Omni-Optimizer A Procedure for Single and Multi-objective Optimization

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Omni-OptimizerA Procedure for Single and
Multi-objective Optimization

• Prof. Kalyanmoy Deb and
• Santosh Tiwari

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Motivation

• Generic Programming Practices
• Unified algorithm for all types of optimization
  problems
• An efficient implementation of NSGA-II framework
  (procedure)
• Developing an efficient and self-adaptive
  optimization paradigm

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Literature Survey

• CHC (Cross generation elitist selection,
  Heterogeneous recombination, Cataclysmic
  mutation) Explicit Diversity
• GENITOR (Steady state GA), more like (µ1)-ES so
  far as selection mechanism is concerned. High
  selection pressure
• NPGA (Niched Pareto Genetic Algorithm), uses
  sharing parameter sshare of niches obtained
  depend on the sharing parameter

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Literature Survey contd

• PESA (Pareto Envelope-based Selection Algorithm),
  Hyper-grid division of phenotypic space,
  selection based on crowding measure
• NSGA-II (Non-dominated Sorting Genetic Algorithm)
• SPEA2 (Strength Pareto Evolutionary Algorithm),
  Fine grained fitness assignment mechanism
  utilizing density information, Only archive
  members participate in mating Excellent
  Diversity in phenotypic space
• NCGA (Neighborhood Cultivation Genetic
  Algorithm), used neighborhood crossover, based on
  NSGA-II and SPEA2
• RPSGAe (Reduced Pareto Set Genetic Algorithm with
  elitism)
• ENORA (Evolutionary Algorithm of Non-dominated
  Sorting with Radial Slots)

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Salient Features of the Algorithm

• Based on NSGA-II framework
• Based on the concept of Pareto dominance
• Incorporates elitism
• Explicit diversity preserving mechanism
• Can be used for single-objective as well as
  multi-objective problems
• Can be used for uni-global as well as
  multi-global problems
• Independent of the number of niches that an
  optimization problems exhibits

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Moving beyond NSGA-II

• Restricted Selection Scheme
• Tournament selection based on usual domination
• Non-dominated sorting based on epsilon dominance
• Crowding Distance Assignment
• Genotypic as well as Phenotypic space niching
• Choose best members from above average population
• Remove worst members from below average
  population
• More robust recombination and variation operators
• Two point crossover for binary variables
• Highly disruptive real variable mutation

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Restricted Selection

• Helps in preserving multi-modality
• Experiments show that it gives faster overall
  convergence

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Epsilon Domination Principle

• A finite percentage (based on function value) of
  solutions assigned a particular rank
• Allows somewhat inferior solutions to remain in
  the population
• Provides guaranteed diversity
• Helps to obtain multi-modal solutions in case of
  single objective problems
• Epsilon is generally taken in the range 10-3
  10-6

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Modified Crowding Distance

• Genotypic as well as Phenotypic space niching

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Highly Disruptive Mutation Operator
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Computational Complexity

• Restricted selection O (nN2)
• Ranking procedure O (MN2)
• Crowding procedure
• max O (MN log N), O (nN log N)
• Overall iteration-wise complexity
• max O (nN2), O (MN2), O (nN log N)

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Implementation Details

• Code written in simple C and strictly conforms to
  ANSI/ISO standard
• Data structure used is a custom doubly linked
  list (gives O(1) insertion and deletion)
• Randomized quick sort used for sorting
• Almost all the functions perform in-place
  operation (addresses are passed, significantly
  decreases stack overheads)

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Simulation Results

• GA parameters for all problems chosen as follows
• ?c 20
• ?m 20
• P (crossover) 0.8
• P (mutation) 1/n, where n of real variables
• d 0.001
• Population size and number of generations
  different for different problems

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Simulation Results contd

• 20 variable Rastrigin function
• of function evaluation
• Least 19260
• Median 24660
• Worst 29120
• 20 variable Schwefel function
• of function evaluation
• Least 54950
• Median 69650
• Worst 103350
• Other single objective problems can be found in
  the paper
• In all cases, better results are found in
  comparison to previous reported studies

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Single objective multi-modal function

• f(x) sin2 (px) x ? 0,20

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Single objective multi-modal function

• Unconstrained Himmelblaus function

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Multi-objective Uni-Global Test Problems

• 30 variable ZDT2 (100100)

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Multi-objective Uni-Global Test Problems

• 10 variable ZDT4 (100250)

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Multi-objective Uni-Global Test Problems

• CTP4 (1007000)

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Multi-objective Uni-Global Test Problems

• CTP8 (100100)

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Multi-objective Uni-Global Test Problems

• DTLZ4 (300100)

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Multi-objective Multi-Global Test Problem

• F1 (x) summation (sin (pxi) ) xi ? 0,6
• F2 (x) summation (cos (pxi) ) xi ? 0,6

Efficient points in phenotypic space
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Multi-objective Multi-Global Test Problem

• Genotypic space plots

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Few Sample Simulations

• F(x) sin2 (10,000pix)
• Himmelblaus Functions
• ZDT Test Problems
• CTP Test Problems
• Test Problem TNK
• Multi-global Multi-objective Test Problem

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Further Ideas and Future Work

• Incorporating PCX instead of SBX for crossover
• Automatically fine-tuning mutation index so as to
  achieve arbitrary precision
• Self-adaptation of parameter d
• Segregating population into niches without the
  introduction of DM
• Dynamic population sizing
• Using hierarchical NDS for the crowding distance
  assignment

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