Vector Optimization

Study Guide for ES205

Yu-Chi HoJonathan T. LeeJan. 12, 2001

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Outline Problem Statement Motivation Pareto Optimality Scalarization Nonconvex S’ Numerical Methods

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Problem Statement

where S is the feasible set in n and J is the objective function, J: S m and ,S’ is the feasible performance region.

)(max xJSx

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xJxJxJ m,...,1

J1

J2

S’

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Motivation Many real world problems require to

optimize multiple criteria at the same time.• Find the optimal path of an airplane

flight while minimizing both the time it takes and the fuel consumption.

• Buying a car with the best quality while spending the lowest amount of money.

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Pareto Optimality x’ S is a Pareto optimum if there is

no x S such that Ji(x) Ji(x’) for all i = 1, …, m and Ji(x) > Ji(x’) for some i.

The “best that could be achieved without disadvantaging at least one group.” (Allan Schick, in Louis C. Gawthrop, 1970, p.32)

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Pareto Optimality (cont.) m = 2

Pareto frontier:the set ofPareto optimum

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J1(x) J1(x’)J2(x) J2(x’)x’

J1

J2

S’

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Scalarization

for i > 0, i = 1, …, m.

To “summarize” the multiple criteria into a single criterion — scalar-valued optimization problem.

m

iii

SxxJ

1

max

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Scalarization (cont.) m = 2

J1

J2x’ 1J1 + 2J2

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S’

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Nonconvex S’ Only two of the optimal points could

be identify through scalarization

J1

J2

S’

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Constraint Optimization Pb.

max J1(x)s.t. J2(x) for different

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J1

J2

S’

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Nonconvex S’ (cont.)

J1

max min [J1, J2]

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J2

S’

x’

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Nonconvex S’ (cont.)

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J1

J2

S’

x:AspirationPoint

x’

xxAxx T

Sx

''min

''

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Numerical Methods Linear Programming: both the

objective function and the constraints are linear

Non-linear programming

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References:

• Heylighen, F., Web Dictionary of Cybernetics and Systems, http://pespmc1.vub.ac.be/ASC/.

• Ho, Y.-C., “Optimization – A Many-Splendored Thing –,” slides presented at IFAC World Congress, 1999.

• Jahn, J., “Theory of Vector Maximization: Various Concepts of Efficient Solutions,” in Chapter 2 of Multicriteria Decision making – Advances in MCDM Models, Algorithms, Theory, and Applications by T. Gal, T.J. Stewart and T. Hanne, Kkuwer, 1999.

• Mas-Colell, A., M. D. Whinston and J. R. Green, Microeconomic Theory, Oxford University Press, 1995.