

Markov Chain Monte Carlo



Many Markov Chain Monte Carlo methods move around the equilibrium distributsmall steps, with no tendency for the steps to proceed in the same direction. Thes

easy to implement and analyse, but unfortunately it can take a long time for the explore all of the space. The walker will often double back and cover ground alHere are some random walk MCMC methods: Metropolis –Hastings algorithm: Generates a random walk using a proposal density a

rejecting proposed moves. Gibbs sampling: Requires that all the conditional distributions of the target distribution

exactly. Popular partly because when this is so, the method does not require any 'tuning Slice sampling: Depends on the principle that one can sample from a distribution by sa

from the region under the plot of its density function. This method alternates uniform samvertical direction with uniform sampling from the horizontal 'slice' defined by the curren

Multiple-try Metropolis: A variation of the Metropolis – Hastings algorithm that allows meach point. This allows the algorithm to generally take larger steps at each iteration, wproblems intrinsic to large dimensional problems.



SIMULATED ANNEA _______________________

METROPOLIS CRITER

METROPOLIS CRITERION98



new c

b

E E

c T P e

If a system is in some current energy state Ecurr and some aspects changed to makepotentially achive a new energy state Enew , than the :

system goes to

if Ecurr < Enew New state with pr

else New state with probability

State Space : Ecurr and Enew can each take one of N values x1 , x2 ,...,, xN and trfrom one state to any others is possible

Spall, Introduction to Stochastic Search and Optimization: Estimation, Simulation a

Wiley & Sons, 2003

cb : Boltzman Constant (k)

T : Temparature

METROPOLIS CRITERION99



Metropolis’s paper describes a Monte Carlo simulation of interacting

Each state change is described by: If E decreases, P = 1, where E is energy

If E increases,

where T = temperature (Kelvin), k = Boltzmann constant, k > 0, energy ch

To summarize behavior:

For large T, P

For small T, P

)(kT

E

e P

01

e

11

0

e

E



SIMULATED ANNEA

_______________________

MARKOV CH

MARKOV CHAINS



Markov Chain:

Sequence of trials where the outcome of each trial depends onoutcome of the previous one

Markov Chain is a set of conditional probabilities:

Pij (k-1,k)

Probability that the outcome of the k-th trial is j, when trial k-1

Markov Chain is homogeneous when the probabilities do not d

MARKOV CHAINS



The result of three Markov chains running on

the 3D Rosenbrock function using theMetropolis-Hastings algorithm.

The algorithm samples from regions where

the posterior probability is high and the

chains begin to mix in these regions.

The approximate position of the maximum hasbeen illuminated.

Note that the red points are the ones that

remain after the burn-in process. The earlier

ones have been discarded.



SIMULATED ANNEA

_______________________

ALGOR

SA: ALGORITHM



The inspiration for simulated annealing is the ancient

process of forging iron. Instead of optimizing profits, itoptimized the metal’s hardness.

The blacksmith’s hammer guided the iron into thedesired shape and density while the heat made the ironmore malleable and responsive to the hammer.

Essentially hill climbing is equivalent to a blacksmithhammering without heat.

Annealing is the word for heating a metal and thencooling it slowly.

SA: ALGORITHM



Simulated Annealing improves on hill climbing. First, here is sombackground to help with the intuition



HILL CLIM

_______________________

ALGOR

Hill-climbing search



function HILL-CLIMBING( problem) return a state that is a local maximum

input: problem, a problem

local variables: current , a node.

neighbor, a node.

current MAKE-NODE(INITIAL-STATE[problem])

loop do neighbor a highest valued successor of current

if VALUE [neighbor] ≤ VALUE[current ] then return STATE[current ]

current neighbor

Hill-climbing search



“a loop that continuously moves in the direction of increasing value”

terminates when a peak is reached

Aka greedy local search

Value can be either

Objective function value

Heuristic function value (minimized)

Hill climbing does not look ahead of the immediate neighbors of the curren

Can randomly choose among the set of best successors, if multiple have the

Characterized as “trying to find the top of Mount Everest while in a thick fo



Other drawbacks



Ridge = sequence of local maxima difficult for greedy algorithms to navigate

Plateau = an area of the state space where the evaluation function is flat.

SA: ALGORITHM



Simulated annealing will

outperform hill climbing whenthe local maximum is near theglobal maximum.

In this case one of the jumps mayget far enough from the local

max to reach the ascendingslope of the global max:

Search using Simulated Annea



Simulated Annealing = hill-climbing with non-deterministic searchBasic ideas:

like hill-climbing identify the quality of the local improvements

instead of picking the best move, pick one randomly

say the change in objective function is

if is positive, then move to that state

otherwise:

move to this state with probability proportional to thus: worse moves (very large negative ) are executed less often

however, there is always a chance of escaping from local maxima

over time, make it less likely to accept locally bad moves

(Can also make the size of the move random as well, i.e., allow “large” ste

Physical Interpretation of Simulated An



A Physical Analogy:

imagine letting a ball roll downhill on the function surface

this is like hill-climbing (for minimization)

now imagine shaking the surface, while the ball rolls, gradually reduamount of shaking this is like simulated annealing

Annealing = physical process of cooling a liquid or metal until part

a certain frozen crystal state simulated annealing:

free variables are like particles

seek “low energy” (high quality) configuration

get this by slowly reducing temperature T, which particles move around



SIMULATED ANNEA _______________________

ALGOR

The SA Algorithm: General Outline



1. generate a neighboured solution / state

2. probabilistically accept the solution / state

3. probability of acceptance depends on the objective function function) difference and an additional parameter called temp

The SA Algorithm: General Out



Solution generation

typically returns a random, neighbored solution

Acceptance criterion

Metropolis acceptance criterion

better solutions are always accepted

worse solutions are accepted with probability

Annealing

Parameter T , called temperature, is slowly decreased

The SA Algorithm: General Out



SA ALGORITHM



Step SA Operation

Step 1: At each iteration, an “atom” is randomly displaced a small amount.Step 2: The energy is calculated for each atom and the difference with the en

location is calculated.

Step 3: The Boltzmann probability factor is calculated

Step 4: If E 0 then the new location is accepted

Step 5: Otherwise, a random number is generated between 0 and 1:Step 6: If the random number is greater than the calculated P, then the higher ener

in the hope that the new location may eventually lead to a better locatilocation.

Step 7: Otherwise, the old atomic location is retained and the algorithm generates a

function SIMULATED ANNEALING( problem schedule) return a solution state

SA ALGORITHM: PSEUODOCO



function SIMULATED-ANNEALING( problem, schedule) return a solution stateinput: problem, a problem

schedule, a mapping from time to temperaturelocal variables: current , a node. next , a node. T , a “temperature” controlling the probability of downward

current MAKE-NODE(INITIAL-STATE[problem])for t 1 to ∞ do

T schedule[t ]

if T = 0 then return currentnext a randomly selected successor of current

∆E VALUE[next ] - VALUE[current ] if ∆E > 0 then current next

else current next only with probability e∆E /T



Initialisation

SA FLOWCHART



Metropolis simulation with fixed

temperature T

Adjust the solution

Evaluate cost function

Improvement

Accept new

solution

Accept new solution

with a probability

Check for equilibrium

Stop criteria at outer loop

Return optimal solution

Cooling

temperature T

Generate new solution

NoYes

Yes

No

Yes

No

SA ALGORITHM



SA ALGORITHM



SA FLOWCHART



SA ALGORITHMThe following is an outline of the steps performed for both the simulated annealing and thresho



algorithms:

1. The algorithm begins by randomly generating a new point. The distance of the new point from textent of the search, is determined by a probability distribution with a scale proportional to the

2. The algorithm determines whether the new point is better or worse than the current point. If the the current point, it becomes the next point. If the new point is worse than the current point, the ait the next point. Simulated annealing accepts a worse point based on an acceptance probabiliaccepts a worse point if the objective function is raised by less than a fixed threshold.

3. The algorithm systematically lowers the temperature and (for threshold acceptance) the thresho

found so far.4. Reannealing is performed after a certain number of points (ReannealInterval) are accepted by

raises the temperature in each dimension, depending on sensitivity information. The search is restemperature values.

5. The algorithm stops when the average change in the objective function is very small, or when anare met.

SIMULATED ANNEA



_______________________

ALGOR

PSEUODOC

SA ALGORITHM: PSEUODOC

PROCEDURE simulated annealing()



InputInstance();Generate randomly an initial solution;

initialize ;DODO thermal equilibrium not reachedGenerate a neighbour state randomly;evaluate ;update current stateIF with new state;

IF with new statewith probability ;OD;Decrease using annealing schedule;OD;RETURN(solution with the lowest energy)

END simulated annealing; A pseudo-code for a simulated annealing procedure

SA ALGORITHM



The following pseudocode implements the simulated annealing heu

described above. It starts from a state s0 and continues to either a maximum of kma

until a state with an energy of emax or less is found.

In the process, the call neighbour(s) should generate a randomly cneighbour of a given state s; the call random() should return a ranthe range [0,1].

The annealing schedule is defined by the call temp(r), which shouldtemperature to use, given the fraction r of the time budget that haexpended so far.

SA ALGORITHM: PSEUODOCOD

s ← s0; e ← E(s) // Initial state, energy.

http://en.wikipedia.org/wiki/Pseudocode

http://en.wikipedia.org/wiki/Pseudocode



sbest ← s; ebest ← e // Initial "best" solution

k ← 0 // Energy evaluation countwhile k < kmax and e < emax // While time left & not g

snew ← neighbour(s) // Pick some neighbour.

enew ← E(snew) // Compute its ene

if P(e, enew, temp(k/kmax)) > random() then // Should we move to it? s ← snew; e ← enew // Yes, change state.

if enew > ebest then // Is this a new best? sbest ← snew; ebest ← enew // Save 'new neighbour' to 'best f

k ← k + 1 // One more evaluation done

return sbest // Return the best solution

SIMULATED ANNEA



_______________________

ALGOR

ACCEPTANCE PROBAB

The SA Algorithm131

Acceptance Probability



Acceptance Probability

T

E

e P

T

E P

1

T

E

P

By Johnson et al.

By Brandimarte et al., need to decide waccept moves for the change of energy

SIMULATED ANNEA



_______________________

ALGOR

STOPPING CRIT

Stopping Criteria133



Stopping Criteria total moves attempted

no improvement over n attempts

no accepted moves over m attempts

minimum temperature

Stopping criteria



The lowest temperature is reached The maximum iteration number is reached (external cycle

The improvement on cost function is not significant in succe

several iterations

Mixed criterion



SIMULATED ANNEA _______________________



DECISION MAK

Decisions to make in SA

Generic decisions



Generic decisions

Initial temperature, t 0

Cooling schedule, a and no of iterations

Stopping condition

Problem specific decisions

Solution space, S Cost function, f

Starting solution, s0

Neighborhood structure, N(s)

SIMULATED ANNEA



_______________________

DECISION MAKGENERAL DEC

SIMULATED ANNEA



_______________________


INITIAL TEMPER

Generic decisions:Initial Temperature



It must be hot enough to allow free exchange of neisolutions and to make the final solution independentstarting solution

Choose it large enough to heat the system rapidly uproportion of accepted moves to rejected moves re

prespecified value Then start the cooling



Generic decisions:Cooling schedule

Th h ld l i i di ib



The system should get close to its stationary distribucurrent temperature before temperature reduction

Temperature (and probability of accepting non-impmoves) should converge to zero

For this, make

(1) a large number of iterations at few temperatures (2) a small number of iterations at many temperature

Generic decisions:Cooling schedule (cont.)

A li h d l f (1) i t i a(t)= t h



A common cooling schedule for (1) is geometric, a (t )=at whe

0.8-0.99 suggested for a, implying slow cooling Increase no. of iterations as temperature decreases to better

local optimum towards the end no of iterations may be dynamic, depending on the number o

acceptances, e.g. iterate until 10 acceptances occur (no of itenaturally increase as t decreases)

no of iterations should not be much smaller than the neighbor

Generic decisions:Cooling schedule (cont.)144

A li h d l f (2) i (t) t/(1+



A common cooling schedule for (2) is a (t )=t /(1+ where b is small, implying very slow cooling

Only one iteration is made at each temperature

Other cooling schedules are proposed but not a

used as the above two

Generic decisions:Cooling schedule (cont.)145

In general:



In general:

Rate of cooling is more important than manner o(geometric or otherwise)

Try to keep rate of acceptance high at the begi

(exploration of the search space), reduce it towend (exploitation of the current solution’s neighb

SIMULATED ANNEA



_______________________


STOPPING COND

Generic decisions:Stopping condition147

As t approaches zero, probability of accepting non-



As t approaches zero, probability of accepting non

moves becomes virtually zero We may stop before that, e.g. after a certain numb

iterations or when

This is to produce a solution within e of optimum withprobability q

]/)1|ln[(| q

e

S t



Problem specific decisions149

Decision criteria



Decision criteria:

Validity of the algorithm should be maintaine

Computation time should be used effectively

Final solution should be close to the optimal s

SIMULATED ANNEA



_______________________

DECISION MAKPROBLEM SP

NEIGHBOURHOOD STRU

Problem specific decisions:Neighborhood structure

151

Every solution should be reachable from every othe



y y

convergence and validity) Neighborhood should be small enough to be search

adequately in fewer iterations, but large enough todrastic cost improvements in a single move (tradeoffbetween computation time and solution quality)

Random generation of a neighboring solution shouldfor effective use of computation time (since this mustat every iteration)

Problem specific decisions:Neighborhood structure

152

Choose the neighborhood (and the cost function) suc



g ( )

difference between the cost for s0 and s can be comfast (since this must be done at every iteration)

Avoid spiky neighborhood topography over the soluspace, as well as deep troughs, to reduce the numbeiterations

Also, avoid large plateau-like areas where the costchange

SIMULATED ANNEA



_______________________

DECISION MAKPROBLEM SP

SOLUTION

Problem specific decisions:Solution space

154

Infeasible solutions may be allowed and can be



y

penalized in the objective function (or repaired) To reduce the number of iterations, no of iterations,

the solution space as small as possible by choosingappropriate moves (may conflict with allowing infeasolutions)

If possible, apply a series of reductions to the problwithout destroying reachability



SIMULATED ANNEA _______________________

FINE TUN

Fine tuning a SA algorithm156

Experiment for best parameter settings (replicat



p p g ( p

are needed) Monitor evolution of the SA algorithm, graphica

possible, for different parameter settings

Monitor temperature, cost, ratio of accepted morejected moves and other statistics (plot these agthe iteration number)



SIMULATED ANNEA

_______________________ENHANCEMENTS AND MODIFICA

Enhancements and modificatio158

Functions other than exp(- /t ) can be used to accept nonimproving moves for speed or solution quality e g 1 /



improving moves for speed or solution quality, e.g. 1- /discrete approximation

Different cooling schedules can be used in different phauseful work is done in the middle of the schedule)

Using a constant temperature may be an option Reheating may be tried if no progress is observed Cooling may take place every time a move (or so many

accepted (no of iterations is dynamic)

Enhancements and modifications

(cont.)159

Neighborhood structure may be adjusted (typically rest



as the temperature decreases Neighborhood size may be changed

Sampling from the neighborhood can be cyclic rather thrandom

An approximate cost function can be used to reduce the

computation time Solution space may be restricted

Control of Annealing Process

Suppose that a step in the search direction produce a differen



performance value. The acceptance criterion, which is often reMetropolis Criterion, is as follows:

If it is a favorable direction, say 0, we always accept it.

Otherwise, this step is accepted with a probability exp(-/T), wparameter.

Control of Annealing Process161

Acceptance of a search step (Metropolis Criterion



Assume the performance change in the search direis E.

Accept a ascending step only if it pass a random te

Always accept a descending step, i.e. E≤0

1,0exp randomT E




Cooling Schedule:



At each temperature, search is allowed to procee

a certain number of steps, L(k ).

T, the annealing temperature, is the parameter tcontrol the frequency of acceptance of ascending

We gradually reduce temperature T (k ).

The choice of parametersis called the cooling schedule.

k Lk T ,


Notation:



T 0 starting (inital) temperatureT F final temperature (T0 > TF, T0,TF ≥ 0)

T t temperature at state t

m number of states

n(t) number of moves at state t

(total number of moves = n * m)

move operator


x0 initial solution



xi solution i xF final solution

f( xi) objective function value of xi

cooling parametera

SA : PROCESS166

Procedure (Minimization)

Select x0, T0, m, n, α

Set x1= x0, T1= T0, xF= x0



for (t = 1,…,m) { for (i = 1,…n) {

xTEMP = σ(xi)

if f(xTEMP) ≤ f(xi), xi+1 = xTEMP

else

if , xi+1 = xTEMP

else xi+1 = xi

if f(xi+1) ≤ f(xF), xF = xi+1 }

Tt+1= α Tt }

return xF

t

iTEMP

T

f f

eU

)x()x(

)1,0(



CONCLUD

REMA



SIMULATED ANNEA _______________________

PERFORMA



Implementation of Simulated

Annealing

Finally, we like to emphasize the interpretation of the algorithm



Simulated annealing is a stochastic algorithm. Because random variables are used in the algorithm, the outco

different trials may vary even for the exact same choice of cooschedule.

Moreover, the convergence to the global optima of simulated a

only achieved when algorithm proceeds to infinite number of it

Implementation of Simulated

Annealing

Understand the result:



• This is a stochastic algorithm. The outcomay be different at different trials.

• Convergence to global optima can onlybe realized in asymptotic sense.

SIMULATED ANNEALING At each iteration of the simulated annealing algorithm, a new point is rand

The distance of the new point from the current point, or the extent of the se



on a probability distribution with a scale proportional to the temperature. The algorithm accepts all new points that lower the objective, but also, with

probability, points that raise the objective.

By accepting points that raise the objective, the algorithm avoids being traminima, and is able to explore globally for more possible solutions.

An annealing schedule is selected to systematically decrease the temperatalgorithm proceeds. As the temperature decreases, the algorithm reduces tsearch to converge to a minimum.

SIMULATED ANNEALING Threshold acceptance uses a similar approach, but instead of accepting ne

raise the objective with a certain probability, it accepts all new points belothreshold.



The threshold is then systematically lowered, just as the temperature is loweannealing schedule.

Because threshold acceptance avoids the probabilistic acceptance calculatannealing, it may locate an optimizer faster than simulated annealing.

SIMULATED ANNEA

_______________________



ADVANT

DISADVANT

Advantages and disadvantag175

Advantages:

SA is historically important



easy to implement

convergence proofs: theoretically interesting, but practical relevance v

good performance often at the cost of substantial run-time

Issues:

Performance

Solution is only as good as the evaluation function Termination Criteria

Whys is SA a META-HEURISTIC SA is not a single algorithm but rather a heuristic strategy

There are several decisions to make and several parameters to set (both gproblem specific) to implement SA for a particular problem



p p ) p p p

The same is true for TS, GA and the others as well (main difficulty of devemetaheuristic algorithm)

SIMULATED ANNEA



_______________________

MA

MATLAB SYNTAX

[x fval] = simulannealbnd(@objfun,x0,lb,ub,options)

Where



@objfun is a function handle to the objective function.

x0 is an initial guess for the optimizer.

lb and ub are lower and upper bound constraints, respectively, on x.

options is a structure containing options for the algorithm. If you doargument, simulannealbnd uses its default options.

The results are given by: x — Final point returned by the solver

fval — Value of the objective function at x

To open the Optimization Tool, enter optimtool('simulannealbnd')

MATLABTEMPERATURE

The temperature is the control parameter in simulated a



is decreased gradually as the algorithm proceeds.

It determines the probability of accepting a worse solut

step and is used to limit the extent of the search in a giv

You can specify the initial temperature as an integer in tInitialTemperature option, and the annealing schedule a

the TemperatureFcn option

MATLAB Annealing Schedule

The annealing schedule is the rate by which the tempera



decreased as the algorithm proceeds. The slower the rate of decrease, the better the chances

finding an optimal solution, but the longer the run time.

You can specify the temperature schedule as a function

the TemperatureFcn option.

MATLABReannealing

Annealing is the technique of closely controlling the temp



cooling a material to ensure that it is brought to an optim Reannealing raises the temperature after a certain num

points have been accepted, and starts the search againtemperature.

Reannealing avoids getting caught at local minima. You specify the reannealing schedule with the Reanneal

MATLAB

options = saoptimset

options = saoptimset('param1',value1,'param2',value2,...)

' '



options = saoptimset(oldopts,'param1',value1,...) options = saoptimset(oldopts,newopts)

options = saoptimset(optimfunction)

MATLAB Temperature Options: Temperature options specify how the temperature will be

iteration over the course of the algorithm.



InitialTemperature — Initial temperature at the start of the algorithm. The defa

TemperatureFcn — Function used to update the temperature schedule. Let i de

number. The options are:

@temperatureexp — The temperature is equal to InitialTemperature * 0.95^i. Thi

@temperaturefast — The temperature is equal to InitialTemperature / i.

@temperatureboltz — The temperature is equal to InitialTemperature / ln(i). @myfun — Uses a custom function, myfun, to update temperature. See the functions

ReannealInterval — Number of points accepted before reannealing. The defa

MATLAB

Algorithm Settings: Algorithm settings define algorithmic specific pa

generating new points at each iteration.

f f



Parameters that can be specified for the simulated annealing and tacceptance algorithms are:

AnnealingFcn — Function used to generate new points for the next

choices are: @annealingfast — The step has length temperature, with direction uniformly at rand

@annealingboltz — The step has length square root of temperature, with direction u

@myfun — Uses a custom annealing algorithm, myfun. The syntax is:

newx = myfun(optimValues,problem)



SIMULATED ANNEA



_______________________

APPLICAT

Applications of SA 187

Aerospace Vehicle Design

Graph partitioning & graph coloring

T li l bl



Traveling salesman problem

Quadratic assignment problem

VLSI and computer design

Scheduling

Image processing

Layout design

A lot more…Find one!...



REFEREN

REFERENCE BOOKS



QU



ASSIGNM

ASSIGNMENT Study any paper on implementation of SQP, GA,

Identify Problem Statement

Identify Problem Formulation

Id tif D i V i bl



Identify Design Variables

Identify Design Objectives

Identify Design Constraints

Identify Control Parameters

Explain Results

Solve any CONSTRAINED optimization problem ofusing MATLAB

USE AIAA Paper format please

SUBMISSION DEADLINE:



THANK YOU FOR YOUR INT

Msdo 2015 Lecture 9 Sa

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