Unit: Orders and Pareto Dominance
MODA Course, LIACS, Michael T.M. Emmerich, 2016
Learning Goals
I.
Recall: Definition of Pareto Dominance orders in Multiobjective
Optimization, and relations that are based on it
II. Learning about different types of ordered sets, in particular preorders and partial orders, and their fundamental properties.
III. How to compare orders? How to represent them in a diagrams?
IV. What is the geometrical interpretation of Pareto orders and the
closely related cone orders?
Recall: Pareto dominance
Orders, why?
Binary relations
Preorders
Minimal elements of a pre-ordered set
The covers relation
Aho, A. V.; Garey, M. R.; Ullman, J. D. (1972), "The transitive reduction of a directed graph", SIAM
Journal on Computing 1 (2): 131-137, doi:10.1137/0201008,MR 0306032.
Topological sorting
Partial orders, posets
Pareto order, invariances
Examples: Preordered and partially ordered sets
≤
Total (linear) orders and anti-chains
Poset that is neither a
chain nor an antichain
antichain
chain
(total order)
What about Pareto fronts? Are they chains, anti-chains or none of these?
Comparing the structure of orders
EXAMPLE
a
b
c
c
b
a
Maps on preorders
Drawing the Hasse* diagram
Example for a Hasse diagram
Is covered by
The art of drawing ordered sets
Three Hasse
Diagrams
of the same
Order –
Subsets of
A={1,2,3,4}
(Linear) extension
d
What about these
orders on {a,b,c,d}?
Identify (linear)
extensions!
The orders are
Represented by their
Hasse diagrams.
Orders on the Euclidean space
Cones
0,0,1
(0,0,1)
0,1
(1,0,0)
0,0
1,0
Examples for cones in 2-D (l) and 3-D (r)
(0,1,0)
Minkowski* sum and scalar multiplication
*Jewish-German mathematician 1864-1909, Goettingen
Some properties of cones (2)
0,0,1
0,1
1,0,0
0,0
1,0
0,1,0
Polyhedral cones
Polyhedral cones: Example
y2
y=λ1d1 +λ2d2
λ 1 d1
1
λ2d2
λ 2 d2
y2
d2
Basis: {d1, d2}
d1
λ1d1
0
y1
0
1
y1
The negative orthant
Definition of Pareto optimality via cones
or, equivalently:
Pareto Optimality and Cones
f2( min)
Non-dominated solutions
f1 ( min)
Attainment curve
Definition of Pareto optimality via cones
Cone-orders and trade-off bounding
What about cones with opening angle 180○ ?
How can we çheck cone dominance?
Example of cone-order: Minkowski’s spacetime
Unit 2: Take home messages(1/2)
1. A formal definition of Pareto Orders on the criterion space and
decision space was given
2. Orders can be introduced as binary relations on a set. The pair
(set, order relation) is then called an ordered set. Axioms are used
to characterize binary relations, and orders.
3. Preorders, partial orders and linear orders are three imporant
classes of orders. Preorders are the most general type of
orders.They are introduced (from the left to the right) by requiring
additional axioms (antisymmetry, totatlity).
4. Incomparability, strict orders, and indifference are often relations
that are defined in conjunction with an preorder.
Unit 2: Take home messages (2/2)
1. Orders can be compared in different ways: Order embedding, order
isomorphisms, equality
2. Hasse diagrams are means to exploit the antisymmetry and
transitivity of partial orders on finite sets in order to compactly
represent them in a diagram
3. Topological sorting can be used to find a linear order that extends
an partial order. It is an recursive algorithm. For a given partial
order there can be different topological sortings.
4. The Pareto order is a special kind of cone order. Cone-orders are
defined on vector spaces by means of a pointed and convex
dominance cone.
5. Using the cone order interpretation we can easily check dominance
and find Pareto fronts of 2-D finite sets, using a geometrical
construction.