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MOEAs

MOEAs. University of Missouri - Rolla Dr. T’s Course in Evolutionary Computation Matt D. Johnson November 6, 2006. Main Topics. Multi Objective Evolutionary Algorithms Example Problem – ZDT1 T-MOEA Preliminary Results. MOEAs. Multi Objective Evolutionary Algorithms

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MOEAs

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  1. MOEAs University of Missouri - Rolla Dr. T’s Course in Evolutionary Computation Matt D. Johnson November 6, 2006

  2. Main Topics • Multi Objective Evolutionary Algorithms • Example Problem – ZDT1 • T-MOEA • Preliminary Results

  3. MOEAs • Multi Objective Evolutionary Algorithms • Based on the concept of the standard EA, but with multiple objectives to optimize • Some of the objectives may conflict with one another

  4. Fitness vs. Dominance • In a standard EA, an individual A is said to be better than an individual B if A has a higher fitness value than B • In a MOEA, an individual A is said to be better than an individual B if AdominatesB (Deb)

  5. Dominance • A solution x1 is said to dominate a solution x2 if both conditions below are true: • The solution x1 is no worse than x2 in all objectives • The solution x1 is strictly better than x2 in at least one objective (Deb) • If x1 dominates x2 (x1 ≤ x2), it can be said that • x2 is dominated by x1 • x1 is non-dominated by x2 • x1 is non-inferior to x2 (Deb)

  6. Pareto Optimality • Non-dominated set: Among a set of solutions P, the non-dominated set of solutions P’ are those that are not dominated by any member of the set P (Deb) • Globally Pareto-optimal set: The non-dominated set of the entire feasible search space S is the globally Pareto-optimal set (Deb)

  7. Deterioration • Deterioration occurs when individuals in the current solution are dominated by individuals that existed in the solution set previously, but have since been terminated • Even “elitist” algorithms can suffer from this condition

  8. Defeating Deterioration • Efficiency Preservation: Property of accepting a new individual only if it dominates an existing individual in the population • Negative Efficiency Preservation: An individual is deleted from the population only if it is being replaced by a superior individual.

  9. Example Problem: ZDT1 • Zitzler-Deb-Thiele’s Test Problems • Minimize f1(x) • Minimize f2(x) = g(x) h(f1(x), g(x))

  10. ZDT1: Pareto Optimal Solution

  11. MOEA Goals • Find the Globally Pareto-Optimal set of solutions • Would like as many solutions as possible • Even distribution of solutions

  12. NSGA-II • A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II • Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, T. Meyarivan • Each generation bounded by a fast nondominated sort - O(MN2)

  13. NSGA-II • Steps before main loop begins • Initialize population P0 • Sort P0 on the basis of nondomination • Fitness of an individual is equal to an individuals nondomination level • Binary Tournament Selection • Mutation and recombination create an offspring population Q0

  14. NSGA-II • Primary loop • Rt = Pt + Qt (t = generation) • Sort Rt on the basis of nondomination • Create Pt+1 by adding individuals from each level of Rt until Pt+1 is of size N • Create Qt+1 by applying Binary Tournament Selection, Mutation, and Recombination to Pt+1 • Increment t

  15. Epsilon MOEA • A Fast Multi-objective Evolutionary Algorithm for Finding Well-Spread Parto-Optimal Solutions • Kalyanmoy Deb, Manikanth Mohan and Shikar Mishra • Steady State, Elitist, Does not suffer from deterioration • Uses epsilon dominance (fuzzy dominance)

  16. Epsilon MOEA • Uses two populations • An individual is only deleted when it is replaced with a superior individual • Uses an identification array B which consists of abbreviated objective values, creating containers – only one individual may occur in a container

  17. Motivation for T-MOEA • Faster – avoid duplicating work (sorting the population again and again, even when it has not changed that much) • Simpler – existing algorithms are difficult to explain and understand • Object Oriented – existing algorithms can be implemented in an OO manner, but are not specifically designed that way

  18. Motivation for T-MOEA • Greater flexibility • Multiple selection approaches • Control size and number of nondomination levels • Elitist • No deterioration

  19. T-MOEA Algorithm • Initialization • while termination condition is false do • Parent Selection • Recombination and Mutation • Insertion • end while

  20. Data Structure • Most EAs and MOEAs use a “pool” of individuals • T-MOEA calls for a vector of Binary Search Trees • No duplicate individuals allowed

  21. Selection – select1 • Parents will be selected from the top level • If all individuals in a level have been selected and more are needed, the next lower level is used • If more parents are needed and all levels have been exhausted, selection is random in the entire population

  22. Selection – select2 • Designed for higher selection pressure • All parents are selected from the top level

  23. Selection – select3 • Randomly select individuals in the population

  24. Recombination and Mutation • Problem specific • Operations are handled within the “Individual” object • Will sometimes use a mutation size that decreases as the number of generations increases

  25. Insert • Check to see if the new individual x belongs in the top level • If x strongly dominates, insert x into a new level above • If x weakly dominates, remove dominated individuals and insert • If x is equivalent, insert • Else, try next lower level

  26. Objective Truncation • Suppose each objective function value is a real number in the range [0, 1] • When an individual is inserted into the data structure, it is compared using less than and equivalency operators • These operators will multiply the objective values by say, 100, and then truncate before doing the comparison

  27. Tree Behavior • Duplicate individuals are not allowed in a tree • Standard Binary Search Tree • Insertion depends on less than and equivalency operators of the individual object

  28. Metrics - Convergence • Calculate a Pareto Optimal solution set • Calculate the distance from an individual in the population to the closest individual in the solution set • The convergence metric is the average of these distances

  29. Metrics - Diversity • Draw a Voronoi Diagram of a population • The Voronoi Diagram will “Decompose space into regions around each point such that all the points in the region around pi are closer to pi than they are to any other point in S” • In a perfectly distributed solution, region will have the same area • The standard deviation of the areas is the metric

  30. ZDT1 Results

  31. Parameters • PopSize:100 • LevelSize: 100 • NumLevels: 4 • NumChildren: 100 • NumGens: 200 • Select1: true • F1_FACTOR: 100 • F2_FACTOR: 500

  32. Questions?

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