SLIDE 1 Evolutionary Multiobjective Optimization: Current and Future Challenges
Carlos A. Coello Coello CINVESTAV-IPN
ıa El´ ectrica Secci´
- n de Computaci´
- n
- Av. Instituto Polit´
ecnico Nacional No. 2508
M´ exico, D. F. 07300, MEXICO
ccoello@cs.cinvestav.mx
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SLIDE 2 Motivation
Most problems in nature have several (possibly conflicting)
- bjectives to be satisfied. Many of these problems are frequently
treated as single-objective optimization problems by transforming all but one objective into constraints.
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SLIDE 3 What is a multiobjective optimization problem?
The Multiobjective Optimization Problem (MOP) (also called multicriteria optimization, multiperformance or vector
- ptimization problem) can be defined (in words) as the problem of
finding (Osyczka, 1985): a vector of decision variables which satisfies constraints and
- ptimizes a vector function whose elements represent the
- bjective functions. These functions form a mathematical
description of performance criteria which are usually in conflict with each other. Hence, the term “optimize” means finding such a solution which would give the values of all the objective functions acceptable to the decision maker.
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SLIDE 4 A Formal Definition
The general Multiobjective Optimization Problem (MOP) can be formally defined as: Find the vector x∗ = [x∗
1, x∗ 2, . . . , x∗ n]T which will satisfy the m
inequality constraints: gi( x) ≥ 0 i = 1, 2, . . . , m (1) the p equality constraints hi( x) = 0 i = 1, 2, . . . , p (2) and will optimize the vector function
x) = [f1( x), f2( x), . . . , fk( x)]T (3)
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SLIDE 5
What is the notion of optimum in multiobjective optimization?
Having several objective functions, the notion of “optimum” changes, because in MOPs, we are really trying to find good compromises (or “trade-offs”) rather than a single solution as in global optimization. The notion of “optimum” that is most commonly adopted is that originally proposed by Francis Ysidro Edgeworth in 1881.
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SLIDE 6 What is the notion of optimum in multiobjective optimization?
This notion was later generalized by Vilfredo Pareto (in 1896). Although some authors call Edgeworth-Pareto optimum to this notion, we will use the most commonly accepted term: Pareto
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SLIDE 7
Definition of Pareto Optimality
We say that a vector of decision variables x∗ ∈ F is Pareto optimal if there does not exist another x ∈ F such that fi( x) ≤ fi( x∗) for all i = 1, . . . , k and fj( x) < fj( x∗) for at least one j.
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SLIDE 8 Definition of Pareto Optimality
In words, this definition says that x∗ is Pareto optimal if there exists no feasible vector of decision variables x ∈ F which would decrease some criterion without causing a simultaneous increase in at least one other criterion. Unfortunately, this concept almost always gives not a single solution, but rather a set of solutions called the Pareto optimal set. The vectors x∗ correspoding to the solutions included in the Pareto optimal set are called
- nondominated. The plot of the objective functions whose
nondominated vectors are in the Pareto optimal set is called the Pareto front.
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SLIDE 9 Sample Pareto Front
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F2 F1
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SLIDE 10
Origins of EMOO
The first actual implementation of what it is now called a multi-objective evolutionary algorithm (or MOEA, for short) was Schaffer’s Vector Evaluation Genetic Algorithm (VEGA), which was introduced in the mid-1980s, mainly aimed for solving problems in machine learning (Schaffer, 1985).
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SLIDE 11 VEGA
2 1 n . . . gene performance parents Generation(t) Generation(t+1) select n subgroups using each dimension of performance in turn popsize 1 shuffle apply genetic
popsize 1 STEP STEP STEP 1 2 3 . . . . . . 1 . . . 2 n
Figure 1: Schematic of VEGA selection.
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SLIDE 12 Early Algorithms
From the second half of the 1980s up to the first half of the 1990s, few
- ther researchers developed MOEAs. Most of the work reported back
then involves the following types of MOEAs:
- Aggregating functions (mainly linear)
- Lexicographic ordering
- Target-vector approaches
Such approaches indicated strong roots in Operations Research. Note however, that these roots dissapeared over time. Algorithms remained relatively simple to implement during these early days of MOEAs.
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SLIDE 13
MOEAs: First Generation
The major step towards the first generation of MOEAs was given by David Goldberg (in 1989) when he proposed a selection scheme based on the concept of Pareto optimality (now called Pareto ranking). He also proposed the use of fitness sharing and niching to maintain diversity (something necessary to avoid converging to a single solution by effect of stochastic noise).
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SLIDE 14 MOEAs: First Generation
The most representative MOEAs of the first generation are the following:
- Nondominated Sorting Genetic Algorithm (NSGA) (Srinivas
& Deb, 1994)
- Niched-Pareto Genetic Algorithm (NPGA) (Horn et al., 1994)
- Multi-Objective Genetic Algorithm (MOGA) (Fonseca &
Fleming, 1993)
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SLIDE 15 MOEAs: First Generation
All the previously indicated algorithms use Pareto ranking, but in subtle different ways. For example, the NSGA uses layers to classify individuals, whereas MOGA ranked using the whole population at the same time. The NPGA, by its side, used tournaments based on
- nondominance. Although there is no theoretical study that indicates
advantages or disadvantages of any of these ranking schemes, several practitioners reported that MOGA seemed to be the most efficient MOEA of the first generation.
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SLIDE 16 The Main Questions of the First Generation
The main questions raised during the first generation were:
- Are aggregating functions (so common before and even during the
golden years of Pareto ranking) really doomed to fail when the Pareto front is non-convex?
- Can we find ways to maintain diversity in the population without
using niches (or fitness sharing), which requires a process O(M 2) where M refers to the population size?
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SLIDE 17 The Main Questions of the First Generation
- If we assume that there is no way of reducing the O(kM 2) process
required to perform Pareto ranking (k is the number of objectives and M is the population size), how can we design a more efficient MOEA?
- Do we have appropriate test functions and metrics to evaluate
quantitatively an MOEA?
- When will somebody develop theoretical foundations for MOEAs?
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SLIDE 18
MOEAs: Second Generation
The second generation of MOEAs was born with the introduction of the notion of elitism. In the context of multiobjective optimization, elitism usually (although not necessarily) refers to the use of an external population (also called secondary population) to retain the nondominated individuals found along the evolutionary process.
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SLIDE 19 MOEAs: Second Generation
The use of this external population (or file) raises several questions:
- How does the external file interact with the main population?
- What do we do when the external file is full?
- Do we impose additional criteria to enter the file instead of just
using Pareto dominance?
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SLIDE 20
MOEAs: Second Generation
Elitism can also be introduced through the use of a (µ + λ)-selection in which parents compete with their children and those which are nondominated (and possibly comply with some additional criterion such as providing a better distribution of solutions) are selected for the following generation.
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SLIDE 21 MOEAs: Second Generation
Some of the most representative MOEAs of the second generation are the following:
- Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler &
Thiele, 1999)
- Strength Pareto Evolutionary Algorithm 2 (SPEA2) (Zitzler
et al., 2001)
- Pareto Archived Evolution Strategy (PAES) (Knowles &
Corne, 2000)
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SLIDE 22 MOEAs: Second Generation
- Nondominated Sorting Genetic Algorithm II (NSGA-II) (Deb
et al., 2000)
- Niched Pareto Genetic Algorithm 2 (NPGA 2) (Erickson et al.,
2001)
- Pareto Envelope-based Selection Algorithm (PESA) (Corne et
al., 2000)
- Micro Genetic Algorithm for Multiobjective Optimization
(microGA) (Toscano & Coello, 2001)
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SLIDE 23
MOEAs: Second Generation
Second generation MOEAs can be characterized by an emphasis on efficiency and by the use of elitism (in the two main forms previously described). During the second generation, some important theoretical work also took place, mainly related to convergence. Also, metrics and standard test functions were developed to validate new MOEAs.
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SLIDE 24 The Main Questions of the Second Generation
- Are our metrics reliable? What about our test functions?
- Are we ready to tackle problems with more than two objective
functions efficiently? Is Pareto ranking doomed to fail when dealing with too many objectives? If so, then what is the limit up to which Pareto ranking can be used to select individuals reliably?
- What are the most relevant theoretical aspects of evolutionary
multiobjective optimization that are worth exploring in the short-term?
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SLIDE 25 Analysis of the Literature
83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 01 02 20 40 60 80 100 120 140 160 Publication Year Number of Publications
Figure 2: MOEA Citations by Year (up to mid-2002)
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SLIDE 26 Types of Applications
Eng Ind Sci Misc 50 100 150 200 250 300 Application Area Number of Applications Reviewed
Figure 3: The labels used are the following: Eng = Engineering, Ind =
Industrial, Sci = Scientific, Misc = Miscellaneous.
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SLIDE 27 Engineering Applications
ENH EE Tel RC SM CC Tra A 10 20 30 40 50 60 Engineering Field Number of Applications Reviewed
Figure 4: ENH = Environmental, Naval, and Hydraulic, EE = Electrical
and Electronics, Tel = Telecommunications and Network Optimization, RC = Robotics and Control, SM = Structural & Mechanical, CC = Civil and Construction, Tra = Transport, A = Aeronautical.
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SLIDE 28 Scientific Applications
GE CH PH MD EC CSE 5 10 15 20 25 30 35 40 45 50 Scientific Field Number of Applications Reviewed
Figure 5: The following labels are used: GE = Geography, CH = Chem-
istry, PH = Physics, MD = Medicine, EC = Ecology, CSE = Computer Science and Computer Engineering.
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SLIDE 29 Industrial Applications
DM SC GR MA 5 10 15 20 25 30 35 40 45 50 Type of Industrial Application Number of Applications Reviewed
Figure 6: The following labels are used: DM = Design and Manufacture,
SC = Scheduling, GR = Grouping and Packing, MA = Management.
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SLIDE 30 Miscellaneous Applications
FI EC MI CP 5 10 15 Type of Miscellaneous Application Number of Applications Reviewed
Figure 7: FI = Finance, CP = Classification and Prediction.
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SLIDE 31 Future Challenges
- Incorporation of preferences in MOEAs
- Dynamic Test Functions
- Highly-Constrained Search Spaces
- Parallelism
- Theoretical Foundations
- Use of More Efficient Data Structures
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SLIDE 32
To know more about evolutionary multiobjective optimization
Please visit our EMOO repository located at: http://delta.cs.cinvestav.mx/˜ccoello/EMOO with mirrors at: http://www.jeo.org/emo and: http://www.lania.mx/˜ccoello/EMOO
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SLIDE 33
To know more about evolutionary multiobjective optimization
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SLIDE 34 To know more about evolutionary multiobjective optimization
The EMOO repository currently contains:
- Over 1000 bibliographic references including 38 PhD theses
- Contact info of about 50 EMOO researchers
- Public domain implementations of MOGA (with elitism),
SPEA, NSGA, NSGA-II, the microGA, and PAES, among
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SLIDE 35
To know more about evolutionary multiobjective optimization
You can consult the following book recently published: Carlos A. Coello Coello, David A. Van Veldhuizen and Gary B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, New York, May 2002, ISBN 0-3064-6762-3.
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