Evolutionary Multi-objective Optimization – A Big Picture

Karthik Sindhya, PhDPostdoctoral Researcher

Industrial Optimization GroupDepartment of Mathematical Information Technology

Karthik.sindhya@jyu.fihttp://users.jyu.fi/~kasindhy/

Objectives

The objectives of this lecture are to:1. Discuss the transition: Single objective optimization

to Multi-objective optimization2. Review the basic terminologies and concepts in use

in multi-objective optimization3. Introduce evolutionary multi-objective optimization4. Goals in evolutionary multi-objective optimization5. Main Issues in evolutionary multi-objective

optimization

Reference

• Books:– K. Deb. Multi-Objective Optimization using

Evolutionary Algorithms. Wiley, Chichester, 2001.

– K. Miettinen. Nonlinear Multiobjective Optimization. Kluwer, Boston, 1999.

Transition

Single objective: Maximize Performance

Maximize: Performance

Min

imiz

e: C

ost

• Multi-objective problem is usually of the form:Minimize/Maximize f(x) = (f1(x), f2(x),…, fk(x))

subject to gj(x) ≥ 0

hk(x) = 0

xL ≤ x ≤ xU

Basic terminologies and concepts

Multiple objectives, constraints and decision variables

Decision space Objective space

• Concept of non-dominated solutions:– solution a dominates

solution b, if• a is no worse than b in

all objectives• a is strictly better than

b in at least one objective.

Basic terminologies and concepts

1 2

3

4

f1 (minimize)

f 2 (m

inim

ize)

2 4 5 6

2

3

5

• 3 dominates 2 and 4• 1 does not dominate 3 and 4• 1 dominates 2

• Properties of dominance relationship– Reflexive: The dominance relation is not reflexive.

• Since solution a does not dominate itself.

– Symmetric: The dominance relation is not symmetric.• Solution a dominates b does not mean b dominated a.• Dominance relation is asymmetric.• Dominance relation is not antisymmetric.

– Transitive: The dominance relation is transitive. • If a dominates b and b dominates c, then a dominates c.

• If a does not dominate b, it does not mean b dominates a.

Basic terminologies and concepts

• Finding Pareto-optimal/non-dominated solutions– 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.• If the set of solutions considered is the entire feasible

objective space, P’ is Pareto optimal.

– Different approaches available. They differ in computational complexities.• Naive and slow

– Worst time complexity is 0(MN2).

• Kung et al. approach– O(NlogN)

Basic terminologies and concepts

• Kung et al. approach– Step 1: Sort objective 1

based on the descending order of importance.• Ascending order for

minimization objective

Basic terminologies and concepts

1 2

3

4

f1 (minimize)

f 2 (m

inim

ize)

2 4 5 6

2

3

5

P = {5,1,3,2,4}

5

Basic terminologies and concepts

P = {5,1,3,2,4}

T = {5,1,3} B = {2,4}

{5,1} {3} {2} {4}

Front = {5} Front = {2,4}

Front(P) = {5}

{5} {1}

Front = {5}

• Non-dominated sorting of population– Step 1: Set all non-dominated fronts Pj , j = 1,2,… as

empty sets and set non-domination level counter j = 1

– Step 2: Use any one of the approaches to find the non-dominated set P’ of population P.

– Step 3: Update Pj = P’ and P = P\P’.– Step 4: If P ≠ φ, increment j = j + 1 and go to Step 2.

Otherwise, stop and declare all non-dominated fronts Pi, i = 1,2,…,j.

Basic terminologies and concepts

Basic terminologies and concepts

5

1 2

3

4

f1 (minimize)

f 2 (m

inim

ize)

Front 1

Front 2

Front 3

f1 (minimize)

f 2 (m

inim

ize)

• Pareto optimal fronts (objective space)– For a K objective problem, usually Pareto front is K-1 dimensional

Basic terminologies and concepts

Min-Max Max-Max

Min-Min Max-Min

• Local and Global Pareto optimal front– Local Pareto optimal front: Local dominance check.

– Global Pareto optimal front is also local Pareto optimal front.

Basic terminologies and concepts

Decision spaceObjective space

Locally Pareto optimal front

• Ideal point: – Non-existent – lower bound of the Pareto front.

• Nadir point: – Upper bound of the Pareto front.

• Normalization of objective vectors:– fnorm

i = (fi - ziutopia )/(zi

nadir - ziutopia )

• Max point:– A vector formed by the maximum objective

function values of the entire/part of objective space.

– Usually used in evolutionary multi-objective optimization algorithms, as nadir point is difficult to estimate.

– Used as an estimate of nadir point and updated as and when new estimates are obtained.

Basic terminologies and concepts

Min-Min

Zideal

Znadir

Zmaximum

Zutopia

ε

ε

Objective space

f1

f 2

• What are evolutionary multi-objective optimization algorithms?– Evolutionary algorithms

used to solve multi-objective optimization problems.

• EMO algorithms use a population of solutions to obtain a diverse set of solutions close to the Pareto optimal front.

Basic terminologies and concepts

Objective space

• EMO is a population based approach– Population evolves to finally converge on to the

Pareto front.• Multiple optimal solutions in a single run.• In classical MCDM approaches– Usually multiple runs necessary to obtain a set of

Pareto optimal solutions.– Usually problem knowledge is necessary.

Basic terminologies and concepts

• Goals in evolutionary multi-objective optimization algorithms– To find a set of solutions as close as possible to the

Pareto optimal front.– To find a set of solutions as diverse as possible.– To find a set of satisficing solutions reflecting the

decision maker’s preferences.• Satisficing: a decision-making strategy that attempts to

meet criteria for adequacy, rather than to identify an optimal solution.

Goal in evolutionary multi-objective optimization

Goal in evolutionary multi-objective optimization

Convergence

Diversity

Objective space

Goal in evolutionary multi-objective optimization

Convergence

Objective space

• Changes to single objective evolutionary algorithms– Fitness computation must be changed– Non-dominated solutions are preferred to

maintain the drive towards the Pareto optimal front (attain convergence)

– Emphasis to be given to less crowded or isolated solutions to maintain diversity in the population

Goal in evolutionary multi-objective optimization

• What are less-crowded solutions ?– Crowding can occur in decision space and/or objective

phase.• Decision space diversity sometimes are needed

– As in engineering design problems, all solutions would look the same.

Goal in evolutionary multi-objective optimization

Min-Min

Decision spaceObjective space

• How to maintain diversity and obtain a diverse set of Pareto optimal solutions?

• How to maintain non-dominated solutions?• How to maintain the push towards the Pareto

front ? (Achieve convergence)

Main Issues in evolutionary multi-objective optimization

• 1984 – VEGA by Schaffer• 1989 – Goldberg suggestion• 1993-95 - Non-Elitist methods– MOGA, NSGA, NPGA

• 1998 – Present – Elitist methods– NSGA-II, DPGA, SPEA, PAES etc.

EMO History