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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 ...

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

    • 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