simulated annealing - a optimisation technique
TRANSCRIPT
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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