SLIDE 1 Unit: Orders and Pareto Dominance
MODA Course, LIACS, Michael T.M. Emmerich, 2016
SLIDE 2 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 pre-
- rders 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?
SLIDE 3
Recall: Pareto dominance
SLIDE 4
Orders, why?
SLIDE 5
Binary relations
SLIDE 6
Preorders
SLIDE 7
Minimal elements of a pre-ordered set
SLIDE 8 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.
SLIDE 9
Topological sorting
SLIDE 10
Partial orders, posets
SLIDE 11
Pareto order, invariances
SLIDE 12
Examples: Preordered and partially ordered sets
≤
SLIDE 13 Total (linear) orders and anti-chains
antichain chain (total order)
What about Pareto fronts? Are they chains, anti-chains or none of these?
Poset that is neither a chain nor an antichain
SLIDE 14 Comparing the structure of orders
a b c c b a EXAMPLE
SLIDE 15
Maps on preorders
SLIDE 16
Drawing the Hasse* diagram
SLIDE 17 Example for a Hasse diagram
Is covered by
SLIDE 18 The art of drawing ordered sets
Three Hasse Diagrams
Order – Subsets of A={1,2,3,4}
SLIDE 19 (Linear) extension
What about these
Identify (linear) extensions! The orders are Represented by their Hasse diagrams. d
SLIDE 20
Orders on the Euclidean space
SLIDE 21 Cones
0,0 1,0 0,1 0,0,1 (0,1,0) (1,0,0) Examples for cones in 2-D (l) and 3-D (r) (0,0,1)
SLIDE 22 Minkowski* sum and scalar multiplication
*Jewish-German mathematician 1864-1909, Goettingen
SLIDE 23 Some properties of cones (2)
0,0 1,0 0,1 0,0,1 0,1,0 1,0,0
SLIDE 24
Polyhedral cones
SLIDE 25 Polyhedral cones: Example
y1 y2
1 1
y=λ1d1 +λ2d2 λ1d1 λ2d2
λ2d2 λ1d1 d1 d2 y1 y2 Basis: {d1, d2}
SLIDE 26
The negative orthant
SLIDE 27 Definition of Pareto optimality via cones
SLIDE 28 Pareto Optimality and Cones
f1 (min) f2(min)
Non-dominated solutions
SLIDE 29
Attainment curve
SLIDE 30
Definition of Pareto optimality via cones
SLIDE 31 Cone-orders and trade-off bounding
What about cones with opening angle 180○ ? How can we çheck cone dominance?
SLIDE 32
Example of cone-order: Minkowski’s spacetime
SLIDE 33 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
- rders.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.
SLIDE 34 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
- rder 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.
