02 dr.r.saravanan - bits
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EvolutionaryMulti-Objective Optimization
Dr .R.Saravanan
Professor and Head
Department of Mechanical Engineering
Bannari Amman Institute of Technology
Compiled from the lecture notes of
Dr. Kalyanmoy Deb
Professor/Mechanical Engineering
I.I.T, Kanpur
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Properties ofPractical Optimization ProblemsNon-differentiable functions and constraints
Discontinuous search space
Discrete search space
Mixed variables (discrete, continuous, permutation)Large dimension (variables, constraints, objectives)
Non-linear constraints
Multi-modalities
Multi-objectivity
Uncertainties in variables
Computationally expensive problems
Multi-disciplinary optimization
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Different Problem Complexities
Mixed variables
Multi-modal Robust solution
Reliablesolution
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Remarks onClassical Optimization Methods
One method not applicable in many problems
Constrained handling sensitive to penalty parameters
Not efficient in handling discrete variablesLocal perspective
Uncertainties in decision and state variables
Noisy/dynamic problems
Multiple objectivesParallel computing
Need for innovative and flexible optimization algorithm
Difficulties:
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Evolutionary Optimization:A Motivation from Nature
Natural evolution +genetics
Guided search procedure
Offspring are created byduplication, mutation,crossover etc.
Good solutions areretained and bad aredeleted
Information is coded
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Computational Intelligence andEvolutionary Algorithms (EAs)
We treat an EA here as a search and optimization tool
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Multi-Objective Optimization
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Mathematical ProgrammingProblem
Multiple objectives, constraints, andvariables
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An Engineering Example
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Which Solutions are Optimal?
Relates to the concept ofdomination
x(1) dominates x(2), if
x(1) is no worse than x(2) inall objectives
x(1) is strictly better than x(2)
in at least one objective
Examples:3 dominates 2
3 does not dominate 5
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Pareto-Optimal Solutions
P=Non-dominated(P)Solutions which are notdominated by any member
of the set PO(N log N) algorithmsexist
Pareto-Optimal set = Non-
dominated(S)A number of solutions areoptimal
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Pareto-Optimal Fronts
Depends on thetype ofobjectives
Always on theboundary offeasible region
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Local Versus Global Pareto-Optimal Fronts
Local Pareto-optimal Front: Domination check is
restricted within a neighborhood (in decision space)of P
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Preference-Based Methods
fgf
Classical Methods follow itResults in a single
solution in each simulation
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Classical Approaches
No Preference methods (heuristic-based)
Posteriori methods (generating solutions)discussed later
A-priori methods (one preferred solution)Interactive methods (involving a decision-maker)
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Classical Approach:
Weighted Sum MethodConstruct a weightedsum of objectives andoptimize
User supplies weightvector w
1
( ) ( )M
i i
i
F x w f x
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Difficulties with Weighted-SumMethod
Need to know w
Non-uniformity in Pareto-optimal solutions
Inability to find somePareto-optimal solutions(those in non-convexregion)
However, a solution ofthis approach is Pareto-optimal
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-Constraint Method
Constrain all but oneobjective
Need to know relevant
vectorsNon-uniformity inPareto-optimal solutions
However, any Pareto-
optimal solutions can befound with this approach
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Difficulties with Most ClassicalApproaches
Need to run a single-objectiveoptimizer many times
Expect a lot of problem
knowledgeEven then, good distributionis not guaranteed
Multi-objective optimizationas an application of single-objective optimization
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Classical Generating Methods
One-at-a-time and repeat
Population approaches
Timmels method
Schafflers method
Absence of parallelsearch is a drawback
EMO finds multiplesolutions with an implicitparallel search
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Ideal Multi-Objective
Optimization
Step 1 :
Find a set ofPareto-optimal
solutions
Step 2 :
Choose one fromthe set
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Ideal Multi-Objective
Optimization
Step 1 :
Find a set ofPareto-optimal
solutions
Step 2 :
Choose one fromthe set
Decision making
Optimization
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Two Goals in Ideal Multi-ObjectiveOptimization
Converge to thePareto-optimal front
Maintain as diverse a
distribution as possible
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Evolutionary Multi-ObjectiveOptimization (EMO)
Principle:
Find multiple Pareto-optimal solutionssimultaneously
Three main reasons:
Help in choosing a particular solutionUnveil salient optimality properties of solutions
Assist in other problem solving
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Why Use Evolutionary
Algorithms?
Population approach suitswell to find multiple solutions
Niche-preservation methodscan be exploited to finddiverse solutions
Implicit parallelism helpsprovide a parallel search
Multiple applications of classical methods do notconstitute a parallel search
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History of Evolutionary Multi-Objective Optimization (EMO)
Early penalty-basedapproachesVEGA (1984)Goldberg's (1989)
suggestionMOGA, NSGA, NPGA(1993-95) usedGoldbergs suggestion
Elitist EMO (SPEA,NSGA-II, PAES,MOMGA etc.) (1998 --Present) Main studies after 1993
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What to Change in a Simple
GA?
Modify the fitnesscomputation
Emphasize non-dominated solutions forconvergence
Emphasize less-crowdedsolutions for diversity
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Non-Dominated Sorting: A NaiveApproach
Identify the best non-dominatedset
Discard them from population
Identify the next-best non-dominated set
Continue till all solutions are
classified
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A Fast Non-Dominated Sorting
Calculate (ni,Si) foreach solution i
ni: Number of
solutions dominating iSi: Set of solutionsdominated by I
Follow an iterative
procedureA faster procedurelater in Lecture L6
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Which are Less-CrowdedSolutions?
Crowding can be in decision variable space or inobjective space
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Non-Elitist EMO Procedures
Vector evaluated GA (VEGA) (Schaffer, 1984)
Vector optimized EA (VOES) (Kursawe, 1990)
Weight based GA (WBGA) (Hajela and Lin, 1993)
Multiple objective GA (MOGA) (Fonseca andFleming, 1993)
Non-dominated sorting GA (NSGA) (Srinivas andDeb, 1994)
Niched Pareto GA (NPGA) (Horn et al., 1994)Predator-prey ES (Laumanns et al., 1998)
Other methods: Distributed sharing GA,neighborhood constrained GA, Nash GA etc.
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Sharing Function
Goldberg and Richardson(1997)
d is a distance measurebetween two solns.
Phenotypic distance: d(xi,xi), x:variable
Genotypic distance:
d(si,si), s: string
Calculate niche count,nci=jSh(dij)
Shared fitness: fi=fi/nciUse proportionate selectionopeartor
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Shortcomings of Non-Elitist EMOProcedures
Elite-preservation is missing
Elite-preservation is important for properconvergence in single-objective EAs
Same is true in EMO proceduresThree tasks
Elite preservation
Progress towards the Pareto-optimal frontMaintain diversity among solutions
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Elitist EMOs
Distance-based Pareto GA (DPGA) (Osyczka andKundu, 1995)Thermodynamical GA (TDGA) (Kita et al., 1996)
Strength Pareto EA (SPEA) (Zitzler and Thiele, 1998)Non-dominated sorting GA-II (NSGA-II) (Deb et al.,1999)Pareto-archived ES (PAES) (Knowles and Corne,1999)
Multi-objective Messy GA (MOMGA) (Veldhuizen andLamont, 1999)Other methods: Pareto-converging GA, multi-objectivemicro-GA, elitist MOGA with co-evolutionary sharing
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Elitist Non-dominated Sorting
Genetic Algorithm (NSGA-II)
NSGA-II can extractPareto-optimal frontier
And find a well-distributed set ofsolutions
Code downloadable
http://www.iitk.ac.in/kangal/soft.htm
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NSGA-II Procedure
Elites are preservedNon-dominated solutions are emphasized
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