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8/8/2019 2008 - Multi-Objective Optimization Inspired by Nature (Slides)
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Projektbro KIT | 14. April 08
Multi-objective OptimizationInspired by Nature
Jrgen BrankeInstitute AIFB
University of Karlsruhe, GermanyKarlsruhe Institute of Technology
1 | Jrgen Branke | 12. April 2008
Menu
Appetizer
Evolutionary algorithms
Main course
Using evolutionary algorithms instead of exact optimizers for MOPs
Multi-objective evolutionary algorithmsIncluding preference information in MOEAs
Desert
Current research
Summary
2 | Jrgen Branke | 12. April 2008
Evolutionary algorithms
Darwins principle of natural evolution:survival of the fittest
in populations of individuals (plants, animals), the better the individual is adapted to theenvironment, the higher its chance for survival and reproduction.
evolving populationenvironment
3 | Jrgen Branke | 12. April 2008
Transfer to optimization
Natural evolution
individual
environment
fitness/how well adapted
survival of the fittest
mutation
crossover
population of
individualsenvironment
Evolutionary algorithms
potential solution
problem
cost/quality of solution
good solutions are kept
small, random perturbations
recombination of partial solutions
population ofsolutions problem
1324513542
15342
13254
31542
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INITIALIZE population(set of solutions)
REPEAT
UNTIL termination-condition
EVALUATE Individualsaccording to goal ("fitness")
SELECT parents
RECOMBINE parents (CROSSOVER)
MUTATE offspring
EVALUATE offspring
2 10
1
7
27
5
12
8
14 9 17 4
910
17
7
47
14
12
8
FORM next population
Basic algorithm
5 | Jrgen Branke | 12. April 2008
Advantages/Disadvantages
+No restriction w.r.t. fitness function (e.g. does not have to be differentiable)
+Universal applicability
+Easy to integrate heuristic knowledge if available
+Easy to parallelize
+Easy to use (usually inexpensive to develop)+Anytime algorithms (available time is fully utilized)
+Can deal with multiple objectives
+User-interaction possible
+Allow for continuous adaptation
+Can work with stochastic fitness functions
- Computationally expensive
- No guaranteed solution quality- Parameter tuning necessary
6 | Jrgen Branke | 12. April 2008
Industrial applications
Warehouse location problem (Locom)
Process scheduling (Unilever)
Job shop scheduling (Deer & Company, SAP, Volvo)
Turbine design (Rolce Royce, Honda) Portfolio optimization (First Quadrant)
Cleaning team assignment (Die Bahn)
Chip design (Texas Instruments)
Roboter movement (Honda)
Nuclear fuel reloading (Siemens)
Design of telephone networks (US West)
Games (creatures)
Military pilot training (British Air Force) Vehicle routing (Pina Petroli)
Coating of fuorescent lamps (Philips)
....
7 | Jrgen Branke | 12. April 2008
Major design decisions
Representation
Genetic operators
Selection mechanism
Crossover/Mutation probability Population size
Stopping criterion
8/8/2019 2008 - Multi-Objective Optimization Inspired by Nature (Slides)
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Simple example: Travelling Salesman Problem
Permutation encoding: 3-1-4-5-7-2-6-8-9
Mutation: Exchange two cities
Order crossover (OX)
select partial sequence from one parent, fill up in order of other parent
123456789 942836157
4567
928 31
9 | Jrgen Branke | 12. April 2008
Menu
Appetizer
Evolutionary algorithms
Main course
Using evolutionary algorithms instead of exact optimizers for MOPs
Multi-objective evolutionary algorithmsIncluding preference information in MOEAs
Desert
Current research
Summary
10 | Jrgen Branke | 12. April 2008
It is not always clear which
solution is better
Letfi, i=1dbe the different
optimization criteria. Then,
a solutionx is said to dominate a solution y ( )if and only if the following
condition is fulfilled:
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
A solution which is not dominated by any other solution in the search space
is called Pareto-optimal.
User preferences are required
AC
B
f1
f2
x f y fi (x) fi (y) i {1...d}j: fj(x) < fj(y)
x f y
Multiple objectives
11 | Jrgen Branke | 12. April 2008
A priori approach
Multi-ObjectiveProblem
Solution
Single-ObjectiveProblem
Specify preferences
Optimize
adjust
preference
informationevolutionary
algorithm
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12 | Jrgen Branke | 12. April 2008
Specifying preferences
Difficult!
Example: Tell me which travel plan you prefer!
Days Paris
Da y s Lo nd o n
Days Paris
Da y s Lo nd o n
13 | Jrgen Branke | 12. April 2008
Specifying preferences
w2
w1
w2
w1
f2
f1
Linear weighting
f2
f1
f2
f1
f2
f1
f2
f1
f2
f1
f2
f1
f2
f1
Constraints
f2
f1
f2
f1
f2
f1
f2
f1
f2
f1
f2
f1
f2
f1
Reference point and
achievement scalarizingfunction
14 | Jrgen Branke | 12. April 2008
Advantages / Disadvantages of EAs
+ Allows to solve problems where no exact methods exist
Metaheuristics do not guarantee (Pareto-) optimality Solutions generated in subsequent iterations may dominate each other
Adjusting preference information may lead to unexpected results
Computationally expensive
15 | Jrgen Branke | 12. April 2008
A posteriori - The power of populations
Multi-ObjectiveProblem
Selected
Solution
Single-ObjectiveProblem
Specify preferences
Specify preferencesPareto Front
Optimize
Optimize
a priori
a posteriorimost EMO approaches
Multi-objective
evolutionaryalgorit
hm
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Multiobjective Evolutionary Algorithms (MOEAs)
Since EAs work with a population of solutions, they can search for all
(a representative subset of) Pareto-optimal solutions in one run
Single EMO run is usually much more effective than multiple runs with
different objectives
17 | Jrgen Branke | 12. April 2008
Demo
18 | Jrgen Branke | 12. April 2008
Basic structure
INITIALIZE population
(set of solutions)
REPEAT
UNTIL termination-condition
EVALUATE Individualsaccording to goal ("fitness")
SELECT parents
RECOMBINE parents (CROSSOVER)
MUTATE offspring
EVALUATE offspring
2 10
1
7
27
5
12
8
14 9 17 4
9 1017
7
47
14
12
8
FORM next population
19 | Jrgen Branke | 12. April 2008
What is a good approximation?
f2
f1
equally distributedwide spread
f2
f1
close to the Pareto front
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Non-dominated Sorting GA (NSGA-II) [Deb et al. 2002]
Based on two ideas:
1. Pareto ranking: based on Pareto-dominance
2. Crowding distance: mechanism to maintain diversity in the population
Other popular approach:
Strength Pareto Evolutionary Algorithm (SPEA) by Zitzler
21 | Jrgen Branke | 12. April 2008
Pareto ranking + crowding distance
1
3
2
Objective 1
Objective2
4
22 | Jrgen Branke | 12. April 2008
Front 1
Front 2
Front 3
Front k
Non-dominated
sorting
old
pop
off-
spring
Diversity
sorting
reject
Form new
population
reject
NSGA-II: Overall algorithm
23 | Jrgen Branke | 12. April 2008
Advantages of finding the complete front
+Not necessary to specify preferences a priori
+Allows DM to choose solution after having seen the alternatives
+ Interactive search of Pareto front
Optimization prior to interaction, thus interaction very fast
Only non-dominated solutions are presented to the user
Direct navigation by user is possible
Additional information on distribution of solutions along the front may be
provided to the user (nadir point, ideal point, ...)
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Advantages of finding the complete front
+Not necessary to specify preferences a priori
+Allows DM to choose solution after having seen the alternatives
+ Interactive search of Pareto front
+Offer different alternatives to different customers
(e.g., mean-variance portfolio optimization)
+Reveal common properties among Pareto-optimal solutions
(some variables are always the same)
+Understand the causes for the trade-off
25 | Jrgen Branke | 12. April 2008
Understanding trade-offs
26 | Jrgen Branke | 12. April 2008
Advantages of finding the complete front
+Allows DM to choose solution after having seen the alternatives
+ Interactive search of Pareto front
+Offer different alternatives to different customers (e.g., mean-varianceportfolio optimization)
+Reveal common properties among Pareto-optimal solutions
(some variables are always the same)
+Understand the causes for the trade-off
+Aid in other optimization tasks (constraints, multi-objectivization)
27 | Jrgen Branke | 12. April 2008
Do we really need the whole front?
Computational overhead
Large set of alternatives, difficult to search by DM
Identify most interesting regions
Take into account partial user preferences
Bias the distribution
Restrict the distribution
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Identifying knees
Solutions where an improvement in either
objective leads to a significant worsening of
the other objective are more likely to be
preferred -> knee Knee
29 | Jrgen Branke | 12. April 2008
Marginal Utilities [Branke et al. 2004]
Assume user has linear utility function
Evaluate each solution with expected loss of utility if solution would not be
there
martinal utility
30 | Jrgen Branke | 12. April 2008
If crowding distance is replaced by marginal utilities,
algorithm focuses on knees
No preference information necessary
31 | Jrgen Branke | 12. April 2008
Menu
Appetizer
Evolutionary algorithms
Main course
Using evolutionary algorithms instead of exact optimizers for MOPs
Multi-objective evolutionary algorithmsIncluding preference information in MOEAs
Desert
Current research
Summary
8/8/2019 2008 - Multi-Objective Optimization Inspired by Nature (Slides)
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32 | Jrgen Branke | 12. April 2008
Motivation
Although a user generally cannot specify his/her preferences exactly before
alternatives are known, he/she usually has some rough idea as to what
solutions are desired A solution should have at least x in objective f1.
f1 of x would be good, f1 of y would be great.
My target solution would look something like this.
If a solution is worse by one unit in objective f1, it should be at least x units
better in objective f2 to be interesting.
Objective f1 is somewhat more important than objective f2.
Hope: Find a larger variety of more interesting solutions more quickly.
33 | Jrgen Branke | 12. April 2008
Considering partial preferences
Multi-ObjectiveProblem
Selected
Solution
Single-ObjectiveProblem
Specify preferences
Specify preferencesPareto Front
Optimize
Optimize
partial
preferences
+EMO
Biased Pareto Front
exactpreferences
a priori
a posteriorimost EMO approaches
34 | Jrgen Branke | 12. April 2008
EMO doesnt need user preferences.Does it?
All EMO approaches attempt to find a representative set of the Pareto
optimal front
Usually, representative means well distributed
But: distribution depends on scaling of the objectives
0
1
10
35 | Jrgen Branke | 12. April 2008
... at least x in objective f1.
Constraint: f1>x
Constraints are easy to integrate into EMO
Lexicographic ordering (feasible solution always dominates infeasible solution)
Penalty
Additional objective
...
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... target solution ...
1. Minimize distance to ideal solution (single objective)
2. Minimize maximal distance in any objective (single objective)
3. Goal Attainment [Fonseca & Fleming, 1993]/Goal Programming [Deb, 1999]
Do not reward improvement over ideal solutionf1 -> max{0, f1-f1*}
T1
T2
37 | Jrgen Branke | 12. April 2008
... at least x units better in objective f2 ...
Maximal and minimal trade-offs
Guided MOEA [Branke et al. 2001]
Modify definition of dominance
A
B
C
Can be achieved by a simple transformation of the objectives Not so easy for more than
2 objectives
1(x) = f1(x)+w12 f2(x)
2(x) = f2(x)+w21 f1(x)
x f y i(x) i (y)i {1,2}j:j(x) < j(y)
38 | Jrgen Branke | 12. April 2008
B
A
4
3
21
f1
f2
Guided dominance criterion
39 | Jrgen Branke | 12. April 2008
Faster convergence and better coverage of the interesting area of the
Pareto-optimal front
standard MOEA guided MOEA
The effect of guidance
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Marginal utility with preferences
With non-uniform distribution of utility functions
41 | Jrgen Branke | 12. April 2008
Biased sharing [Branke & Deb 2004]
User specifies weights and spread parameter
Crowding distance calculation is modified
42 | Jrgen Branke | 12. April 2008
43 | Jrgen Branke | 12. April 2008
Light Beam Search [Deb&Kumar 2007]
Specify aspiration and reservation point
Determine projection on Pareto front
Identify interesting local area using outranking approaches
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Interactive MOEA
Narrow down / refocus search during MOEA run
Explicitly by
adjusting constraints
moving the target
modifying the max/min trade-offs
...
Implicitly by comparing solutions
learn user preferences
45 | Jrgen Branke | 12. April 2008
46 | Jrgen Branke | 12. April 2008
Menu
Appetizer
Evolutionary algorithms
Main course
Using evolutionary algorithms instead of exact optimizers for MOPsMulti-objective evolutionary algorithms
Including preference information in MOEAs
Desert
Current research
Summary
47 | Jrgen Branke | 12. April 2008
Current research
Many-objective problems (difficulty: almost all solutions non-dominated)
Multiobjectivization (influence diversity and search space structure)
Noisy objective functions (e.g., stochastic simulation)
Worst-case multi-objective optimization
Using metamodels in case of expensive fitness evaluation
Individual = Set of solutions
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Summary
Evolutionary algorithms open new possibilities in multi-objective
optimization because
they are very general problem solvers
they work on a population of solutions and can thus search for a whole set ofsolutions simultaneously
Different ways to use EAs in MOO:
1. As single-objective optimizer in classical MOO techniques2. To generate an approximation to the whole Pareto front3. With partial user preferences resulting in a partial front or biased distribution4. Interactively guided by the user
49 | Jrgen Branke | 12. April 2008
EMO resources
Books:
K. Deb: Multi-objective optimization using evolutionary algorithms. Wiley,
2001
C. Coello Coello, D. A. Van Veldhuizen and G. B. Lamont: Evolutionary
algorithms for solving multi-objective problems. Kluwer, 2002 J. Branke, K. Deb, K. Miettinen, R. Slowinski: Multi-objective optimization -
interactive and evolutionary approaches. Springer, to appear
Websites:
http://www.lania.mx/~coello
50 | Jrgen Branke | 12. April 2008
References
[Branke et al. 2001] J. Branke, T. Kauler, H. Schmeck: Guidance in evolutionarymulti-objective optimization. Advances in Engineering Software, 32:499-507
[Branke et al. 2004] J. Branke, K. Deb, H. Dierolf, M. Osswald: Finding knees in multi-
objective optimization. Parallel Problem Solving from Nature, Springer, pp. 722-731
Branke and Deb Biased[Branke and Deb 2004] J. Branke and K. Deb: Integrating user preferences into
evolutionary multi-objective optimization. In Y. Jin, editor, Knowledge Incorporationin Evolutionary Computation, Springer, pages 461477
[Deb 1999]: Solving goal programming problems using multi-objective geneticalgorithms. Congress on Evolutionary Computation, IEEE, pp. 77-84
[Deb et al. 2002] K. Deb, S. Agrawal, A. Pratap, T. Meyarivan: A fast and Elitist multi-objective Genetic Algorithm: NSGA-II. IEEE Transactions on Evolutionary
Computation 6(2):182-197[Deb and Kumar 2007] K. Deb and A. Kumar: Light beam search based multi-
objective optimization using evolutionary algorithms. Congress on EvolutionaryComputation, IEEE, pp. 2125-2132
[Fonseca and Fleming 1993] C. M. Fonseca and P. J. Fleming: Genetic algorithms formultiobjective optimization: Formulation, discussion, and generatization.International Conference on Genetic Algorithms, Morgan Kaufmann, pp. 416-423