Silicon Cluster Optimization Using
Extended Compact Genetic Algorithm
(ECGA)
Kumara Sastry Guanghua Xiao
University of Illinois at Urbana-Champaign
{ksastry, gxiao}@uiuc.edu
December 12, 2000
MatSE 385
ATOMIC SCALE SIMULATION

Si Cluster Optimization Using ECGA 1
Outline
Motivation
•
Overview of ECGA
•
Marginal Product Models and BBs
•
Silicon Potential
•
Algorithm Implementation
•
Results
•
Conclusions
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 2
Motivation
Existing algorithms use “not-so-good” operators
•
– Proportionate selection
– Single point crossover
Increased interest in competent GAs
•
– Solves hard problems quickly reliably and accurately
An interesting competent GA is ECGA (Harik, 1999)
•
– builds models of good data as linkage groups
Cluster optimization is a NP-hard problem (Wille and
•
Vennik, 1985)
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 3
Overview of ECGA
Key Idea: Good probability distribution ≡ Linkage
•
learning
Probability distribution: Marginal Product Models
•
(MPM)
Quantified based on Minimum Description Length
•
(MDL)
MDL Concept: Simpler distributions are better
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 4
Flowchart of Optimization Algorithm
Initialize Perform Have the
Evaluate Yes
N clusters Nelder-Mead clusters
potential energy End
randomly simplex search converged
of each cluster
No
Create new Perform
Replace N*Pc Build MPM
clusters using tournament
using MDL
old clusters MPM selection
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 5
MPM and Building Blocks
y
y1 y2 y3 z4
x1 x2 z2 x3 z3 x4
z1 4
5

























































4




























Generation










3























































2















































1


















































0
















4 9 14 19 24 29 34 39 44 49 54 59
Bit position
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 6
Marginal Product Models
Product of marginal distributions on a partition of
•
genes
Similar to CGA (Harik et. al;1998) and
•
PBIL(Baluja;1994)
Represent more than one gene in a partition
•
Make exposition simpler
•
Gene partition maps to linkage groups
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 7
Minimum Description Length Models
Hypothesis: Good distributions are those for which
•
– Representation of the distribution is minimum (model
complexity, Cm ).
– Representation of population compressed is minimum
(compressed population complexity, Cp ).
Penalize complex models
•
Penalize inaccurate models
•
Combined complexity, Cc = Cm + Cp
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 8
Building MPM using MDL
Uses a steepest ascent search:
1. Assume all the genes to be independent ([1],[2],· · ·,[L]) and
compute Cc .
2. Form all possible combinations (Nbb (Nbb -1)/2) of merging two
subsets. eg., ([1,2],[3],· · ·,[L]), · · ·, ([1],[2],[3],· · ·,[L-1,L]).
3. Select the set with minimum combined complexity (Cc ).
4. If Cc > Cc go to step 6.
5. Use the set with Cc as the current MPM and go to step 2.
6. Merging is not possible, exit with set from step 2 as MPM.
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 9
Generation of New Population
Transfer Np *(1-Pc ) best individuals to the next
•
generation
The rest Np *Pc individuals are generated as follows:
•
– Take each of the subset of MPM from one of the individuals.
– Similar to multiple point crossover.
– Number of crossover points = Nbb .
– Instead of two parents we have Nbb parents.
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 10
Silicon Potential
Gong Potential
Gong, X.G. Phys. Rev. B 47, 2329 (1993)
•
Empirical three body potential
•
Based on Stillinger Weber potential
•
Reflects both tetrahedral (∼ 109◦ ) and preferred bond
•
angles (∼ 60◦ )
Accurate for predicting structural properties
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 11
Gong Potential: Equations
n n
Utot = v2 (i, j) + v3 (i, j, k)
i<j i<j<k
−1
−q
−p
A Brij − rij exp (rij − a) |rij | < a
,
v2 (i, j) =
v3 (i, j, k) = h (rji , rki ) + h (rkj , rij ) + h (rik , rjk )
−1 −1
λ exp γ (rij − a) + (rki − a) |rij | < a
h (rji , rki ) =
12 2
|rki | < a
cos θjik + (cos θjik + c0 ) + c1 ,
3
• A = 7.0496, B = 0.6022, a = 1.8, p = 4, q = 0.
• λ = 25, γ = 1.2, c0 = -0.5, c1 = 0.45
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 12
Algorithm Implementation
Variables are fixed-space Cartesian coordinates
•
Each coordinate is encoded by 5-bit binary
•
25 independent runs, pc = 0.8, 4-11 atoms
•
Nelder-Mead simplex: Press et al
•
Termination criteria:
•
– Fitness variance ≤ 0.1
– Population variance ≤ 0.1
At most one failure allowed
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 13
Results: Single GA Run
Gen 1
Gen 0
Gen 2 Gen 3
Gen 7 Gen 6
Gen 5
Gen 4
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 14
Results: Single GA Run
Gen 0
Gen 1,2
Gen 5,6,7 Gen 4
Gen 8
Gen 3
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 15
Results: Optimal Structures
Compressed trigonal
Tetrahedron Octahedron Pentagonal bipyramid
bipyramid, U = -5.7518
U = -4.0016 U = -7.6696
U = -9.6240
Unicapped distorted
Tricapped trigonal Bicapped tetragonal
pentagonal bipyramid
prism, U = -13.5999 antiprism, U = - 15.6487
U = -11.5346
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 16
Results: Population Size
5
10
Simul
0.7
Avg. O(n )
Worst. O(n)
4
10
Minimum population size
3
10
2
10
1
10
4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9
Number of atoms, n
Average Case: O n0.7 , Worst Case: O(n).
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 17
Results: Convergence Time
20
18
16
14
No. of generations
12
10
8
6
4
2
0
3 4 5 6 7 8 9 10
No. of atoms, n
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 18
Results: Function Evaluation
9
10
Simul
Avg. O(n1.3)
Worst, O(n1.7)
8
10
No. of function evaluations
7
10
6
10
5
10
4
10
3 4 5 6 7 8 9 10
No. of atoms, n
Average Case: O n1.3 , Worst Case: O n1.7 .
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao

Si Cluster Optimization Using ECGA 19
Conclusions
Optimal structures of small Si clusters found
•
Results agree with literature (Iwamtsu, M. J. Chem.
•
Phy. 112 (2000))
Convergence is very fast (≤ 25 generations).
•
Population size increases linearly with cluster size
•
Polynomial increase of function evaluation
•
Results to be confirmed with bigger clusters
•
Illinois Genetic Algorithms Laboratory
Department of General Engineering
MatSE 385, Dec 12, 2000
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA. K. Sastry, G. Xiao