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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/. Objectives - PowerPoint PPT Presentation
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Evolutionary Multi-objective Optimization – A Big Picture
Karthik Sindhya, PhDPostdoctoral Researcher
Industrial Optimization GroupDepartment of Mathematical Information Technology
[email protected]://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
imize
: Cos
t
• 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
34
f1 (minimize)
f 2 (m
inim
ize)
2 4 5 6
23
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 (
min
imize
)2 4 5 6
23
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