Transcript Omni-Optimizer A Procedure for Single and Multi

Omni-Optimizer
A Procedure for Single and
Multi-objective Optimization
Prof. Kalyanmoy Deb and
Santosh Tiwari
Motivation
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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
Literature Survey
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CHC (Cross generation elitist selection,
Heterogeneous recombination, Cataclysmic
mutation) – Explicit Diversity
GENITOR (Steady state GA), more like (µ+1)-ES
so far as selection mechanism is concerned. –
High selection pressure
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NPGA (Niched Pareto Genetic Algorithm), uses
sharing parameter σshare – # of niches obtained
depend on the sharing parameter
Literature Survey contd…
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PESA (Pareto Envelope-based Selection Algorithm), Hypergrid 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
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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)
Salient Features of the Algorithm
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Based on NSGA-II framework
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Based on the concept of Pareto dominance
Incorporates elitism
Explicit diversity preserving mechanism
Can be used for single-objective as well as multiobjective problems
Can be used for uni-global as well as multiglobal problems
Independent of the number of niches that an
optimization problems exhibits
Moving beyond NSGA-II
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Restricted Selection Scheme
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Crowding Distance Assignment
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Tournament selection based on usual domination
Non-dominated sorting based on epsilon dominance
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
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Two point crossover for binary variables
Highly disruptive real variable mutation
Restricted Selection
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Helps in preserving multi-modality
Experiments show that it gives faster overall
convergence
Epsilon Domination Principle
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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
Modified Crowding Distance
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Genotypic as well as Phenotypic space niching
Highly Disruptive Mutation Operator
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x
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p
 xl 
 xu  xl 
, 2
x
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u
 xp 
 xu  xl 
1
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m 1  1
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 m  1
2
r
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(1
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2
r
)*(1
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)
if r  0.5
1
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q  
1
1   2(1  r )  2(r  0.5)*(1   )m 1 m 1 otherwise
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xc  x p   q ( x p  xl )
Computational Complexity
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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)}
Implementation Details
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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)
Simulation Results
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GA parameters for all problems chosen as follows
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ηc = 20
ηm = 20
P (crossover) = 0.8
P (mutation) = 1/n, where n = # of real variables
δ = 0.001
Population size and number of generations different for
different problems
Simulation Results contd…
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20 variable Rastrigin function
 # of function evaluation
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20 variable Schwefel function
 # of function evaluation
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Least = 19260
Median = 24660
Worst = 29120
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
Single objective multi-modal function
f(x) = sin2 (πx)
x є [0,20]
Single objective multi-modal function
Unconstrained Himmelblau’s function
Multi-objective Uni-Global Test Problems
30 variable ZDT2 (100×100)
Multi-objective Uni-Global Test Problems
10 variable ZDT4 (100×250)
Multi-objective Uni-Global Test Problems
CTP4 (100×7000)
Multi-objective Uni-Global Test Problems
CTP8 (100×100)
Multi-objective Uni-Global Test Problems
DTLZ4 (300×100)
Multi-objective Multi-Global Test Problem
F1 (x) = summation (sin (πxi) )
F2 (x) = summation (cos (πxi) )
xi є [0,6]
xi є [0,6]
Efficient points in phenotypic space
Multi-objective Multi-Global Test Problem
Genotypic space plots
Few Sample Simulations
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F(x) = sin2 (10,000*pi*x)
Himmelblau’s Functions
ZDT Test Problems
CTP Test Problems
Test Problem TNK
Multi-global Multi-objective Test Problem
Further Ideas and Future Work
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Incorporating PCX instead of SBX for
crossover
Automatically fine-tuning mutation index
so as to achieve arbitrary precision
Self-adaptation of parameter δ
Segregating population into niches without
the introduction of DM
Dynamic population sizing
Using hierarchical NDS for the crowding
distance assignment