02 dr.r.saravanan - bits

8/4/2019 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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