Simulated Annealing

Term Paper on-Global Optimization Search

Techniques (A I – Soft computing Domain)

PRESENTED BY -JOY DUTTA

ROLL – 91/CSE/101006BTECH 6 TH SEM

COMPUTER SCIENCE & ENGINEERINGCALCUTTA UNIVERSITY

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MAIN PROBLEM -> OPTIMIZATION

Local

GlobalOptimiza

tion search

techniques

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TABU SEARCH , GREEDY APPROACH , STEEPEST DESCEND,

ETC

SIMMULATED ANNEALING,PARTICLE SWARM OPTIMIZATION (PSO),GRADIENT DESCENT ETC

Difficulty in Searching Global Optima

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startingpoint

descenddirection

local minima

global minima

barrier to local search

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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Consequences of the Occasional Ascents

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Help escaping the local optima.

desired effect

Might pass global optima after reaching it

adverse effect

Background: AnnealingSimulated 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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Background (cont..)Solid is heated to melting point

High-energy, high-entropy stateRemoves defects/irregularities

Temp is very slowly reducedRecrystallization occurs (regular structure)New internal state of diffused atomsFast cooling induces fragile structure

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Control of Annealing Process

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Acceptance of a search step (Metropolis Criterion):

Assume the performance change in the search direction is .

Accept a ascending step only if it pass a random test,

Always accept a descending step, i.e. 0

1,0exp randomT

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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Stopping Criterion • A given minimum value of the temperature

has been reached.

• A certain number of iterations (or temperatures) has passed without acceptance of a new solution.

• A specified number of total iterations has been executed

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11

Flow Chart:Start With an initial solution

Add new random stand at random period

Improvement?

Accept new Solution

Stop criteria?

Stop

P(delta)>rand?

yesyes

yes

no

no

noP(delta) 1 when c is very high.P(delta) 0 when c is very smallrand (0,1)

Simulated Annealing Algorithm

• Initial temperature (TI)• Temperature length (TL) : number of

iterations at a given temperature• cooling ratio (function f): rate at which

temperature is reduced . f(T) = aT , where a is a constant, 0.8 ≤ a ≤ 0.99 (most often closer to 0.99)stopping criterion

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Simulated Annealing Algorithm construct initial solution x0; xnow = x0 set initial temperature T = TI repeat for i = 1 to TL do generate randomly a neighbouring solution x′ ∈

N(xnow) compute change of cost ΔC = C(x′) - C(xnow) if ΔC ≤ 0 then xnow = x′ (accept new state) else Generate q = random(0,1) if q < exp(-ΔC /T) then xnow = x′ end if end if end for set new temperature T = f(T) until stopping criterion return solution corresponding to the minimum cost

function04/12/2023 13

Convergence of simulated annealing

HILL CLIMBING

HILL CLIMBING

HILL CLIMBINGCO

ST

FU

NC

TIO

N,

C

NUMBER OF ITERATIONS

AT INIT_TEMP

AT FINAL_TEMP

Move accepted withprobability= e-(^C/temp)

Unconditional Acceptance

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Implementation of Simulated Annealing

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Understand the result:

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

• Convergence to global optima can only be realized in asymptotic sense.

Qualitative AnalysisRandomized local search.

Is simulated annealing greedy?Controlled greed.

Is a greedy algorithm better? Where is the difference?Explain with - The ball-on-terrain example.

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Ball on terrain example – Simulated Annealing vs Greedy Algorithms

• The ball is initially placed at a random position on the terrain. From the current position, the ball should be fired such that it can only move one step left or right. What algorithm should we follow for the ball to finally settle at the lowest point on the terrain?

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Ball on terrain example – SA vs. Greedy Algorithms

Greedy Algorithmgets stuck here!Locally Optimum

Solution.

Simulated Annealing exploresmore. Chooses this move with asmall probability (Hill Climbing)

Upon a large no. of iterations,SA converges to this solution.

Initial positionof the ball

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Jigsaw puzzles – Intuitive usage of Simulated Annealing

• Given a jigsaw puzzle such that one has to obtain the final shape using all pieces together.

• Starting with a random configuration, the human brain unconditionally chooses certain moves that tend to the solution.

• However, certain moves that may or may not lead to the solution are accepted or rejected with a certain small probability.

• The final shape is obtained as a result of a large number of iterations.

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ApplicationsCircuit partitioning and placement. Hardware/Software PartitioningGraph partitioningVLSI: Placement, routing.Image processingStrategy scheduling for capital products with

complex product structure.Umpire scheduling in US Open Tennis

tournament! Event-based learning situations. etc

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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.

• Heuristic methods, which are problem-specific or take advantage of

extra information about the system, will often be better than general methods, although SA is often comparable to heuristics.

• 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.

Disadvantages

ConclusionsSimulated Annealing algorithms are usually

better than greedy algorithms, when it comes to problems that have numerous locally optimum solutions.

Simulated Annealing is not the best solution to circuit partitioning or placement. Network flow approach to solving these problems functions much faster.

Simulated Annealing guarantees a convergence upon running sufficiently large number of iterations.

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Reference:

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• 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!