Multi-Objective Optimization Using
Evolutionary Algorithms
1
Short Review (I)
• Evolution Strategies (ESs) were developed in Germany and have been
extensively studied in Europe
1. ESs use real-coding of design parameters since they model the organic
evolution at the level of individual’s phenotypes.
2. ESs depend on deterministic selection and mutation for its evolution.
3. ESs use strategic parameters such as on-line self-adaptation of mutability
parameters.
The selection of parents to form offspring is less constrained than it is in genetic
algorithms and genetic programming. For instance, due to the nature of the
representation, it is easy to average vectors from many individuals to form a
single offspring.
In a typical evolutionary strategy,  parents are selected uniformly randomly (i.e.,
not based upon fitness), more than  offspring are generated through the use of
recombination (considering ), and then  survivors are selected deterministically.
The survivors are chosen either from the best  offspring (i.e., no parents survive,
(,)-ES) or from the best  +  parents and offspring - (+)-ES.
2
Short Review (II)
• Genetic programming and Genetic algorithms are similar in most
other aspects, except that the reproduction operators are tailored to a
tree representation.
– The most commonly used operator is subtree crossover, in which an
entire subtree is swapped between two parents.
– In a standard genetic program, the representation used is a variablesized tree of functions and values. Each leaf in the tree is a label
from an available set of value labels. Each internal node in the tree is
labeled from an available set of function labels.
– The entire tree corresponds to a single function that may be
evaluated. Typically, the tree is evaluated in a left-most depth-first
manner. A leaf is evaluated as the corresponding value. A function is
evaluated using as arguments the result of the evaluation of its
children.
3
Overview
• Principles of Multi-Objective Optimization.
• Difficulties with the classical multi-objective
optimization methods.
• Schematic of an ideal multi-objective optimization
procedure.
• The original Genetic Algorithm (GA).
• Why using GA?
• Multi-Objective Evolutionary Algorithm (MOEA).
• An example of using a MOEA for solving
engineering design problem.
4
Multiobjective algorithms classification based on
how the objectives are integrated within
We will use the following simple classification of Evolutionary
Multi-Objective Optimization (EMOO) approaches:
• Non-Pareto Techniques
– Aggregating approaches
– Lexicographic ordering
– VEGA (Vector Evaluated Genetic Algorithm)
• Pareto Techniques
– Pure Pareto ranking
– MOGA
– NSGA
• Recent Approaches
– PAES
– SPEA
• Bio-inspired Approaches
– PSO
– Ant-colony based
5
Principles of Multi-Objective Optimization
• Real-world problems have more than one objective
function, each of which may have a different
individual optimal solution.
• Different in the optimal solutions corresponding to
different objectives because the objective functions
are often conflicting (competing) to each other.
• Set of trade-off optimal solutions instead of one
optimal solution, generally known as “ParetoOptimal” solutions (named after Italian economist
Vilfredo Pareto (1906)).
• No one solution can be considered to be better than
any other with respect to all objective functions. The
non-dominant solution concept.
6
Multi-Objective Optimization
• Is the optimization of different objective functions at the
same time, thus at the end the algorithm return n different
optimal values which is different to return one value in a
normal optimization problem.
• Thus, there are more than one objective function
Pareto - optimal solutions and Pareto - optimal front
• Pareto - optimal solutions: The optimal solutions
found in a multiple-objective optimization problem
• Pareto - optimal front: the curve formed by joining all
these solution (Pareto - optimal solutions)
7
Nondominated and dominated solutions
• Non-dominated -> given two objectives, a nondominated solution is when none of both solution
are better than the other with respect to two
objectives. Both objectives are equally important.
e.g. speed and price.
• Dominated: when solution a is no worse than b in
all objectives, and solution a is strictly better than b
in at least one objective, then solution a dominate
solution b.
• A weakly dominated solution: when solution a is no
worse than b in all objectives.
8
Multi-Objective Problems: Dominance
• we say x dominates y if it is at least as
good on all criteria and better on at least
one
f1
Pareto front
x
Dominated by x
f2
Principles of Multi-Objective Optimization
(cont.)
•
Simple car design example: two objectives - cost and
accident rate – both of which are to be minimized.
A, B, D - One objective
can only be improved at
the expense of at least
one other objective!
•
A multi-objective optimization algorithm must achieve:
1. Guide the search towards the global Pareto-Optimal front.
2. Maintain solution diversity in the Pareto-Optimal front.
10
Non-Pareto Classification Techniques
(Traditional Approaches)
 Aggregating the objectives into a single and parameterized
objective function and performing several runs with different
parameter settings to achieve a set of solutions approximating
the Pareto-optimal set.
Weighting Method (Cohon, 1978)
Constraint Method (Cohon, 1978)
Goal Programming (Steuer, 1986)
Minimax Approach (Koski, 1984)
Vector Evaluated Genetic Algorithm
• Proposed by Schaffer in the mid-1980s (1984,1985).
• Only the selection mechanism of the GA is modified
so that at each generation a number of subpopulations
was
generated
by
performing
proportional selection according to each objective
function in turn.
• Thus, for a problem with k objectives and a population
size of M, k sub-populations of size M/k each would be
generated.
• These sub-populations would be shuffled together to
obtain a new population of size M, on which the GA
would apply the crossover and mutation operators in the
usual way.
12
Schematic of VEGA selection
13
Advantages and Disadvantages of VEGA
• Efficient and easy to implement.
• If proportional selection is used, then the
shuffling and merging of all the subpopulations corresponds to averaging the
fitness components associated with each of
the objectives.
• In other words, under these conditions, VEGA
behaves as an aggregating approach and
therefore, it is subject to the same problems of
such techniques.
14
Problems in Multiobjectives Optimization
Weighting Method example
Fitness Function = w1 F1(x) + w2 F2(x)
Consider the problem for minimize response time,
maximize throughput
When F1(x) = response time,
F2(x) = throughput
Wi = weight value
Then,
It is hard to find the values of W1 and W2.

It is hard to form a fitness function.
Traditional Approaches
 Difficulties with classical methods:
 Being sensitive to the shape of the Pareto-optimal front (e.g.
weighting method).
 Need for problem knowledge which may not be available.
 Restrictions on their use in some application areas.
 Need to several optimization runs to achieve the best parameter
setting to obtain an approximation of the Pareto-optimal set.
Difficulties with the classical multi-objective
optimization methods
•
Such as weighted sum, є-perturbation, goal
programming, min-max, and others:
1. Repeat many times to find multiple optimal solutions.
2. Require some knowledge about the problem being solved.
3. Some are sensitive to the shape of the Pareto-optimal front
(e.g. non-convex).
4. The spread of optimal solutions depends on efficiency of
the single-objective optimizer.
5. Not reliable in problems involving uncertainties or
stochastic.
6. Not efficient for problems having discrete search space.
17
Lexicographic Ordering (LO)
• In this method, the user is asked to rank the
objectives in order of importance. The
optimum solution is then obtained by
minimizing the objective functions, starting
with the most important one and proceeding
according to the assigned order of importance
of the objectives.
• It is also possible to select randomly a single
objective to optimize at each run of a GA.
18
Advantages and Disadvantages of LO
• Efficient and easy to implement.
• Requires a pre-defined ordering of objectives and its
performance will be affected by it.
• Selecting randomly an objective is equivalent to a
weighted combination of objectives, in which each
weight is defined in terms of the probability that each
objective has of being selected. However, if tournament
selection is used, the technique does not behave like
VEGA, because tournament selection does not require
scaling of the objectives (because of its pair-wise
comparisons). Therefore, the approach may work
properly with concave Pareto fronts.
• Inappropriate when there is a large amount of objectives.
19
Schematic of an ideal Multi-Objective
optimization procedure
Step 1
Multi-objective
optimization problem
Minimize f1
Minimize f2
…
Minimize fn
subject to constraints
IDEAL
Multi-objective Optimizer
In 1967, Rosenberg hinted the potential of
Genetic
Algorithms
in
multi-objective
optimization
No significant study until in 1989 Goldberg
outlined
a new non-dominated sorting
procedure
A lot of interest recently because a GA is
capable of
finding multiple optimum
solutions in one single run (more than 630
publications in this research area)
Multiple trade-off solutions
found
Choose one solution
Higher-level
Information
Step 2
20
Pareto-based Techniques
• Suggested by Goldberg (1989) to solve the
problems with Schaffer’s VEGA.
• Use of non-dominated ranking and selection to
move the population towards the Pareto front.
• Requires a ranking procedure and a technique to
maintain diversity in the population (otherwise,
the GA will tend to converge to a single solution,
because of the stochastic noise involved in the
process).
21
The original Genetic Algorithm (GA)
•
•
Initially introduced by Holland in 1975.
General-purpose heuristic search algorithm that mimic the natural
selection process in order to find the optimal solutions.
1. Generate a population of random individuals or candidate
solutions to the problem at hand.
2. Evaluate of the fitness of each individual in the population.
3. Rank individuals based on their fitness.
4. Select individuals with high fitness to produce the next
generation.
5. Use genetic operations crossover and mutation to generate a
new population.
6. Continue the process by going back to step 2 until the problem’s
objectives are satisfied.
•
The best individuals are allowed to survive, mate, and
reproduce offspring.
Evolving solutions over time leads to better solutions.
•
22
The original Genetic Algorithm (GA) –
Flow Chart
A real coded GA represents
parameters without coding,
which makes representation of
the solutions very close to the
natural formulation of many
problems.
Special crossover and
mutation operators are
designed to work with
real parameters.
Multi-objective Fitness:
1.Non-dominated (best)
2.Dominated but feasible
(average)
3.Infeasible points (worst)
23
Why using GA?
• Using a GA when the search space is large and not
so well understood and unstructured.
• A GA can provide a surprisingly powerful heuristic
search.
• Simple, yet it performs well on many different types of
problems:
– optimization of functions with linear and nonlinear
constraints,
– the traveling salesman problem,
– machine learning,
– parallel semantic networks,
– simulation of gas pipeline systems,
– problems of scheduling, web search, software testing,
financial forecasting, and others.
24
Multi-Objective Evolutionary Algorithm
(MOEA)
• An EA is a variation of the original GA.
• An MOEA has additional operations to maintain multiple
Pareto-optimal solutions in the population.
Advantages:
• Deal simultaneously with a set of possible solutions.
• Enable of finding several members of the Pareto optimal set in
a single run of the algorithm.
• Explore solutions over the entire search space.
• Less susceptible to the shape or continuity of the Pareto front.
Disadvantages:
• Not completely supported theoretically yet (compared to
another method such as Stochastic Approximation which has
been around for half a century).
25
Multi-Objective Genetic Algorithm (MOGA)
• Proposed by Fonseca and Fleming (1993).
• The approach consists of a scheme in which the
rank of a certain individual corresponds to the
number of individuals in the current
population by which it is dominated.
• It uses fitness sharing and mating restrictions.
26
Advantages and Disadvantages of MOGA
• Efficient and relatively easy to implement.
• Its performance depends on the
appropriate selection of the sharing factor.
• MOGA has been very popular and tends
to perform well when compared to other
EMOO approaches.
Some Applications
 Fault diagnosis
 Control system design
 Wings plan form design
27
Nondominated Sorting Genetic Algorithm (NSGA)
• Proposed by Srinivas and Deb (1994).
• It is based on several layers of classifications of
the individuals.
• Nondominated individuals get a certain dummy
fitness value and then are removed from the
population. The process is repeated until the
entire population has been classified.
• To maintain the diversity of the population,
classified individuals are shared (in decision
variable space) with their dummy fitness values.
28
NSGA – Flow Chart
Multi-objective Fitness:
1.Non-dominated (best)
2.Dominated but feasible (average)
3.Infeasible points (worst)
Before selection is performed, the population is
ranked on the basic of domination: all nondominated individuals are classified into one
category (with a dummy fitness value, which is
proportional to the population size).
To maintain the diversity of the population, these
classified individuals are shared (in decision
variable space) with their dummy fitness values.
Then this group of classified individuals is removed
from the population and another layer of nodominated individuals is considered (the
remainder of the population is re-classified).
The process continues until all the individuals in
the population are classified. Since individuals in
the first front have maximum fitness value, they
always get more copies than the rest of the
population. This allow us to search for nondominated regions, and results in convergence of
the population toward such regions. Sharing, on its
part, helps to distribute the population over this
region.
29
Demo – NSGA II
0.5
0.45
CPI
[Gellert et al., 2012]
Multi-Objective
Optimizations for a
Superscalar
Architecture with
Selective Value
Prediction, IET
Computers & Digital
Techniques, Vol. 6, No. 4
(July), pp. 205-213, ISSN:
1751-8601
0.4
0.35
0.3
0.25
7.00E+09
1.20E+10
1.70E+10
2.20E+10
2.70E+10
3.20E+10
3.70E+10
4.20E+10
4.70E+10
Energy
Run without fuzzy
Run with fuzzy
Manual
- Features of NSGA II
http://webspace.ulbsibiu.ro/adrian.florea/html/docs/
IET_MultiObjective.pdf
30
The research area
Problems:
1. The so called “standard” settings (De Jong, 1990) : population
size of 50-100, crossover rate of 0.6 – 0.9, and mutation rate of
0.001 do not work for complex problems.
2. For complex real-world problems, GAs require parameter
tunings in order to achieve the optimal solutions.
3. The task of tuning GA parameters is not trivial due to the
complex and nonlinear interactions among the parameters and
its dependency on many aspects of the particular problem
being solved (e.g. density of the search space).
Research:
1. Self-Adaptive MOEA: use information fed-back from the MOEA
during its execution to adjust the values of parameters attached
to each individual in the population.
2. Improve the performance of MOEA: finding wide spread
Pareto-Optimal solutions and reducing computing resources.
3. Make them easier to use and available to more users.
31
Multi-Objective Evolutionary Algorithms references (MOEAs)

Some representatives of MOEAs in operational
research through past years:
a)
Non-Dominated Sorting genetic Algorithm (NSGA), Srinivas
et Deb, 1995.
b)
NSGA-II, Deb et al., 2002.
c)
Strength Pareto Evolutionary Algorithm (SPEA), Zitzler and
Thiele, 1999.
d)
SPEA2, Zitzler et al., 2001.
e)
Epsilon-NSGAII, Kollat and Reed, 2005.
f)
Multi-objective Shuffled Complex Evolution Metropolis
Algorithm (MOSCEM-UA), Vrugt et al., 2003.