Evolutionary multi-objective optimization

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Tutorial on
Tutorial on
Evolutionary Multiobjective Optimization
Evolutionary Multiobjective Optimization
GECCO 2007
GECCO 2007
Eckart Zitzler, Kalyanmoy Deb
Computer Engineering (TIK), ETH Zurich, Switzerland
Computer Engineering
Computer Engineering
and
and NetworksNetworks LaboratoryLaboratory
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Introductory Example: The Knapsack Problem
Introductory Example: The Knapsack Problem
?
weight = 750g
profit= 5
weight = 1500g
profit = 8
weight = 300g
profit= 7
weight = 1000g
profit= 3
Single objective:
choose subset that
maximizes overall profit
w.r.t. a weight limit
Multiobjective:
choose subset that
maximizes overall profit
minimizes overall weight
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
500g1000g1500g2000g2500g3000g3500g
weight
profit
5
10
15
20
The Search Space
The Search Space
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
500g1000g1500g2000g2500g3000g3500g
weight
profit
5
10
15
20
The Trade
The Trade- -off Frontoff Front
Observations:
?
? some solutions ( ) are better than others ( )
GECCO 2007 Tutorial / Evolutionary Multiobjective Optimization



ACM 978-1-59593-698-1/07/0007.


Copyright is held by the author/owner(s).
GECCO’07, July 7–11, 2007, London, England, United Kingdom.
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© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
finding the good
solutions
500g1000g1500g 2000g 2500g3000g3500g
weight
profit
5
10
15
20
The Trade
The Trade- -off Frontoff Front
Observations:
? there is no single optimal solution, but
? some solutions ( ) are better than others ( )
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
finding the good
solutions
500g1000g1500g2000g 2500g3000g3500g
weight
profit
5
10
15
20
The Trade
The Trade- -off Front off Front
Observations:
? there is no single optimal solution, but
? some solutions ( ) are better than others ( )
selecting a
solution
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Approaches:
profit more important than cost (ranking)
500g1000g1500g2000g2500g3000g3500g
weight
profit
5
10
15
20
Decision Making: Selecting a Solution
Decision Making: Selecting a Solution
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Approaches:
profit more important than cost (ranking)
weight must not exceed 2400g (constraint)
too heavy
500g1000g1500g2000g2500g3000g3500g
weight
profit
5
10
15
20
Decision Making: Selecting a Solution
Decision Making: Selecting a Solution
GECCO 2007 Tutorial / Evolutionary Multiobjective Optimization
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© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
When to Make the Decision
When to Make the Decision
Before Optimization:
ranks objectives,
defines constraints,…
searches for one
(green) solution
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
When to Make the Decision
When to Make the Decision
Before Optimization:
ranks objectives,
defines constraints,…
searches for one
(green) solution
too heavy
500g1000g1500g 2000g2500g3000g 3500g
weight
profit
5
10
15
20
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
After Optimization:
searches for a set of
(green) solutions
selects one solution
considering constraints, etc.
decision making often easier
EAs well suited
When to Make the Decision
When to Make the Decision
Before Optimization:
ranks objectives,
defines constraints,…
searches for one
(green) solution
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Outline
Outline
1.
Introduction: Why multiple objectives make a difference
2.
Basic Principles:
3.
Algorithm Design: Do it yourself
4.
Performance Assessment:
5.
Applications Domains: Where EMO is useful
6.
Further Information:
GECCO 2007 Tutorial / Evolutionary Multiobjective Optimization
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© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Optimization Problem: Definition
Optimization Problem: Definition
A general optimization problem is given by a quadruple
(X, Z, f, rel)
X denotes the decision space containing the elements among
which the best is sought; elements of are called decision
vectors or simply solutions;
Z denotes the objective space, the space within which the
decision vectors are evaluated and compared to each other;
elements of Z are denoted as objective vectors;
f represents a function f: X ?Z
vector a corresponding objective vector; f is usually neither
injective nor surjective;
rel is a binary relation over Z, i.e., rel?Z ?Z , which represents
a partial order over Z.
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Objective Functions
Objective Functions
Usually, consists of one or several functions1, ..., fn
assign each solution a real number. Such a function
fi: X ??
cost, size, execution time, etc.
In the case of a single objective function (n=1), the problem is
denoted as a single-objective optimization problem; a
multiobjective optimization problem involves several (n ? 2)
objective functions:
performance
performance
cost
single objectivemultiple objectives
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Comparing
Comparing ObjectiveObjective VectorsVectors
The pair (Z, rel) forms a partially ordered set, i.e., for any two objective
vectors a, b ?Z
a and b are equal: a relb andb rela
a is better than b: a relb and not(b rela)
a is worse than b: not(a relb) andb rela
a and b are incomparable: neither a relb norb rela
Example: Z = ?2, (a1, a2) rel(b1, b2) :? a1?b1?a2?b2
Often, (Z, rel) is a totally ordered set, i.e., for all a, b ?Z either a relb or b
rela or both holds (no incomparable elements).
worse
better
incomparable
incomparable
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Preference Structures
Preference Structures
The function f together with the partially ordered set (Z, rel)
a preference structure on the decision space X
solutions the decision maker / user prefers to other solutions:
x1prefrelx2:?f(x1) relf(x2)
One says:
Two solutions x1, x2are equal iff
A solution is indifferent to a solution
prefrelx1 and x1?x2;
A solution x1is preferred to a solutionx2iff x1prefrelx2;
A solution x1is strictly preferred to a solutionx2iff x1prefrelx2
and not ();
A solution x1is incomparable to a solution
x2nor x2prefrelx1.
= x2;
iffand
iff neither
GECCO 2007 Tutorial / Evolutionary Multiobjective Optimization
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© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
The Notion of Optimality
The Notion of Optimality
A solution x ?X
no solution x’?S is strictly preferred to x, i.e., for allx’?S: x’
prefrelx ?x prefrelx’.
In other words, f(x) is a minimal element of f(S) regarding the
partially ordered set (Z, rel).
optimal solutions w.r.t. X
S ?X
4
6
x1
x2
x3
x4
f
f
minimal element
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Pareto Dominance
Pareto Dominance
Assumption:
n
all objectives are to be maximized
fi: X ??
Z = ?n
Usually considered relation: weak Pareto dominance
(X, ?n, (f1, ..., fn), )
weak Pareto dominance:
Pareto dominance: strict version of weak Pareto dominance
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Illustration of Pareto Optimality
Illustration of Pareto Optimality
Maximize (y1, y2, …, yk) = f(x1, x2, …, xn)
Pareto(-optimal) set
y2
y1
worse
better
incomparable
incomparable
y2
y1
Pareto optimal = not dominated
dominated
© Eckart Zitzler
ETH Zürich
GECCO 2007
Tutorial on EMO
Decision and Objective Space
Decision and Objective Space
y2
y1
x2
x1
(x1, x2, …, xn) f
(y1, y2, …, yk)
decision
space
objective
space
Pareto set
non-optimal decision vector
Pareto front
non-optimal objective vector
search evaluation
GECCO 2007 Tutorial / Evolutionary Multiobjective Optimization
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