Efficient Atomic Cluster Optimizer

(5-§_‘;  Kumara Sastry
‘ . ./ Illinois Genetic Algorithms Laboratory
. « University of...
Hybrid ECGA Based Efficient Cluster Optimizer

Overview

 Iii Background & Motivation
’ e Objective
 ii Overview of ECGA
l...
Hybrid ECGA Based Efficient Cluster Optimizer 2

Background:  GA Design

.  Design of competent GAS:  A key challenge

— S...
Hybrid ECGA Based Efficient Cluster Optimizer 3

Background:  Cluster Optimization

.  Used in Surface & atomic Simulation...
Hybrid ECGA Based Efficient Cluster Optimizer 4

.  ECGA:  (’)(€") function Evals
— for small clusters,  1.1% 8.2 (Sastry&...
Hybrid ECGA Based Efficient Cluster Optimizer 5

Objective

.  Employ ECGA to optimize atomic clusters

— Hybridize with a...
Hybrid ECGA Based Efficient Cluster Optimizer 6

Overview of ECGA

($. {; . . Probability distribution
C’ is Prob.  dist. ...
Hybrid ECGA Based Efficient Cluster Optimizer 7

Encoding & Fitness Function

.  Variables:  Fixed-space Cartesian coords
...
Hybrid ECGA Based Efficient Cluster Optimizer 8

Seeding Initial Population

.  Initial population generated through seedi...
Hybrid ECGA Based Efficient Cluster Optimizer 9

Hybridization

.  Nelder-Mead simplex (Press et al,  1989)
— Requires 371...
Hybrid ECGA Based Efficient Cluster Optimizer 10

Creating New Individuals

.  Create np individuals using MPM
— Generate ...
Hybrid ECGA Based Efficient Cluster Optimizer 11

Algorithm Flowchart

 

  
 
 

 

 N
(“  .  '   
‘ '  Ff‘:  WON  h
- -'...
Hybrid ECGA Based Efficient Cluster Optimizer

Results:  Minimum Energy

 

D l l l l _
xii ECGA Seeded
'1-‘ % + SGA Seede...
Hybrid ECGA Based Efficient Cluster Optimizer

Results:  Population size

I

0 scan with seeding.  olnml
9 SGA will seedin...
Hybrid ECGA Based Efficient Cluster Optimizer 14

Results:  Convergence Time

 

   

 ‘V
j,  16 l l l I l I
:   .  - U EC...
Hybrid ECGA Based Efficient Cluster Optimizer 15

Results:  Function Evaluations

-9- com wall seeding.  oin’ ‘5l

it stil...
Hybrid ECGA Based Efficient Cluster Optimizer

Results:  Scale Up

I

Avoraafi case.  O(rf“5l
Worst case.  Olnz ‘S
lo‘ < - ...
Hybrid ECGA Based Efficient Cluster Optimizer 17

0 An efficient hybrid cluster optimizer
— Solves larger clusters (Up to ...
Hybrid ECGA Based Efficient Cluster Optimizer 18

Acknowledgments

(f-§j;  ll!  David E.  Goldberg & David Ceperley

 iii ...
Hybrid ECGA Based Efficient Cluster Optimizer 19

Building MPM using MDL

Uses a steepest ascent search: 
1. Compute Cc.  ...
Hybrid ECGA Based Efficient Cluster Optimizer 20

Gong Potential Equations

f:7"2(iij)+ i v3(il.7-wk)

i'. <j i<j<k

» «Kw...
Hybrid ECGA Based Efficient Cluster Optimizer 21

Results:  Single GA Run

 

F .  _ llll--lcllmrlltfb-tlmnnlulnruan
l—"“"...
Hybrid ECGA Based Efficient Cluster Optimizer

Results:  Optimal Stru ctu res

' ‘~.  / ,
 ‘‘i_‘_‘ i Teiralicdmn

. .‘. .‘...
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Efficient Cluster Optimization Using A Hybrid Extended Compact Genetic Algorithm with A Seeded Population

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A recent study Sastry and Xiao (2001) proposed a highly reliable cluster optimization algorithm which employed extended compact genetic algorithm (ECGA) along with Nelder-Mead simplex search. This study utilizes an efficiency enhancement technique for the ECGA based cluster optimizer to reduce the population size and the number of function evaluation requirements, yet retaining the high reliability of predicting the lowest energy structure. Seeding of initial population with lowest energy structures of smaller cluster has been employed as the efficiency enhancement technique. Empirical results indicate that the population size and total number of function evaluations scale up with the cluster size are reduced from O(n4.2) and O(n8.2) to O(n0.83) and O(n2.45) respectively.

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Efficient Cluster Optimization Using A Hybrid Extended Compact Genetic Algorithm with A Seeded Population

  1. 1. Efficient Atomic Cluster Optimizer (5-§_‘; Kumara Sastry ‘ . ./ Illinois Genetic Algorithms Laboratory . « University of Illinois at Urbana—Champaign 0'; - Urbana, IL 61801 http: // www-illigal . ge . uiuc . edu K Genetic and Evolutionary Computation Conference ~ teecco-2oo1) , [‘_/ G€CC= j-, . July 7-11, 2001 " A San Francisco. CA
  2. 2. Hybrid ECGA Based Efficient Cluster Optimizer Overview Iii Background & Motivation ’ e Objective ii Overview of ECGA l ii Algorithm Description - Results & Conclusions GECCO. July 7~11. 2001 K. Sastry
  3. 3. Hybrid ECGA Based Efficient Cluster Optimizer 2 Background: GA Design . Design of competent GAS: A key challenge — Solve hard problems Quickly, Reliably and Accurately 0 Much progress made (Goldberg, 1999) . Existing competent GAS: - Render intractable problems tractable — Require subquadratic function evaluations . ECGA is a competent GA (Harik, 1999) Illa-h UPIIIK laturu-rV . """""‘. ... ... .'. '.‘. .‘. ':'. :'. '."""°"“. ... ..: ... ... GECCO. July 7-11. 2001 L. ,—. i~1‘. .'1.’ili'.1i". l.‘. T.; ... i. K. Sastry
  4. 4. Hybrid ECGA Based Efficient Cluster Optimizer 3 Background: Cluster Optimization . Used in Surface & atomic Simulations . Simplest problem is NP hard . Local minima grows as exp(n2) . 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 Illodn tientlk laturu-ry . """""‘. ... ... ..‘. '.‘. .‘. '.". :'. '."""°"". ..4.: ... ... oecco. July 7-11. 2001 L. '.: .."i. .'. “.. 'l. ’.l. ",: .", §§T. ;., ,.. . K. Sastry
  5. 5. Hybrid ECGA Based Efficient Cluster Optimizer 4 . ECGA: (’)(€") function Evals — for small clusters, 1.1% 8.2 (Sastry& Xiao, 2001) 0 Clusters with large no. of atoms - function evaluations is high — Need Efficiency Enhancement Techniques (EET) . Hybridization and Seeding are EETs Illudn (icnuk lnh-tn-ry . ""“"“. ... ... ..‘. '.‘. .‘. ;.". :': '.”"“"". ... .4.: ... ... GECCO. July 7-11. 2001 L'. rt. 'l£. }.", §.", ,‘, §T. ’.', ,.. . K. Sastry
  6. 6. Hybrid ECGA Based Efficient Cluster Optimizer 5 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 Ilhdn (irnuk [churn-ry . ‘""""‘. ... ... ..‘. '.‘. .‘. '.". :'. '.'. ”"‘°"". ... .4.: ... ... GECCO. July 7-11. 2001 L'. L‘i. .'. “.. 'l. ’.l. ',: .', §§T. ’.', ,.. . K. Sastry
  7. 7. Hybrid ECGA Based Efficient Cluster Optimizer 6 Overview of ECGA ($. {; . . Probability distribution C’ is Prob. dist. : Marginal Product Models Linkage learning aim — Maps models of good data as linkage groups E:77l'f" — Groups linked variables as a single variable Kr — Eg. [1], [2,5,9], [3,8]. [4,6], [7], [10] " ti Quantified by Minimum Description Length / — Penalize inaccurate distributions / ‘ [41 — Penalize complex distributions GECCO. July 7-11. 2001 K. Sastry
  8. 8. Hybrid ECGA Based Efficient Cluster Optimizer 7 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 Ilhdn (ienuk [churn-ry . ‘""""‘. ... ... ..‘. '.‘. .‘. '.". :'. '.'. ”"‘°"". ... .4.: ... ... GECCO. July 7-11. 2001 L'. L‘l. .'. “.. 'l. ’.l. ',: .', §§T. ’.', ,.. . K. Sastry
  9. 9. Hybrid ECGA Based Efficient Cluster Optimizer 8 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 Illudn (icnuk lnh-tn-ry . ""“"“. ... ... ..‘. '.‘. .‘. ;.". :': '.”"“"". ... .4.: ... ... GECCO. July 7-11. 2001 L'. ;“l. t.. 'l. ’.}. ",§. ",, ‘,. ‘T. ’.', ,.. . K. Sastry
  10. 10. Hybrid ECGA Based Efficient Cluster Optimizer 9 Hybridization . Nelder-Mead simplex (Press et al, 1989) — Requires 371. +1 initial points - The individual accounts for one point — Perturb an atom in one coordinate — Creates 312. points . Local search for every individual . Use fully lamarckian approach — Local search solution replaces the individual Illudn (icnuk lnh-tn-ry . ""“"“. ... ... ..‘. '.‘. .‘. ;.". :': '.”"“"". ... .4.: ... ... GECCO. July 7-11. 2001 L'. ;“l. t.. 'l. ’.}. ",§. ",, ‘,. ‘T. ’.', ,.. . K. Sastry
  11. 11. Hybrid ECGA Based Efficient Cluster Optimizer 10 Creating New Individuals . Create np individuals using MPM — Generate each partition independently — Assign values proportional to the frequency — rn. point crossover between rip individuals. . Elitist replacement scheme - Select top up individuals from * np new individuals, and * vn. ,, old individuals Ilbiu (Bunk kg-minim hhutuory . ‘”""“. ... ... .‘. ‘.‘. l.‘. .‘*. :'. '.‘. "“‘“"'. ... .J. :.. ... GECCO. July 7-11. 2001 L'. rt. .'l£. L’,2." . .;, ,.. . K. Sastry
  12. 12. Hybrid ECGA Based Efficient Cluster Optimizer 11 Algorithm Flowchart N (“ . ' ‘ ' Ff‘: WON h - -' nilia ize er nrm El, -alualc avcl c . . _ / N clusters Nelder-Mead poiemial energy +4 clusters Y“ End W ’ by seeding simplex search of each chm“: eonverged/ ,.. ~ :2 . _ '"_'; '., ,l “ . :.)_,1 . . N0 . [ ' v’ Replace N“Pc , Create nfiw , Build MPM Perform _: - clusters using ~ um, MDL touniaincnl K * ~ old clusters MPM ( 3 selection .1 , .4 LJ k . - 3' x - -/ V . A‘ F . _ llllmldimr _u. --rlmuniul-man l—"“"f _ ~ - l. ’:l. ".. ‘.'. '.‘. “'. ... .': :‘. '.‘. ":: ‘.‘: ‘.‘: ‘:: ... i.. . GECCO, July 7-11. 2001 11“ L'. ",£? £Jl3i7.'l.5.ZL. .i. K. Sastry
  13. 13. Hybrid ECGA Based Efficient Cluster Optimizer Results: Minimum Energy D l l l l _ xii ECGA Seeded '1-‘ % + SGA Seeded 5‘ . —1o~ = - W « S to E‘ -20- a Q 0 ii. I E i. E -30- -.1. T ‘T I is 5 . Ci I: 0 -40 — E . 9 _ E 8 1:: _ E _ _ _. 3 5° , P . 1 ‘ii _6° _ . _ _. i; : 70 _l_ —_J_— _A— _l— —J 0 5 l0 15 20 25 30 35 No of atoms. n llllmk liuullt A i. --mun» | .al-rum gumn .1('. e.m . i I. r.‘iIu1i'n; ' . '~wII| it man. u l rl-up-4 n—p-il. - (W4. l rhaln. II II mplswm . ... .‘-mi. 12 GECCO. July 7-11. 2001 K. Sastry
  14. 14. Hybrid ECGA Based Efficient Cluster Optimizer Results: Population size I 0 scan with seeding. olnml 9 SGA will seeding. oln‘ "1 , * - +— ECGA. om‘-Zl 10’ IIHIUCCNR Abflllnnlnhu-nary dficnevflllmhl LIL-uni illlhd-all In-g mu. lL|1A. Hfiallwvlv-UIl| flp. uhr. nlI 13 GECCO. July 7-11. 2001 K. Sastry
  15. 15. Hybrid ECGA Based Efficient Cluster Optimizer 14 Results: Convergence Time ‘V j, 16 l l l I l I : . - U ECGA with seeding »‘ + it SGA wllh seeding _ , -- M > - ECGA 7 . - -4 . i2— + — sn‘ . __‘ Q--‘-. ‘.. l'; ." yo E I _ . _ 8 + B x 8 -: .~ 2 h . -» .3, 31 .5 ‘,1’ ’*—; ,~‘'' -.1 n . ~ -_ 5- - Q 5 _ ~_. _ -. . i . , E »= -1 k _ / L *5 f’ ‘ av H 1: I __ , . 2 l 7' i . l l A . l 4 6 8 10 12 I4 16 18 20 ‘ * No olatoms_ n 1'. " . -— iiii. .l. i;uml« Av. --«mu» la? -orunr_ ‘ (2 ul I. ." l gi--~~r A, » - . .. .': :‘. ...7.: :‘.1'. ':_. .i. _. GECCO. July 7-11. 2001 301- L. '“, :?: '.'l. ‘.l . ‘;. L’. '.. ... K. sasiry
  16. 16. Hybrid ECGA Based Efficient Cluster Optimizer 15 Results: Function Evaluations -9- com wall seeding. oin’ ‘5l it still win seeding O(n"") - +- ECGA. Din”) II“IU(INkAflNIHI| lJNI'1I'y {". l.". .'. '.. "‘: .‘l. ... ... ‘7""". .'-f. P“. "‘D. :.. ... .. GECCO. July 7-11. 2001 k'. ‘,. i?"‘i. ... .i‘i'ii'. '.'. ‘.‘. “.‘. i‘. ‘.. .i. K. Sastry
  17. 17. Hybrid ECGA Based Efficient Cluster Optimizer Results: Scale Up I Avoraafi case. O(rf“5l Worst case. Olnz ‘S lo‘ < - acsi sac. om‘-“i No.01 Iuncllon evaluations 3. ‘O. .’x. ‘'[ IIHIUCCNR Abflllnnlnhu-nary dficnevflllmhl LIL-uni illlhd-all In-g mu. lL|1A. Hfiallwvlv-UIl| flp. uhr. nlI No. of Home n O0 16 GECCO. July 7-11. 2001 K. Sastry
  18. 18. Hybrid ECGA Based Efficient Cluster Optimizer 17 0 An efficient hybrid cluster optimizer — Solves larger clusters (Up to 40 atoms) - High reliability: 96% - Minimum population size: 0 (‘ll»0‘83) — Total No. of func. evals. : O(n? -45) . Successfully predicts global optimum 0 Iwamatsu (2000): 15 atoms . Niesse & Mayne (1996): O('ll3'3) Illa-h (hulk labor: -ry . """""‘. ... ... .'. '.‘. .‘. ':'. :'. '.“""°"“. ... .:. ... .. GECCO. July 7-11. 2001 L. ,—. l~1‘. .'1.’ili'. L". }.‘. T.; ... i. K. Sastry
  19. 19. Hybrid ECGA Based Efficient Cluster Optimizer 18 Acknowledgments (f-§j; ll! David E. Goldberg & David Ceperley iii Air Force Office of Scientific Research, Air Force 0'__‘j Materiel Command, USAF, under grant F49620-00-1-0163. a National Science Foundation under grant DMI-9908252. fl :1 a U. 5. Army Research Laboratory under the f'{ Federated Laboratory Program, Cooperative / »---‘ Agreement DAAL01—96—2-0003. GECCO. Jilly 7-11. 2001 K. Sastry
  20. 20. Hybrid ECGA Based Efficient Cluster Optimizer 19 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]), Select set with minimum combined complexity (cg). If (1,, > C], go to step 6. MPM is the set with 0;. Go to step 2. . °‘$". “.°° Merging is not possible, exit. Ilhii (Bunk hhutu-r) . ‘”"""‘. ... ... .‘. ‘.‘. .‘. ;.". :'. '.‘. ""““". ... .J. :.. ... GECCO. July 7-11. 2001 L'. rt. .'i£. },’,2.", L‘T. ’.', ,.. . K. Sastry
  21. 21. Hybrid ECGA Based Efficient Cluster Optimizer 20 Gong Potential Equations f:7"2(iij)+ i v3(il.7-wk) i'. <j i<j<k » «Kw 5 II v2(i, j) = A (Brig; -P — 1'27]-q) exp [(r, -j — a)_1] , |1=. -j| < a 1-"i('i= J}k) = h(Tji: Tki) + h(TkjlTij) + h(rik: rjk) Aexp ~i (7'i'Ta)—1+(7'k-iITa)—1 Ir. -il < a h(7‘ji, ?'ln) = [ ( J J (cos 0,-. ,:k + §)2 [(003 0,4-; ,~ + co)2 + cl] , |1:k. .,-| < a 0 A = 7.0496, B = 0.6022, a = 1.8, p = 4, q = 0 / = 25, 7 = 1.2, Co = -0.5, C1 = 0.45 iin-Mia mcniumunornory i. .l-. n‘ny.1uia. .'. u: GECCO. July 7-11. 2001 l'. II. " “HI-lL » flyarwv-1-ilIigAp. nn'—. :niu K - 535‘ fy
  22. 22. Hybrid ECGA Based Efficient Cluster Optimizer 21 Results: Single GA Run F . _ llll--lcllmrlltfb-tlmnnlulnruan l—"“"’ , ~ 1 l. ;:‘. f:‘l': f:: .‘: ‘2f_. ... , GECCO. July 7-11. 2001 uni. :, ,'; ;:; ''‘'*'_; _*, ,;9_-, ,_, _ K. sasiry
  23. 23. Hybrid ECGA Based Efficient Cluster Optimizer Results: Optimal Stru ctu res ' ‘~. / , ‘‘i_‘_‘ i Teiralicdmn . .‘. .‘__a'. . ‘ [I : —4.00|6 > H _ Uniczipped disioned ~’ ‘ pcniaciinal til yramid U: -I 1.5. 46 F . lllln--lditmllt K‘: -dlnirn I. -h-nun - Ihv-(hull . :l. c.. -ml I --mm . i.. l -' “ " f ‘ ‘ - i. ..i-mm .4 mi. .. :1 I rlnla-4 -. -pulp N -—. _ [Hindu II AINOI I‘§. Illpzlhvv-clIIrl. u'. uiu: .nlI Conlpncsscd lri anal hip)'l’i1ll1ld. U = - .75|8 ‘ 2 h . . Pcnizigon. -ll blpy rumid _ U : -_9.624() C‘ if I Q0 / _l_ _ _/ ‘N ‘L; K 2 lr V‘ . _H O l ‘W ’ 0 Kay " '5 rm‘ 0 [x] l, ,'l . K _/ ' xx "-—-‘ If 1 / "“w i . - . V. r -—-' 2 -1- Biciippcd lclrimonal Ti'ii: :ip ~dlri until 8‘ uniipriuu. ll : - 75.6137 prixili, = -l. .5999 22 GECCO. July 7~l1. 2001 K. Sastry

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