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MAIN PROBLEM -> OPTIMIZATION

Local

Global

Optimization search

techniques

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TABU SEARCH , GREEDY

APPROACH , ETC

SIMMULATED ANNEALING,PARTICLE SWARM OPTIMIZATION

(PSO),GRADIENT DESCENT ETC

Difficulty in Searching Global Optima3

starting

point

descend

direction

local minima

global minima

barrier to local search

Background: Annealing

Simulated annealing is so named because of its analogy to

the process of physical annealing with solids,.

A crystalline solid is heated and then allowed to cool very

slowly

until it achieves its most regular possible crystal lattice

configuration (i.e., its minimum lattice energy state), and

thus is free of crystal defects.

If the cooling schedule is sufficiently slow, the final

configuration results in a solid with such superior structural

integrity.

Simulated annealing establishes the connection between this

type of thermodynamic behaviour and the search for global

minima for a discrete optimization problem.

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Simulated Annealing(SA)

SA is a global optimization technique.

SA distinguishes between different local optima.

SA is a memory less algorithm, the algorithm does not use any information gathered during the search

SA is motivated by an analogy to annealing in solids.

Simulated Annealing – an iterative improvement algorithm.

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Simulated Annealing6

Local Search

Solution space

Cost

fun

ction

?

Analogy

Slowly cool down a heated solid, so that all particles arrange

in the ground energy state

At each temperature wait until the solid reaches its thermal

equilibrium

Probability of being in a state with energy E :

Pr { E = E } = 1/Z(T) . exp (-E / kB.T)

E Energy

T Temperature

kB Boltzmann constant

Z(T) Normalization factor (temperature dependant)

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Simulation Of Cooling (Metropolis 1953)

At a fixed temperature T :

Perturb (randomly) the current state to a new state

E is the difference in energy between current and new state

If E < 0 (new state is lower), accept new state as current state

If E 0 , accept new state with probability

Pr (accepted) = exp (- E / kB.T)

Eventually the systems evolves into thermal equilibrium at

temperature T .

When equilibrium is reached, temperature T can be lowered and

the process can be repeated

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Relationship Between Physical

Annealing And Simulated Annealing

Thermodynamic

Simulation

Combinatorial

Optimization

System states Solutions

Energy Cost

Change of State Neighbouring Solutions

Temperature Control Parameter T

Frozen State Heuristic Solution

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Simulated Annealing

Same algorithm can be used for combinatorial optimization

problems:

Energy E corresponds to the Cost function C

Temperature T corresponds to control parameter c

Pr { configuration = i } = 1/Q(c) . exp (-C(i) / c)

C Cost

c Control parameter

Q(c) Normalization factor (not important)

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Ball On Terrain Example – SA Vs.

Greedy Algorithms

Greedy Algorithm

gets stuck here!

Locally Optimum

Solution.

Simulated Annealing explores

more. Chooses this move with a

small probability (Hill Climbing)

Upon a large no. of iterations,

SA converges to this solution.

Initial position

of the ball

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12 Advantages

Can deal with arbitrary systems and cost functions.

Statistically guarantees finding an optimal solution.

Is relatively easy to code, even for complex problems.

Generally gives a ``good'' solution

This makes annealing an attractive option for Optimization

problems where heuristic (specialized or problem specific)

methods are not available.

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Repeatedly annealing with a 1/log k schedule is very

slow, especially if the cost function is expensive to

compute.

For problems where the energy landscape is smooth, or

there are few local minima, SA is overkill - simpler, faster

methods (e.g., gradient descent) will work better. But

generally don't know what the energy landscape is for a

particular problem.

The method cannot tell whether it has found an optimal

solution. Some other complimentary method (e.g. branch

and bound) is required to do this.

Conclusions

Simulated Annealing algorithms areusually better than greedy algorithms,when it comes to problems that havenumerous locally optimum solutions.

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References15

P.J.M. van Laarhoven, E.H.L. Aarts, Simulated Annealing:

Theory and Applications, Kluwer Academic Publisher,

1987.

A. A. Zhigljavsky, Theory of Global Random Search,

Kluwer Academic Publishers, 1991.

Thank You