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Metaheuristics and Combinatorial Optimization: Ideas, Insights, and Observations. by Bruce L. Golden Robert H. Smith School of Business University of Maryland Presented at the Seventh Metaheuristics International Conference Montreal, June 2529, 2007. Some of My Favorite Metaheuristics.
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by
Bruce L. Golden
Robert H. Smith School of Business
University of Maryland
Presented at the Seventh Metaheuristics International Conference
Montreal, June 2529, 2007
The Elegance of Simulated Annealing
Step 1. Compute an initial feasible configuration T and choose an initial temperature tand a repetition factor r
Step 2. As long as the stopping criterion is not satisfied, perform the substeps
(i) Perform a random modification of the current configuration to obtain the configuration T' and let = c(T') – c(T), the difference in configuration lengths
(ii) If < 0 (improvement), then set T = T'. If not, set T = T' with probability = e / t
Step 3. Output the final (or best) solution
Significance of Simulated Annealing
Why should we only consider downhill moves?
Alternatives to Simulated Annealing
Can we mimic SA without generating random numbers?
Similar to Simulated Annealing
Metaheuristics and Tabu Search
Why leave things to chance?
The Evolution of Genetic Algorithms
How can one build a metaheuristic based on the
notion of competition?
What would a metaheuristic based on cooperation (as well as competition) look like?
Tabu Search with Strategic Oscillation
Why not consider infeasible intermediate solutions?
Finetuned Learning Metaheuristic
0. Start with original inputs
1. Average, aggregate, approximate, or smooth the inputs
2. Perform PROCEDURE
3. If inputs are the original inputs, then stop
Otherwise, disaggregate or refine the inputs
4. Go to step 2
Overview of Finetuned Learning
Iterations: a = 6, 3, 2, 1
This transform clusters distances in the center
The ranking of the distances is preserved



d + (dij – d)a, dij d
{
dij (a) =



d – (d – dij)a, dij< d
Marginal Improvements Rejected in the Concave Region of the Transformation
Unfavorable Moves in the Transitional Region of the Transformation
When is a downhill move not what it seems?
A OneParameter Genetic Algorithm
For each j between 0 and p – 1, do:
t [ j ] = crossover { s [ j ], s [ (j + k) mod p ] }
t [ j ] = mutation { t [ j ] }
s [ j ] = the better solution of s [ j ] and t [ j ]
End For
S[2]
S[3]
Generation 0
S[1]
S[0]
S[2]
S[1]
S[3]
S[0]
Generation 1
S[2]
S[1]
S[0]
S[3]
Generation 2
S[1]
S[2]
S[3]
S[0]
Generation 3
a. Let S = s [ 1 ] s [ 2 ] and T be the empty set
b. Sort S in decreasing order of the frequency of labels in G
c. Add labels of S, from the first to the last, to T until T represents a feasible solution
d. Output T
s [ 1 ] = { a, b, d }
s [ 2 ] = { a, c, d }
a
a
a
a
d
b
b
d
b
a
a
a
a
c
d
d
c
c
T = { }
S = { a, b, c, d }
Ordering: a, b, c, d
a. Randomly select c not in S and let T = S c
b. Sort T in decreasing order of the frequency of the labels in G
c. From the last label on the above list to the first, try to remove one label from T and keep T as a feasible solution
d. Repeat the above step until no labels can be removed
e. Output T
a
a
a
b
b
a
a
a
a
c
c
c
c
c
c
An Example of Mutation
S = { a, b, c }
S = { a, b, c, d }
d
b
b
d
b
b
Add { d }
Ordering: a, b, c, d
a
b
a
a
c
An Example of Mutation
Remove { a }
S = { b, c }
Remove { d }
S = { a, b, c }
b
b
b
b
b
c
c
c
c
c
T = { b, c }
Some Interesting Open Research Questions
Research Questions and a Conclusion
Bean (1994) : ORSA JOC
Chandran, Golden, and Wasil (2003): in book
Coy, Golden, Runger, and Wasil (1998): IEEE Trans. on SMC
Coy, Golden, and Wasil (2000): EJOR
Dorigo (1992): Ph.D. dissertation
Dueck (1993): J. Comput. Phys.
Dueck and Scheur (1990): J. Comput. Phys.
Glover (1986): C&OR
9. Glover (1989): ORSA JOC
Gu and Huang (1994): IEEE Trans. on SMC
Kelly, Golden, and Assad (1993): AOR
Kirkpatrick, Gelatt, and Vecchi (1983): Science
Mingjun and Huanwen (2004): Chaos, Solitons & Fractals
Pepper, Golden, and Wasil (2002): IEEE Trans. on SMC
Suman and Kumar (2006): JORS
Tovey (1998): Am. J. Math. and Man. Sci.
van Hemert (2006): Evolutionary Computation
Wood and Downs (1998): two Proceedings articles
Xiong, Golden, and Wasil (2005): IEEE Trans. Evol. Comp.
Xiong, Golden, and Wasil (2006): IEEE Trans. Evol. Comp.