Efficient Atomic Cluster Optimizer
Kumara Sastry
Illinois Genetic Algorithms Laboratory
University of Illinois at Urbana-Champaign
Urbana, IL 61801
http://www-illigal.ge.uiuc.edu
Genetic and Evolutionary Computation Conference
(GECCO-2001)
July 7-11, 2001
San Francisco, CA

1
Hybrid ECGA Based Efficient Cluster Optimizer
Overview
• Background & Motivation
• Objective
• Overview of ECGA
• Algorithm Description
• Results & Conclusions
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

2
Hybrid ECGA Based Efficient Cluster Optimizer
Background: GA Design
• Design of competent GAs: A key challenge
– Solve hard problems Quickly, Reliably and
Accurately
• Much progress made (Goldberg, 1999)
• Existing competent GAs:
– Render intractable problems tractable
– Require subquadratic function evaluations
• ECGA is a competent GA (Harik, 1999)
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

3
Hybrid ECGA Based Efficient Cluster Optimizer
Background: Cluster Optimization
• Used in Surface & atomic simulations
• Simplest problem is NP hard
exp (n2 )
• Local minima grows as
• GAs for cluster optimization:
– Hartke (1993,1995); Zeiri et al, 1995; Deaven &
Ho, 1995; Gregurick & Alexander, 1996; Niesse
& Mayne, 1996; Zeiri, 1997; Iwamatsu, 2000
• They use “not-so-good” operators
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

4
Hybrid ECGA Based Efficient Cluster Optimizer
Motivation
O( k ) function Evals
• ECGA:
– for small clusters, k ≈ 8.2 (Sastry& Xiao, 2001)
• Clusters with large no. of atoms
– function evaluations is high
– Need Efficiency Enhancement Techniques (EET)
• Hybridization and Seeding are EETs
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

5
Hybrid ECGA Based Efficient Cluster Optimizer
Objective
• Employ ECGA to optimize atomic clusters
– Hybridize with a local search
∗ Nelder-Mead simplex used as local search
– Seed initial population
• Obtain better scale-up
• Solve larger clusters
• Silicon clusters used as test case
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

6
Hybrid ECGA Based Efficient Cluster Optimizer
Overview of ECGA
≡ Linkage learning
• Probability distribution
• Prob. dist.: Marginal Product Models
– Maps models of good data as linkage groups
– Groups linked variables as a single variable
– Eg. [1], [2, 5, 9], [3, 8], [4, 6], [7], [10]
• Quantified by Minimum Description Length
– Penalize inaccurate distributions
– Penalize complex distributions
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

7
Hybrid ECGA Based Efficient Cluster Optimizer
Encoding & Fitness Function
• Variables: Fixed-space Cartesian coords
– Each atom is coded by three variables
– Each coordinate is encoded by 5-bit binary
• Fitness Function: Cluster potential energy
• Silicon Potential:
– Gong, X.G. Phys. Rev. B 47, 2329 (1993)
– Empirical two & three body potential
– Also includes angular terms
– Accurate for predicting structural properties
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

8
Hybrid ECGA Based Efficient Cluster Optimizer
Seeding Initial Population
• Initial population generated through seeding
– Hoare (1979), Niesse & Mayne (1986)
– Use optimal structure of n − 1 atom cluster
– Insert an atom to the n − 1 atom cluster
– Randomly generate its position
• Considerably reduces the population size
• Initial structures have better fitness
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

9
Hybrid ECGA Based Efficient Cluster Optimizer
Hybridization
• Nelder-Mead simplex (Press et al, 1989)
– Requires 3n + 1 initial points
– The individual accounts for one point
– Perturb an atom in one coordinate
– Creates 3n points
• Local search for every individual
• Use fully lamarckian approach
– Local search solution replaces the individual
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

10
Hybrid ECGA Based Efficient Cluster Optimizer
Creating New Individuals
np individuals using MPM
• Create
– Generate each partition independently
– Assign values proportional to the frequency
– m point crossover between np individuals.
• Elitist replacement scheme
– Select top np individuals from
∗ np new individuals, and
∗ np old individuals
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

11
Hybrid ECGA Based Efficient Cluster Optimizer
Algorithm Flowchart
Initialize Perform Have the
Evaluate Yes
N clusters Nelder-Mead clusters
potential energy End
by seeding 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
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

12
Hybrid ECGA Based Efficient Cluster Optimizer
Results: Minimum Energy
0
ECGA Seeded
SGA Seeded
−10
Total potential energy, (units of e=2.17eV)
−20
−30
−40
−50
−60
−70
0 5 10 15 20 25 30 35 40
No. of atoms, n
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

13
Hybrid ECGA Based Efficient Cluster Optimizer
Results: Population Size
ECGA with seeding, O(n0.83)
1.83
SGA with seeding, O(n )
4.2
ECGA, O(n )
3
10
p
Minimum population size, N
2
10
1
10
No. of atoms, n
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

14
Hybrid ECGA Based Efficient Cluster Optimizer
Results: Convergence Time
16
ECGA with seeding
SGA with seeding
ECGA
14
12
No. of generations
10
8
6
4
2
4 6 8 10 12 14 16 18 20
No. of atoms, n
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

15
Hybrid ECGA Based Efficient Cluster Optimizer
Results: Function Evaluations
ECGA with seeding, O(n2.45)
SGA with seeding, O(n3.68)
ECGA, O(n8.2)
7
10
No. of function evaluations
6
10
5
10
4
10
1
10
No. of atoms, n
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

16
Hybrid ECGA Based Efficient Cluster Optimizer
Results: Scale Up
2.45
Average case, O(n )
Worst case, O(n2.46)
2.44
6
Best case, O(n )
10
No. of function evaluations
5
10
4
10
1
10
No. of atoms, n
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

17
Hybrid ECGA Based Efficient Cluster Optimizer
Summary
• An efficient hybrid cluster optimizer
– Solves larger clusters (Up to 40 atoms)
– High reliability: 96%
– Minimum population size: O n0.83
– Total No. of func. evals.: O n2.45
• Successfully predicts global optimum
• Iwamatsu (2000): 15 atoms
O (n3.3 )
• Niesse & Mayne (1996):
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

18
Hybrid ECGA Based Efficient Cluster Optimizer
Acknowledgments
• David E. Goldberg & David Ceperley
• Air Force Office of Scientific Research, Air Force
Materiel Command, USAF, under grant
F49620-00-1-0163.
• National Science Foundation under grant
DMI-9908252.
• U. S. Army Research Laboratory under the
Federated Laboratory Program, Cooperative
Agreement DAAL01-96-2-0003.
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

19
Hybrid ECGA Based Efficient Cluster Optimizer
Building MPM using MDL
Uses a steepest ascent search:
1. Compute Cc for independent genes ([1],[2],· · ·,[L])
2. Form all possible combinations (m(m − 1)/2) of
merging two subsets. eg., ([1,2],[3],· · ·,[L]), · · ·,
([1],[2],[3],· · ·,[L-1,L]).
3. Select set with minimum combined complexity (Cc ).
4. If Cc > Cc go to step 6.
5. MPM is the set with Cc . Go to step 2.
6. Merging is not possible, exit.
Illinois Genetic Algorithms Laboratory
Department of General Engineering
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

20
Hybrid ECGA Based Efficient Cluster Optimizer
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
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

21
Hybrid ECGA Based Efficient Cluster Optimizer
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
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu

22
Hybrid ECGA Based Efficient Cluster Optimizer
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
GECCO, July 7-11, 2001
University of Illinois at Urbana-Champaign
Urbana, IL 61801. USA.
K. Sastry
http://www-illigal.ge.uiuc.edu