Evolutionary Multi-objective Optimization A Big Pictureusers.jyu.fi/~jhaka/uppsala/Lecture8a_Bigpicture.pdf · –Usually used in evolutionary multi-objective optimization algorithms,

Evolutionary Multi-objective Optimization – A Big Picture

Karthik Sindhya, PhD

Postdoctoral Researcher

Industrial Optimization Group

Department of Mathematical Information Technology

[email protected] http://users.jyu.fi/~kasindhy/

mailto:[email protected]

http://users.jyu.fi/~kasindhy/

Objectives

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

to Multi-objective optimization 2. Review the basic terminologies and concepts in use

in multi-objective optimization 3. Introduce evolutionary multi-objective optimization 4. Goals in evolutionary multi-objective optimization 5. 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.


1 2

3

4

f1 (minimize)

f 2 (

min

imiz

e)

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.


• 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)


• Kung et al. approach

– Step 1: Sort objective 1 based on the descending order of importance.

• Ascending order for minimization objective


1 2

3

4

f1 (minimize)

f 2 (

min

imiz

e)

2 4 5 6

2

3

5

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

5


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.



5

1 2

3

4

f1 (minimize)

f 2 (

min

imiz

e)

Front 1

Front 2

Front 3

f1 (minimize)

f 2 (

min

imiz

e)

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


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.



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.


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.


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.


• 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


Convergence

Diversity

Objective space


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


• 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.


Min-Min


• 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

Documents

Evolutionary Multi-objective Optimization A Big Pictureusers.jyu.fi/~jhaka/uppsala/Lecture8a_Bigpicture.pdf · –Usually used in evolutionary multi-objective optimization algorithms,