Optimization by Simulated Annealing(2)

20 Pages·2006·0.19 MB·English

Optimization by Simulated
Annealing
S. Kirkpatrick, C. D. Gelatt Jr., M. P. Vecchi
Science, Volume 220 (1983), Number 4598: 671-679
Presented by Ryan Cheng
Reviewed by Amanda McCoy
Combinatorial Optimization
Central to science, engineering, CS
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Optimizing objective (cost) functions of complicated
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systems
Determination of global extremum of objective
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functions
Deterministic methods unrealistic as number of
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parameters becomes very large
Traveling Salesman Problem
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Computer Design
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Introduction to the Traveling
Salesman Problem
Problem 1: Minimize cost
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function (E) of a salesman
traveling between N number
of cities and back
Cost to travel between
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cities proportional to
distance between cities
Simplest form :
Intro. Traveling Salesman Problem
continued…
Number of possible path configurations = N!
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Exact solutions to minimization of E computationally determined for
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magnitudes of N<~102 (as of 1983)
Non-deterministic polynomial time complete (NP-C) problem
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Computing effort for exact soln. increases exponentially with N
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Heuristic methods for near-optimal solutions
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“Divide and Conquer” and “Iterative Improvements”
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(cid:198)Monte Carlo (MC) & Simulated Annealing (SA)
Metropolis Monte Carlo Algorithm
Start with known configuration, objective function (ie, energy),
1)
some Temperature value
Random change configuration (ie, add small random
2)
displacement)
Calculate new energy value (E )
3)
2
Compare to energy at previous position (E ):
4)
1
If E < E , keep new position
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2 1
If E > E , keep new position if the Boltzmann factor for transition is
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2 1
greater or equal to a random number between 0 and 1
Repeat steps 2) – 4) K number of times
5)
Simulated Annealing (SA)
Concept of SA from annealing process
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Slowly cooling melt to form perfect crystals
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SA provides a temperature schedule for the Metropolis method
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Start at effectively high temperature and gradually decrease the
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temperature by increments until T slightly above 0 (<1)
At every temperature, Metropolis algorithm is run (nested-loop)
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Benefits
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Ability to escape local minima at non-zero temperatures
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“Divide and Conquer” -> Gross features of final state appear at high
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temp. while fine details appear at low temp.
SA Application: The Traveling
Salesman
Kirkpatrick et al solved problem where N=400
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Re-arrangement involved random selection of string of cities and
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reversal of order (Lin-Kernighan method)
Side of square boundary has length of N1/2
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Cities grouped into nine clusters
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Solved problem in “Manhattan” metric space so thus,
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Solved problem several times and averaged optimal step
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lengths (α)
SA Results: Traveling Salesman
Results at T = 1.2 (α = 2.0567)
a)
Results at T = 0.8 (α = 1.515)
b)
SA Results: Traveling Salesman
continued…
c) Results at T = 1.2 (α = 2.0567)
d) Results at T = 0.8 (α = 1.515)
Physical Design of Computers
Optimization problems in Comp.
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design
Partitioning circuits into groups
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to fit on chip
Placement of circuits on chip
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Wiring of circuits on chip
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Goal to optimize performance of
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system without compromising
any design stage