Multiobjective Optimization

Athens 2005

Multiobjective Optimization

Athens 2005

Department of Architecture and TechnologyUniversidad Politécnica de Madrid

Santiago González Tortosa

Department of Architecture and TechnologyUniversidad Politécnica de Madrid

Santiago González Tortosa

ContentsContents

Introduction Multiobjective Optimization MO Non-Heuristic

Linear Nonlinear

Handling Constraints Techniques EMOO Using Constraints in EMOO Conclusions References

Introduction Multiobjective Optimization MO Non-Heuristic

Linear Nonlinear

Handling Constraints Techniques EMOO Using Constraints in EMOO Conclusions References

IntroductionIntroduction

Optimization Problem Find a solution in the feasible region which

has the minimum (or maximum) value of the objective function

Possibilities Unique objective (function) Multiobjective

multiple optimal solutions selection by preference function Solving:

Non-Heuristic (deterministic) Heuristic

Optimization Problem Find a solution in the feasible region which

has the minimum (or maximum) value of the objective function

Possibilities Unique objective (function) Multiobjective

multiple optimal solutions selection by preference function Solving:

Non-Heuristic (deterministic) Heuristic

Multiobjective Optimization

Multiobjective Optimization

Find a solution that:Minimize (objectives)

f(x) = (f1(x), f2(x), ..., fn(x))Subject to (constraints)

g(x) = (g1(x), g2(x), ... ,gm(x)) ≤ 0

x = (x1, …, xn) f, g linear/nonlinear functions

Find a solution that:Minimize (objectives)

f(x) = (f1(x), f2(x), ..., fn(x))Subject to (constraints)

g(x) = (g1(x), g2(x), ... ,gm(x)) ≤ 0

x = (x1, …, xn) f, g linear/nonlinear functions

Multiobjective Optimization

Multiobjective Optimization

Δ (The searching space): set of all possible solutions of x.

Ð (The feasible space): set of all solutions that satisfy all the constraints.

Δ (The searching space): set of all possible solutions of x.

Ð (The feasible space): set of all solutions that satisfy all the constraints.

Multiobjective Optimization

Multiobjective Optimization

Pareto Optimal x ℮ Ð is said to be Pareto Optimal if there

does not exist another solution x’ ℮ Ð that fi(x) = fi(x’) i = 1, …, m fi(x) < fi(x’) i = 1, …, m at least one i.

x solution dominate x’ solution Pareto Front

The maximal set of non-dominated feasible solutions.

Pareto Optimal x ℮ Ð is said to be Pareto Optimal if there

does not exist another solution x’ ℮ Ð that fi(x) = fi(x’) i = 1, …, m fi(x) < fi(x’) i = 1, …, m at least one i.

x solution dominate x’ solution Pareto Front

The maximal set of non-dominated feasible solutions.

MO Non-HeuristicMO Non-Heuristic

Multiobjective Optimization Linear (f and g)

Multiobjective Simplex Method Techniques:

Multiparametric Decomposition (weights a priori) Fractional Program (ratio objectives) Goals Program (goal deviations)

Nonlinear (f or g) Compromise Programming

Ideal Solution for each objective function Distance between solutions Objective weights Compromised Solution Compensation of objectives Competitive Objectives

Multiobjective Optimization Linear (f and g)

Multiobjective Simplex Method Techniques:

Multiparametric Decomposition (weights a priori) Fractional Program (ratio objectives) Goals Program (goal deviations)

Nonlinear (f or g) Compromise Programming

Ideal Solution for each objective function Distance between solutions Objective weights Compromised Solution Compensation of objectives Competitive Objectives

MO Non-HeuristicMO Non-Heuristic

Actual Basic

Variables

Basic Variables

NonbasicVariables

Values of Basic

Variables

X1

.

.

.Xm

1 … 0. .. .. .0 … 1

Y1(m+1) … Y1p

. .

. .

. .Ym(m+1) … Ymp

X10

.

.

.Xm

0

Objetives

0 … 0. .. .. .0 … 0

Z1(m+1) … Z1p

. .

. .

. .Zn(m+1) … Znp

f1(x0)

.

.

.fn(x0)

Constraints-Handling Techniques

Constraints-Handling Techniques

Penalty Function Very easy Depends on problem Problems with strong constraints

Repair Heuristic Useful when it’s difficult to find feasible problems Depends on problem

Separation between objectives & constraints No depends on problem Extend to multiobjective optimization problems

Hybrid Methods Use of numerical optimization problem Excessive computational cost

Others

Penalty Function Very easy Depends on problem Problems with strong constraints

Repair Heuristic Useful when it’s difficult to find feasible problems Depends on problem

Separation between objectives & constraints No depends on problem Extend to multiobjective optimization problems

Hybrid Methods Use of numerical optimization problem Excessive computational cost

Others

EMOOEMOO

Evolutionary MultiObjective Optimization

Techniques: A priori

Preferences before executing Reduce the problem to a unique objective Unique solution

A posteriori Preferences after executing Multiple solutions Methods:

Non-based on Pareto Optimal concept based on Pareto Optimal concept

Non-elitist elitist

Evolutionary MultiObjective Optimization

Techniques: A priori

Preferences before executing Reduce the problem to a unique objective Unique solution

A posteriori Preferences after executing Multiple solutions Methods:

Non-based on Pareto Optimal concept based on Pareto Optimal concept

Non-elitist elitist

EMOOEMOO

A posteriori: Non-based on Pareto Optimal concept VEGA algorithm (Vector Evaluated Genetic

Algorithm) k objectives, population size N Subpopulations size N/k Calculate fitness function and select t best

individuals (create new subpopulation) Shuffle all subpopulations Apply GA operators and create new populations

of size N

Speciation Problem: select individuals depending on 1 objective only

A posteriori: Non-based on Pareto Optimal concept VEGA algorithm (Vector Evaluated Genetic

Algorithm) k objectives, population size N Subpopulations size N/k Calculate fitness function and select t best

individuals (create new subpopulation) Shuffle all subpopulations Apply GA operators and create new populations

of size N

Speciation Problem: select individuals depending on 1 objective only

EMOOEMOO

A posteriori: non-elitist based on Pareto Optimal concept MOGA algorithm (MultiObjective Genetic

Algorithm) range (x) = 1 + p(x) (p(x) number of individuals

that dominate it) Sorting by minimal range Create a dummy fitness (lineal or non-linear) and

calculate (interpolate) depending on individual range

Select t best individuals (niches) Apply GA operators and create new population

Others: NSGA, NPGA, …

A posteriori: non-elitist based on Pareto Optimal concept MOGA algorithm (MultiObjective Genetic

Algorithm) range (x) = 1 + p(x) (p(x) number of individuals

that dominate it) Sorting by minimal range Create a dummy fitness (lineal or non-linear) and

calculate (interpolate) depending on individual range

Select t best individuals (niches) Apply GA operators and create new population

Others: NSGA, NPGA, …

EMOOEMOO

A posteriori: elitist based on Pareto Optimal concept NSGA-II algorithm (Non-dominated Sorting

Genetic Algorithm) Population P (size N) Create new population P’ (size N) using GA

operators Merge both populations and create new Population

R (size 2N) Sort by range of domination Select t individuals (tournament & niches) and

create a new population R’ Others: DPGA, PESA, PAES, MOMGA, …

A posteriori: elitist based on Pareto Optimal concept NSGA-II algorithm (Non-dominated Sorting

Genetic Algorithm) Population P (size N) Create new population P’ (size N) using GA

operators Merge both populations and create new Population

R (size 2N) Sort by range of domination Select t individuals (tournament & niches) and

create a new population R’ Others: DPGA, PESA, PAES, MOMGA, …

Using Constraints in EMOO

Using Constraints in EMOO

Eliminate non-feasible solutionsUse Penalty functionsSeparate solutions feasible and

non-feasibleDefine problem with Goals

Eliminate non-feasible solutionsUse Penalty functionsSeparate solutions feasible and

non-feasibleDefine problem with Goals

ConclusionsConclusions

Multiobjective OptimizationNon-Heuristic

Multiobjective Simplex Method (Linear) or Compromised Programming (Non-Linear)

Using Constraints Goals, Penalties, Weights…

HeuristicUsing GA (EMOO)

A priori (unique objective) A posteriori

Using GA and Constraints

Multiobjective OptimizationNon-Heuristic

Multiobjective Simplex Method (Linear) or Compromised Programming (Non-Linear)

Using Constraints Goals, Penalties, Weights…

HeuristicUsing GA (EMOO)

A priori (unique objective) A posteriori

Using GA and Constraints

ReferencesReferences

• Gracia Sánchez Carpena. Diseño y Evaluación de Algoritmos Evolutivos Multiobjetivo en Optimización y Modelación Difusa, PhD Thesis, Departamento de Ingeniería de la Información y las Comunicaciones, Universidad de Murcia, Murcia, Spain, November, 2002 (in Spanish).

• Carlos A. Coello Coello, David A. Van Veldhuizen and Gary B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, New York, March 2002, ISBN 0-3064-6762-3. David A. Van Veldhuizen. Multiobjective Evolutionary algorithms: Classifications, Analyses, and New Innovations. PhD thesis, Department of Electrical and Computer Engineering. Graduate School of Engineering. Air Force Institute of Technology, Wright-Patterson AFB, Ohio, May 1999.

• J. David Schaffer. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. PhD thesis, Vanderbilt University, 1984.

• Tadahiko Murata. Genetic Algorithms for Multi-Objective Optimization. PhD thesis, Osaka Prefecture University, Japan, 1997.

• Kalyanmoy Deb, Associate Member, IEEE, Amrit Pratap, Sameer Agarwal, and T. Meyarivan A Fast and Elitist Multiobjective Genetic Algorithm:NSGA-II

• Kaisa Miettinen Nonlinear Multiobjective Optimization Kluwer Academic Publishers, Boston, 1999

• Gracia Sánchez Carpena. Diseño y Evaluación de Algoritmos Evolutivos Multiobjetivo en Optimización y Modelación Difusa, PhD Thesis, Departamento de Ingeniería de la Información y las Comunicaciones, Universidad de Murcia, Murcia, Spain, November, 2002 (in Spanish).

• Carlos A. Coello Coello, David A. Van Veldhuizen and Gary B. Lamont, Evolutionary Algorithms for Solving Multi-Objective Problems, Kluwer Academic Publishers, New York, March 2002, ISBN 0-3064-6762-3. David A. Van Veldhuizen. Multiobjective Evolutionary algorithms: Classifications, Analyses, and New Innovations. PhD thesis, Department of Electrical and Computer Engineering. Graduate School of Engineering. Air Force Institute of Technology, Wright-Patterson AFB, Ohio, May 1999.

• J. David Schaffer. Multiple Objective Optimization with Vector Evaluated Genetic Algorithms. PhD thesis, Vanderbilt University, 1984.

• Tadahiko Murata. Genetic Algorithms for Multi-Objective Optimization. PhD thesis, Osaka Prefecture University, Japan, 1997.

• Kalyanmoy Deb, Associate Member, IEEE, Amrit Pratap, Sameer Agarwal, and T. Meyarivan A Fast and Elitist Multiobjective Genetic Algorithm:NSGA-II

• Kaisa Miettinen Nonlinear Multiobjective Optimization Kluwer Academic Publishers, Boston, 1999

Multiobjective Optimization

Athens 2005

Multiobjective Optimization

Athens 2005

Department of Architecture and TechnologyUniversidad Politécnica de Madrid

Santiago González Tortosa

Department of Architecture and TechnologyUniversidad Politécnica de Madrid

Santiago González Tortosa