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

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

    1. 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
    2. 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
    3. 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
    4. 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
    5. 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
    6. 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
    7. 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
    8. 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
    9. 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
    10. 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
    11. 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
    12. 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
    13. 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
    14. 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
    15. 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
    16. 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
    17. 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
    18. 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
    19. 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
    20. 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
    21. 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
    22. 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
    23. 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

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