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Iterative Prototype Optimisation with Evolved Improvement Steps

Iterative Prototype Optimisation with Evolved Improvement Steps. Ji ří Kubalík and Jan Faigl Czech Technical University Department of Cybernetics Technicka 2, 166 27, Prague, Czech Republic email: {kubalik,xfaigl}@labe.felk.cvut.cz. Motivation. Typically ,

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Iterative Prototype Optimisation with Evolved Improvement Steps

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  1. Iterative Prototype Optimisation with Evolved Improvement Steps Jiří Kubalík and Jan Faigl Czech Technical University Department of Cybernetics Technicka 2, 166 27, Prague, Czech Republic email: {kubalik,xfaigl}@labe.felk.cvut.cz

  2. Motivation • Typically, • evolutionary optimisation framework considers an evolution of a population of candidate solutions to the problem at hand • candidate solution encodes a complete solution (a complete set of problem control parameters, a complete schedule in JSP, a complete tour for TSP, etc.) • the optimal or well-fit solution is hard to find for large problem instances • Here, • EA does not handle the solved problem as a whole • EA is employed within the iterative optimisation framework • its role is to evolve the best modification of the current solution prototype in each iteration

  3. Outline of POEMS algorithm generate(Prototype); repeat BestSequence  run_EA(Prototype); if(apply_to(BestSequence, Prototype) is_better_thanPrototype); thenPrototype  apply_to(BestSequence, Prototype); until(POEMS termination condition); returnPrototype; • Prototype initialised randomly or using some problem-specific knowledge

  4. Implementation Issues • Action Sequence Representation • Actions – set of primitive actions • determines the explorative power of the algorithm • linear chromosomes of maximal length MaxGenes << ProblemSize • gene = (action_type, parameters) • nop(no operation) • void action with no effect on the prototype, regardless of the values of its parameters • one or more nop actions allowed in a chromosome • variable effective length of chromosomes • At least one action must be active Ex.: MaxGenes =5, Actions ={action1, action2, action3} Ch = (action3, 0.8),(nop, 0.7),(action2, 0.2),(nop, 0.5),(action2, 0.2)

  5. Implementation Issues • Genetic Operators • Tournament selection • Crossover – generalised uniform • Given two parental chromosomes, any combination of their actions forms a valid offspring Ex.: Par1 = (action3, 0.8),(nop, 0.7),(action1, 0.2),(nop, 0.5),(action2, 0.2) Par2 = (nop, 0.8),(action1, 0.8),(action3, 0.6),(nop, 0.5),(nop, 0.5) Off1 = (action1, 0.2),(action3, 0.6),(nop, 0.7),(action1, 0.2),(nop, 0.5) Off2 = (nop, 0.8),(action2, 0.2),(action3, 0.6),(nop, 0.5),(action_1, 0.8) • Mutation – action_type or parameters changed (1 gene per sequence) • Optionally, multiple duplicates of the same genes are eliminated Off1 = (action1, 0.2),(action3, 0.6),(nop, 0.7),(action1, 0.2),(nop, 0.5) Off1’ = (action1, 0.2),(action3, 0.6),(nop, 0.7),(nop, 0.2),(nop, 0.5)

  6. Implementation Issues • Evaluation criterion • maximizes improvement of the prototype • penalizes sequences that do not change the prototype • Evolutionary Model • EA should converge fast, since it runs many times through the whole run • Generational / Steady-state

  7. Standard Generational EA initialize(OldPop); BestSequence  best_of(OldPop); repeat NewPop  BestSequence; // elitism repeat Parents  select(OldPop); Children  cross_over(Parents); mutate(Children); evaluate(Children); NewPop  Children; until(NewPop is completed); BestSequence= best_of(NewPop); switch(OldPop, NewPop); until(EA termination condition); return BestSequence;

  8. Steady-state EA initialize(Population); repeat Parents  select(Population); Children  cross_over(Parents); mutate(Children); evaluate(Children); Replacement = find replacement(Population); Population[Replacement]  Child1; Replacement = find replacement(Population); Population[Replacement]  Child2; until(EA termination condition); BestSequence  best of(Population); return BestSequence;

  9. TSP: Experimental Setup • Primitive Actions • move(city1, city2) moves city1 right after city2 in the tour • invert(city1, city2) inverts a subtour between city1 and city2 • swap(city1, city2) swaps city1 and city2 • POEMS configuration: • chromosome length: 10 • population size: Cities ×2 • EA evaluations: 10.000 • POEMS evaluations: Cities ×1000 • Other algorithms • parameters set as recommended • use the same number of fitness evaluations, if possible

  10. Heuristic used for tour initialisation • Only links between cities from specified neighborhood are considered when generating • the prototype in POEMS, • tours form the initial population (E-R), and • the initial tour for 2-opt. • No heuristic used for initialising the ring of neurons for SOM

  11. 100 cities 200 cities 1000 cities 2000 cities 500 cities TSP: Test Datasets • TSP Data – uniformly sampled points

  12. POEMS: Random vs. Heuristic Init. • 100 cities 4738.5  831.9 1029.4  818.6

  13. Random vs. Heuristic initialization • 200 cities 9574.2  1190.1 1357.4  1132.9

  14. Random vs. Heuristic initialization • 500 cities 25520.1  6234.9 2134.1  1746.3

  15. Random vs. Heuristic initialization • 1000 cities 51117.9  17997.0 2946.8  2523.0

  16. Random vs. Heuristic initialization • 2000 cities 102331.6  34878.0 4344.5  3692.2

  17. Example of POEMS execution: TSP Initial tour 965.134 after iteration 1 after iteration 2 after iteration 3 Final tour 824.8

  18. Example of Evolved Solution Initial prototype of length 4403 Initial prototype of length 3655

  19. POEMS using Single Elem. Function

  20. Deceptive function– 100 bits • Hierarchical If and only If • Ch_length = 128 bits • Royal Road Problem • Ch_length = 96 bits Binary String Opt.: Test Problems • OneMax – counts 1s in a chromosome • Ch_length = 100 • Rosenbrock • Ch_length = 4 × 2 × 12 • F(x’1, x’2)< 0.001 is optimum • F103 • Ch_length = 5 × 2 × 10 • F(x’1, x’2)< 0.001 is optimum

  21. Binary String Opt.: Experimental Setup • Primitive Actions • invert(gene) … inverts the specified gene of the prototype • Linkage • tight – groups of dependent genes form compact clusters within the chromosome • loose – dependent genes are loosely coupled within the chromosome • POEMS configuration: • chromosome length: 10 • population size: 200 • EA evaluations: 3.000 • POEMS evaluations: 106 • Evolutionary model: generational, steady-state • Other algorithms • parameters set as recommended • use the same number of fitness evaluations, if possible

  22. Results: Loose Linkage

  23. Real-valued Opt.: Implementation Issues • Primitive Actions • add(value) … adds value(maxixi) to the respective prototype variable xi, where maxi is an upper bound of variable xi. • value is initialised from interval (0.0, 1.0). • sub(value) … subtracts value(ximini) from xi. • sample_right(value) … picks a new value of the respective prototype variable xi, from interval (xi , maxi). • samples the interval (xi , maxi) non-linearly preferring smaller changes to the larger ones. • sample_left(value) … picks a new value of the respective prototype variable xi, from interval (mini , xi). • Chromosome structure – an ordered list of D actions, each of them operating on the respective prototype variable, where D is a dimension of the problem.

  24. Real-valued Opt.: Implementation Issues • Reuse of the evolved code • Initial population of EA in later iterations a portion of action from the previous EA • to keep the search direction that appeared beneficial in the past, • to reuse the unexploited yet possibly useful material evolved in the last iteration. • POEMS configuration: • chromosome length: D • population size: 200 • EA evaluations: 1500 (10D), 2000 (30D) • POEMS evaluations: D×104

  25. Real-valued Opt.: Experiments • 10D and 30D functions collected for the Special Session on Real-parameter Optimization of the IEEE Congress on Evolutionary Computation 2005. • 25 functions in 4 groups • P. N. Suganthan, N. Hansen, J. J. Liang, K. Deb, Y.-P. Chen, A. Auger, and S. Tiwari,“Problem definitions and evaluation criteria for the CEC 2005 Special Session onReal-Parameter Optimization”, tech. rep., Nanyang Technological University, Singapore,May 2005. http://www.ntu.edu.sg/home/epnsugan • POEMS compared to the algorithms presented at the Special Sesion. • Observations: • POEMS exhibits about-average performance among the compared alg. • It is competitive to most of the algorithms on multimodal prob. • Improvement step size tends to decrease during the course of the run

  26. F1 Shifted Sphere Function

  27. F9 Shifted Rastrigin’s Function

  28. 13 Shifted Expanded Griewank’s plus Rosenbrock’s Function (F8F2)

  29. 23 Non-Continuous Rotated Hybrid Composition Function 3

  30. Conclusions • Pros and Cons • General optimisation approach for combinatorial, discrete, and real-valued optimisation problems • Linkage independent • Inability to treat problems with building blocks of bigger size • Local (point-to-point) search approach – prone to get stuck in a local optimum • The latter stage of the iterational algorithm is the harder it is to find any improvement modification ( local search) • Scope of possible applications • Rescheduling – insert a new order into the current schedule • SAT problem – dependencies among variables are necessary for effective searching for the optimum solution (but not known a priori)

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