Combinatorial Optimization for Weighing Matrices with the Ordering ...
SEA2011 D. E. Simos Weighing Matrices Competent GAs Conclusion
Combinatorial Optimization for Weighing Matrices with the Ordering Messy Genetic Algorithm
Dimitris E. Simos (joint work with C. Koukouvinos) Department of Mathematics National Technical University of Athens
May 5, 2011 10th International Symposium on Experimental Algorithms Orthodox Academy of Crete, Chania, Greece 1 / 22
Outline of the Talk SEA2011 D. E. Simos Weighing Matrices Competent GAs
1
Weighing Matrices
Conclusion
Motivation Recent Progress
2 / 22
Outline of the Talk SEA2011 D. E. Simos Weighing Matrices Competent GAs
1
Weighing Matrices
Conclusion
Motivation Recent Progress
2
Competent GAs Messy GA OmeGA
2 / 22
Weighing Matrices SEA2011 D. E. Simos Weighing Matrices Motivation Recent Progress Competent GAs Conclusion
Definition of a Weighing Matrix A square n × n matrix with elements from {−1, 0, 1} that satisfies W W T = wIn , and is denoted by W (n, w)
Properties of a Weighing Matrix 1 2
w non-zero entries per row and column Inner product of distinct rows is zero
Construction of Weighing Matrices If there exist two circulant matrices of order n each, satisfying AAT + BB T = wIn , then there exists a W (2n, w) A B W (2n, w) = −B T AT • Reference: Koukouvinos and Seberry, JSPI (1999) 3 / 22
Reduce the Search Space for Weighing Matrices Sequences with Zero Periodic Autocorrelation SEA2011
How to Locate the Circulant Matrices?
D. E. Simos
From sequences with zero periodic autocorrelation function Weighing Matrices Motivation Recent Progress
Periodic Autocorrelation Function of a Sequence
Competent GAs Conclusion
For a {0, ±1} sequence A = [a1 , a2 , . . . , an ] of length n we define, 1 the periodic autocorrelation function, PAF, PA (s) as n X P AFA (s) ≡ PA (s) = ai ai⊕s , s = 0, 1, . . . , n − 1 i=1 2
where by ⊕ we mean addition mod n P AFA : {0, 1, . . . , n − 1} → Z
Periodic Complementary Sequences Let two sequences as above, A = [a1 , . . . , an ] and B = [b1 , . . . , bn ]: 1 zero PAF: if P AFA (s) + P AFB (s) = 0 for s = 1, . . . , n − 1 2 Notation: DC(n, w) (double circulant pairs) 4 / 22
Reduce the Search Space for Weighing Matrices Sequences with Zero Periodic Autocorrelation generate Circulant Matrices SEA2011
Sum of PAF is Zero
D. E. Simos Weighing Matrices Motivation Recent Progress Competent GAs
DC(3, 5)
P AFS1 (0) + P AFS2 (0) = 2 + 3 = 5 P AFS1 (1) + P AFS2 (1) = 0 + 0 = 0
S1 = [1, 0, 1] S2 = [1, 1, −1]
P AFS1 (2) + P AFS2 (2) = 1 + (−1) = 0
Conclusion
Circulant Matrices
1 A= 1 0 1 B = −1 1
The two Circulant Construction D=
0 1 1 0 1 1 1 −1 1 1 −1 1
AAT + BB T = w · In (w = 5)
5 / 22
B AT
D = W (2n, w) where w is the weight of D
Weighing Matrix: W (2 · 3, 5) 1 1 0 −1 −1 1
Additive Property
A −B T
0 1 1 1 −1 −1
1 0 1 −1 1 −1
1 −1 1 1 0 1
1 1 −1 1 1 0
−1 1 1 0 1 1
Why Interested in Complementary Sequences? SEA2011 D. E. Simos Weighing Matrices
Applications of Periodic Complementary Sequences 1
Motivation Recent Progress Competent GAs
2
Conclusion 3 4
Pivotal role in Combinatorial Design Theory (Seberry and Yamada, 1992, Craigen and Koukouvinos, 2001) Construct sequences with desirable properties for radar applications (Weathers and Holiday, 1983) Design of cryptographic systems (Schneier, 1996) Intervene in coded aperture imaging, (Fenimore and Cannon, 1976) and higher-dimensional signal processing applications, (Golomb and Taylor, 1982)
Our Goal 1
2
6 / 22
Express the search for weighing matrices as an instance of a Combinatorial Optimization problem Design competent Genetic Algorithms (GAs) to solve it
Recent Upsurge for Searching Weighing Matrices SEA2011 D. E. Simos Weighing Matrices
Computational Optimization Algorithms
Motivation Recent Progress
String Sorting (SS) • Reference: Kotsireas, Koukouvinos and Seberry, AC (2009) Power Spectral Density (PSD) Test
Competent GAs Conclusion
• Reference: Kotsireas, Koukouvinos and Pardalos, JOCO (2011)
Combinatorial Optimization Algorithms 1
2
3
7 / 22
Tabu Search (TS) • Reference: Kotsireas, Koukouvinos, Pardalos and Shylo, JOCO (2010) Messy GA (mGA), (this talk) • Reference: Kotsireas, Koukouvinos, Pardalos and Simos, JOCO (2011) Ordering Messy GA (OmeGA), (this talk)
A Messy Genetic Algorithm (mGA) for Weighing Matrices SEA2011 D. E. Simos Weighing Matrices
Linkage Problem Building blocks (partial solutions) disruption in first generation GAs
Competent GAs Messy GA OmeGA Conclusion
Motivation The ability of the mGA to put tight genes together in a solution e.g. (0000 ∗ ∗) (Goldberg, Deb and Korb, 1989, 1990)
Plan of Attack to the Weighing Matrices Problem Exploit structural patterns of weighing matrices in terms of messy coding Take advantage of the tight coding of mGAs Correspond certain properties of sequences with zero PAF to the theory of messy GAs
8 / 22
Schemata and Building Blocks SEA2011 D. E. Simos
Schemata Similarity templates describing a subset of chromosomes
Weighing Matrices Competent GAs Messy GA OmeGA
Weighing Matrices Representation From {−1, 0, 1} and an additional “do not care” symbol (“*”)
Conclusion
A Schema of Length 8 and Order 7 H(−101∗; 1100) which matches the three concatenated strings {-1010;1100,-101-1;1100,-1011;1100}
Building Block (BB) A schema is expected to grow in subsequent generations if: 1 It has above average fitness 2 It is relatively short 3 Is of low order 9 / 22
Schemata and Building Blocks for developing Weighing Matrices SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA
Near-DC pairs A near-DC of length n, weight w and error ε, N DC(n, w, ε) to be a pair of ternary sequences of length n with w non-zero entries where ε is the sum of squares of its autocorrelation coefficients
Conclusion
An N DC(3, 3, 2) Pair [1, 0, 0]; [1, −1, 0] represented from H(1 ∗ 0; 1 ∗ 0) of order 4 • BBs of order k correspond to near-DC pairs of length bk/2c
Optimal BBs correspond to embedded DC Pairs The fixed positions [1, 0]; [1, 0] in schema H form a DC(2, 2) pair
Design Theory analogue to “Building Block Hypothesis” Combine small DC pairs to form a larger solution • Reference: Seberry and Yamada, 1992 10 / 22
Messy Representation for Weighing Matrices SEA2011
Messy Genes for sequences with zero PAF
D. E. Simos
messy gene g: (sequence index, position, value)
Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Figure: Messy encoding of a DC(4, 4)
Searching for DC(n, w) is a 2n bit Problem gmp : Λ2n → {1, 2} × S` × S` × Λ` , gmp([a1 , . . . , an ], [b1 , . . . , bn ]) = g 1 2 11 / 22
Λ = {−1, 0, 1} S` is the set of all different permutations of integers from 1 to `
Why mGAs are called “Messy”? Handling Over- and Underspecification of Messy Genes SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Handling Overspecification Overspecification requires that we choose between conflicting genes in a solution string The solution g1 = ((1, 1, 1), (1, 2, 1), (2, 1, 1), (2, 2, −1), (2, 2, 0)) is valid for a 4-bit problem Conflicting genes are (2, 2, −1) and (2, 2, 0) Left-to-right-scan: gpm(g1) → [1, 1]; [1, −1] → DC(2, 4)
Figure: Usage of a competitive template that two tight genes together. Underspecified chromosomes are evaluated by taking the missing genes from the template. 12 / 22
An Objective Function for Weighing Matrices based on Sequences with zero PAF SEA2011 D. E. Simos Weighing Matrices
The Objective Function Composed from certain subfunctions of the form fs (s) = (P AFA (s) + P AFB (s))2 , s = 1, . . . , n − 1:
Competent GAs Messy GA OmeGA Conclusion
OF ([a1 , . . . , an ], [b1 , . . . , bn ])
=
n−1 X
(P AFA (s) + P AFB (s))2
s=1
=
=
n−1 X
n X
s=1
i=1
n−1 X
n X
s=1
i=1
ai ai+s +
n X
2 13 / 22
bi bi+s
i=1
!2 (ai ai+s + bi bi+s )
Minimization Problem 1
!2
When OF equals zero we have a DC pair For OF = x > 0, we have a near-DC pair of error x
Messy Operators for Sequences with zero PAF SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Figure: BB disruption and preservation in a 8-bit weighing matrices problem. The one-point crossover operator will definitely disrupt the loose BB, 00**;**00, while in the mGA it would be probably be preserved after cut and splice due to the flexibility of the messy coding. 14 / 22
Mechanics of the Fast Messy GA (FmGA) SEA2011
Implementation
D. E. Simos Weighing Matrices
We used an adaptation of the mGA, the Fast Messy GA (FmGA) in order to avoid initialization bottlenecks
Competent GAs Messy GA OmeGA Conclusion
How the FmGA Works 1
2
3
Probabilistically complete initialization phase: Generate N fully specified chromosomes of length 2n − k, where 2n is the problem size of weighing matrices and k is the size of the BB Building-block filtering phase: Identify the BBs and filter out the “bad” genes by reducing the chromosome’s length Juxtapositional phase: Apply the cut-and-splice operators to the population and evaluate the OF
Guided Level-wise Processing Usage of competitive templates at each cycle of FmGA • The best individual found so far that has k tight genes 15 / 22
Illustration of the BB Filtering Phase SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Figure: After ns = 2 selections are applied nd = 3 genes are deleted at each chromosome. gpm((2, 3, 1), (1, 3, 0)) → [0]; [1] → DC(1, 1), FmGA scans the BB of order 2. Selection increases the proportion of this schema, the embedded DC(1, 1) survives the gene deletion. 16 / 22
An Added Level of Sophistication for searching Weighing Matrices SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA
OmeGA The Ordering Messy Genetic Algorithm (OmeGA) is a fmGA specialized for permutation problems • Represent the chromosomes by vectors of real numbers, the so-called random keys introduced by Bean (1994)
Conclusion
Similar to Other Competent GAs Random Key-based Simple GA (RKGA) • Reference: Bean, ORSA JoC (1994) Biased Random Key-based Simple GA (BRKGA) • Reference: Goncalves and Resende, JoH (2010)
Design of the OmeGA All mechanisms of the fmGA are applied The alleles are (long) integer numbers The alleles are treated as random keys to encode permutations 17 / 22
A New Criterion for PAF Verification that enables the Usage of Random-Keys SEA2011 D. E. Simos
DC(11, 9) A=+00000+0-0- B=0+0++00+0-0
Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Positive and Negative Support P OS(A) = {1, 7}, N EG(A) = {9, 11} P OS(B) = {2, 4, 5, 8}, N EG(B) = {10}
Signed and Cross Differences + − + − DA = [6, 5], DA = [2, 9], DB = [2, 3, 6, 1, 4, 3, 9, 8, 5, 10, 7, 8], DB =[]
CA = [8, 2, 10, 4, 3, 9, 1, 7], CB = [8, 6, 5, 2, 3, 5, 6, 9]
Verification + − + − D = DA ] DA ] DB ] DB = [1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10]
C = CA ] CB = [1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10] [D]s = [C]s = 1, s ∈ {1, 4, 7, 9, 10}, [D]s = [C]s = 2, s ∈ {2, 3, 5, 6, 8, 9} 18 / 22
Using Random-Keys for Representation of DC pairs and First Results of the Application of OmeGA SEA2011 D. E. Simos
Ordering Messy Genes for sequences with zero PAF Ordering messy gene g: (sequence index, position, random-key)
Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Formulation of the Weighing Matrices Problem Represent a permutation of length ` as an integer vector r = (r1 , r2 , . . . , r` ) where r ∈ [−n, n]` such that rφ(1) ≤ rφ(2) ≤ . . . ≤ rφ(`) holds, where φ : {1, . . . , `} −→ {1, . . . , `} is the corresponding mapping function arranging the keys in ascending order For W (2n, w) we have that ` = w since the integers represent the support of the candidate DC(n, w) The OF is the minimum number of random keys that have to be changed to transform one permutation into another
A New DC(61, 72) used to form a W (122, 72) --000+00--0+---+---+--0+000-++000000---0++00+00-+-++-+-0+000--000-00--0---+-+++-++0-000+--000000++-0--00+00-+-++-+-0+00019 / 22
Summary SEA2011 D. E. Simos Weighing Matrices
Highlights 1
Competent GAs Conclusion 2
3
20 / 22
We formulated the weighing matrices problem to an instance of a combinatorial optimization problem We employed competent GAs (mGA, OmeGA) to search for weighing matrices We resolved open cases of weighing matrices
Summary SEA2011 D. E. Simos Weighing Matrices
Highlights 1
Competent GAs Conclusion 2
3
We formulated the weighing matrices problem to an instance of a combinatorial optimization problem We employed competent GAs (mGA, OmeGA) to search for weighing matrices We resolved open cases of weighing matrices
Future Work Consider other classes of competent GAs for searching weighing matrices and similar combinatorial objects 1 2
Gene Expression Messy GA (GEMGA) Biased Random-Key Based Simple GA (BRKGA)
Find more new weighing matrices
20 / 22
References SEA2011 D. E. Simos Weighing Matrices Competent GAs Conclusion
Bean, J.C.: Genetic algorithms and random keys for sequencing and optimization. ORSA J. on Computing 6, 154-160 (1994) Goldberg, D.E., Deb, K., Korb, B.: Messy genetic algorithms: Motivation, analysis, and first results. Complex Systems 5, 493-530 (1989) Goncalves, J.F., Resende, M.G.C.: Biased random-key genetic algorithms for combinatorial optimization (to appear in Journal of Heuristics) Knjazew, D.: OmeGA: A Competent Genetic Algorithm for Solving Permutation and Scheduling Problems. Kluwer, Norwell (2002) Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Simos, D.E.: Competent genetic algorithms for weighing matrices (to appear in J. Comb. Optim.) Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Shylo, O.: Periodic complementary binary sequences and combinatorial optimization algorithms. J. Comb. Optim. 20, 63-75 (2010)
21 / 22
Questions - Comments SEA2011 D. E. Simos Weighing Matrices Competent GAs Conclusion
22 / 22
Thanks for your Attention!
Combinatorial Optimization for Weighing Matrices with the Ordering Messy Genetic Algorithm
Dimitris E. Simos (joint work with C. Koukouvinos) Department of Mathematics National Technical University of Athens
May 5, 2011 10th International Symposium on Experimental Algorithms Orthodox Academy of Crete, Chania, Greece 1 / 22
Outline of the Talk SEA2011 D. E. Simos Weighing Matrices Competent GAs
1
Weighing Matrices
Conclusion
Motivation Recent Progress
2 / 22
Outline of the Talk SEA2011 D. E. Simos Weighing Matrices Competent GAs
1
Weighing Matrices
Conclusion
Motivation Recent Progress
2
Competent GAs Messy GA OmeGA
2 / 22
Weighing Matrices SEA2011 D. E. Simos Weighing Matrices Motivation Recent Progress Competent GAs Conclusion
Definition of a Weighing Matrix A square n × n matrix with elements from {−1, 0, 1} that satisfies W W T = wIn , and is denoted by W (n, w)
Properties of a Weighing Matrix 1 2
w non-zero entries per row and column Inner product of distinct rows is zero
Construction of Weighing Matrices If there exist two circulant matrices of order n each, satisfying AAT + BB T = wIn , then there exists a W (2n, w) A B W (2n, w) = −B T AT • Reference: Koukouvinos and Seberry, JSPI (1999) 3 / 22
Reduce the Search Space for Weighing Matrices Sequences with Zero Periodic Autocorrelation SEA2011
How to Locate the Circulant Matrices?
D. E. Simos
From sequences with zero periodic autocorrelation function Weighing Matrices Motivation Recent Progress
Periodic Autocorrelation Function of a Sequence
Competent GAs Conclusion
For a {0, ±1} sequence A = [a1 , a2 , . . . , an ] of length n we define, 1 the periodic autocorrelation function, PAF, PA (s) as n X P AFA (s) ≡ PA (s) = ai ai⊕s , s = 0, 1, . . . , n − 1 i=1 2
where by ⊕ we mean addition mod n P AFA : {0, 1, . . . , n − 1} → Z
Periodic Complementary Sequences Let two sequences as above, A = [a1 , . . . , an ] and B = [b1 , . . . , bn ]: 1 zero PAF: if P AFA (s) + P AFB (s) = 0 for s = 1, . . . , n − 1 2 Notation: DC(n, w) (double circulant pairs) 4 / 22
Reduce the Search Space for Weighing Matrices Sequences with Zero Periodic Autocorrelation generate Circulant Matrices SEA2011
Sum of PAF is Zero
D. E. Simos Weighing Matrices Motivation Recent Progress Competent GAs
DC(3, 5)
P AFS1 (0) + P AFS2 (0) = 2 + 3 = 5 P AFS1 (1) + P AFS2 (1) = 0 + 0 = 0
S1 = [1, 0, 1] S2 = [1, 1, −1]
P AFS1 (2) + P AFS2 (2) = 1 + (−1) = 0
Conclusion
Circulant Matrices
1 A= 1 0 1 B = −1 1
The two Circulant Construction D=
0 1 1 0 1 1 1 −1 1 1 −1 1
AAT + BB T = w · In (w = 5)
5 / 22
B AT
D = W (2n, w) where w is the weight of D
Weighing Matrix: W (2 · 3, 5) 1 1 0 −1 −1 1
Additive Property
A −B T
0 1 1 1 −1 −1
1 0 1 −1 1 −1
1 −1 1 1 0 1
1 1 −1 1 1 0
−1 1 1 0 1 1
Why Interested in Complementary Sequences? SEA2011 D. E. Simos Weighing Matrices
Applications of Periodic Complementary Sequences 1
Motivation Recent Progress Competent GAs
2
Conclusion 3 4
Pivotal role in Combinatorial Design Theory (Seberry and Yamada, 1992, Craigen and Koukouvinos, 2001) Construct sequences with desirable properties for radar applications (Weathers and Holiday, 1983) Design of cryptographic systems (Schneier, 1996) Intervene in coded aperture imaging, (Fenimore and Cannon, 1976) and higher-dimensional signal processing applications, (Golomb and Taylor, 1982)
Our Goal 1
2
6 / 22
Express the search for weighing matrices as an instance of a Combinatorial Optimization problem Design competent Genetic Algorithms (GAs) to solve it
Recent Upsurge for Searching Weighing Matrices SEA2011 D. E. Simos Weighing Matrices
Computational Optimization Algorithms
Motivation Recent Progress
String Sorting (SS) • Reference: Kotsireas, Koukouvinos and Seberry, AC (2009) Power Spectral Density (PSD) Test
Competent GAs Conclusion
• Reference: Kotsireas, Koukouvinos and Pardalos, JOCO (2011)
Combinatorial Optimization Algorithms 1
2
3
7 / 22
Tabu Search (TS) • Reference: Kotsireas, Koukouvinos, Pardalos and Shylo, JOCO (2010) Messy GA (mGA), (this talk) • Reference: Kotsireas, Koukouvinos, Pardalos and Simos, JOCO (2011) Ordering Messy GA (OmeGA), (this talk)
A Messy Genetic Algorithm (mGA) for Weighing Matrices SEA2011 D. E. Simos Weighing Matrices
Linkage Problem Building blocks (partial solutions) disruption in first generation GAs
Competent GAs Messy GA OmeGA Conclusion
Motivation The ability of the mGA to put tight genes together in a solution e.g. (0000 ∗ ∗) (Goldberg, Deb and Korb, 1989, 1990)
Plan of Attack to the Weighing Matrices Problem Exploit structural patterns of weighing matrices in terms of messy coding Take advantage of the tight coding of mGAs Correspond certain properties of sequences with zero PAF to the theory of messy GAs
8 / 22
Schemata and Building Blocks SEA2011 D. E. Simos
Schemata Similarity templates describing a subset of chromosomes
Weighing Matrices Competent GAs Messy GA OmeGA
Weighing Matrices Representation From {−1, 0, 1} and an additional “do not care” symbol (“*”)
Conclusion
A Schema of Length 8 and Order 7 H(−101∗; 1100) which matches the three concatenated strings {-1010;1100,-101-1;1100,-1011;1100}
Building Block (BB) A schema is expected to grow in subsequent generations if: 1 It has above average fitness 2 It is relatively short 3 Is of low order 9 / 22
Schemata and Building Blocks for developing Weighing Matrices SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA
Near-DC pairs A near-DC of length n, weight w and error ε, N DC(n, w, ε) to be a pair of ternary sequences of length n with w non-zero entries where ε is the sum of squares of its autocorrelation coefficients
Conclusion
An N DC(3, 3, 2) Pair [1, 0, 0]; [1, −1, 0] represented from H(1 ∗ 0; 1 ∗ 0) of order 4 • BBs of order k correspond to near-DC pairs of length bk/2c
Optimal BBs correspond to embedded DC Pairs The fixed positions [1, 0]; [1, 0] in schema H form a DC(2, 2) pair
Design Theory analogue to “Building Block Hypothesis” Combine small DC pairs to form a larger solution • Reference: Seberry and Yamada, 1992 10 / 22
Messy Representation for Weighing Matrices SEA2011
Messy Genes for sequences with zero PAF
D. E. Simos
messy gene g: (sequence index, position, value)
Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Figure: Messy encoding of a DC(4, 4)
Searching for DC(n, w) is a 2n bit Problem gmp : Λ2n → {1, 2} × S` × S` × Λ` , gmp([a1 , . . . , an ], [b1 , . . . , bn ]) = g 1 2 11 / 22
Λ = {−1, 0, 1} S` is the set of all different permutations of integers from 1 to `
Why mGAs are called “Messy”? Handling Over- and Underspecification of Messy Genes SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Handling Overspecification Overspecification requires that we choose between conflicting genes in a solution string The solution g1 = ((1, 1, 1), (1, 2, 1), (2, 1, 1), (2, 2, −1), (2, 2, 0)) is valid for a 4-bit problem Conflicting genes are (2, 2, −1) and (2, 2, 0) Left-to-right-scan: gpm(g1) → [1, 1]; [1, −1] → DC(2, 4)
Figure: Usage of a competitive template that two tight genes together. Underspecified chromosomes are evaluated by taking the missing genes from the template. 12 / 22
An Objective Function for Weighing Matrices based on Sequences with zero PAF SEA2011 D. E. Simos Weighing Matrices
The Objective Function Composed from certain subfunctions of the form fs (s) = (P AFA (s) + P AFB (s))2 , s = 1, . . . , n − 1:
Competent GAs Messy GA OmeGA Conclusion
OF ([a1 , . . . , an ], [b1 , . . . , bn ])
=
n−1 X
(P AFA (s) + P AFB (s))2
s=1
=
=
n−1 X
n X
s=1
i=1
n−1 X
n X
s=1
i=1
ai ai+s +
n X
2 13 / 22
bi bi+s
i=1
!2 (ai ai+s + bi bi+s )
Minimization Problem 1
!2
When OF equals zero we have a DC pair For OF = x > 0, we have a near-DC pair of error x
Messy Operators for Sequences with zero PAF SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Figure: BB disruption and preservation in a 8-bit weighing matrices problem. The one-point crossover operator will definitely disrupt the loose BB, 00**;**00, while in the mGA it would be probably be preserved after cut and splice due to the flexibility of the messy coding. 14 / 22
Mechanics of the Fast Messy GA (FmGA) SEA2011
Implementation
D. E. Simos Weighing Matrices
We used an adaptation of the mGA, the Fast Messy GA (FmGA) in order to avoid initialization bottlenecks
Competent GAs Messy GA OmeGA Conclusion
How the FmGA Works 1
2
3
Probabilistically complete initialization phase: Generate N fully specified chromosomes of length 2n − k, where 2n is the problem size of weighing matrices and k is the size of the BB Building-block filtering phase: Identify the BBs and filter out the “bad” genes by reducing the chromosome’s length Juxtapositional phase: Apply the cut-and-splice operators to the population and evaluate the OF
Guided Level-wise Processing Usage of competitive templates at each cycle of FmGA • The best individual found so far that has k tight genes 15 / 22
Illustration of the BB Filtering Phase SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Figure: After ns = 2 selections are applied nd = 3 genes are deleted at each chromosome. gpm((2, 3, 1), (1, 3, 0)) → [0]; [1] → DC(1, 1), FmGA scans the BB of order 2. Selection increases the proportion of this schema, the embedded DC(1, 1) survives the gene deletion. 16 / 22
An Added Level of Sophistication for searching Weighing Matrices SEA2011 D. E. Simos Weighing Matrices Competent GAs Messy GA OmeGA
OmeGA The Ordering Messy Genetic Algorithm (OmeGA) is a fmGA specialized for permutation problems • Represent the chromosomes by vectors of real numbers, the so-called random keys introduced by Bean (1994)
Conclusion
Similar to Other Competent GAs Random Key-based Simple GA (RKGA) • Reference: Bean, ORSA JoC (1994) Biased Random Key-based Simple GA (BRKGA) • Reference: Goncalves and Resende, JoH (2010)
Design of the OmeGA All mechanisms of the fmGA are applied The alleles are (long) integer numbers The alleles are treated as random keys to encode permutations 17 / 22
A New Criterion for PAF Verification that enables the Usage of Random-Keys SEA2011 D. E. Simos
DC(11, 9) A=+00000+0-0- B=0+0++00+0-0
Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Positive and Negative Support P OS(A) = {1, 7}, N EG(A) = {9, 11} P OS(B) = {2, 4, 5, 8}, N EG(B) = {10}
Signed and Cross Differences + − + − DA = [6, 5], DA = [2, 9], DB = [2, 3, 6, 1, 4, 3, 9, 8, 5, 10, 7, 8], DB =[]
CA = [8, 2, 10, 4, 3, 9, 1, 7], CB = [8, 6, 5, 2, 3, 5, 6, 9]
Verification + − + − D = DA ] DA ] DB ] DB = [1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10]
C = CA ] CB = [1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 8, 8, 9, 9, 10] [D]s = [C]s = 1, s ∈ {1, 4, 7, 9, 10}, [D]s = [C]s = 2, s ∈ {2, 3, 5, 6, 8, 9} 18 / 22
Using Random-Keys for Representation of DC pairs and First Results of the Application of OmeGA SEA2011 D. E. Simos
Ordering Messy Genes for sequences with zero PAF Ordering messy gene g: (sequence index, position, random-key)
Weighing Matrices Competent GAs Messy GA OmeGA Conclusion
Formulation of the Weighing Matrices Problem Represent a permutation of length ` as an integer vector r = (r1 , r2 , . . . , r` ) where r ∈ [−n, n]` such that rφ(1) ≤ rφ(2) ≤ . . . ≤ rφ(`) holds, where φ : {1, . . . , `} −→ {1, . . . , `} is the corresponding mapping function arranging the keys in ascending order For W (2n, w) we have that ` = w since the integers represent the support of the candidate DC(n, w) The OF is the minimum number of random keys that have to be changed to transform one permutation into another
A New DC(61, 72) used to form a W (122, 72) --000+00--0+---+---+--0+000-++000000---0++00+00-+-++-+-0+000--000-00--0---+-+++-++0-000+--000000++-0--00+00-+-++-+-0+00019 / 22
Summary SEA2011 D. E. Simos Weighing Matrices
Highlights 1
Competent GAs Conclusion 2
3
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We formulated the weighing matrices problem to an instance of a combinatorial optimization problem We employed competent GAs (mGA, OmeGA) to search for weighing matrices We resolved open cases of weighing matrices
Summary SEA2011 D. E. Simos Weighing Matrices
Highlights 1
Competent GAs Conclusion 2
3
We formulated the weighing matrices problem to an instance of a combinatorial optimization problem We employed competent GAs (mGA, OmeGA) to search for weighing matrices We resolved open cases of weighing matrices
Future Work Consider other classes of competent GAs for searching weighing matrices and similar combinatorial objects 1 2
Gene Expression Messy GA (GEMGA) Biased Random-Key Based Simple GA (BRKGA)
Find more new weighing matrices
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References SEA2011 D. E. Simos Weighing Matrices Competent GAs Conclusion
Bean, J.C.: Genetic algorithms and random keys for sequencing and optimization. ORSA J. on Computing 6, 154-160 (1994) Goldberg, D.E., Deb, K., Korb, B.: Messy genetic algorithms: Motivation, analysis, and first results. Complex Systems 5, 493-530 (1989) Goncalves, J.F., Resende, M.G.C.: Biased random-key genetic algorithms for combinatorial optimization (to appear in Journal of Heuristics) Knjazew, D.: OmeGA: A Competent Genetic Algorithm for Solving Permutation and Scheduling Problems. Kluwer, Norwell (2002) Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Simos, D.E.: Competent genetic algorithms for weighing matrices (to appear in J. Comb. Optim.) Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Shylo, O.: Periodic complementary binary sequences and combinatorial optimization algorithms. J. Comb. Optim. 20, 63-75 (2010)
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Questions - Comments SEA2011 D. E. Simos Weighing Matrices Competent GAs Conclusion
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