Decision Support Systems SYSTeMS Ghent University Utrecht - - PowerPoint PPT Presentation

SLIDE 1

SYSTeMS Ghent University Gert de Cooman Arthur Van Camp Jasper De Bock Stavros Lopatatzidis Decision Support Systems Utrecht University Linda van der Gaag Silja Renooij Marjan van den Akker Steven Woudenberg Merel Rietbergen Algorithms & Complexity Centrum Wiskunde & Informatica Peter Grünwald Erik Quaeghebeur

SLIDE 2

SYSTeMS Ghent University Gert de Cooman Arthur Van Camp Jasper De Bock Stavros Lopatatzidis Decision Support Systems Utrecht University Linda van der Gaag Silja Renooij Marjan van den Akker Steven Woudenberg Merel Rietbergen Algorithms & Complexity Centrum Wiskunde & Informatica Peter Grünwald Erik Quaeghebeur

SLIDE 3

SYSTeMS Ghent University Gert de Cooman Arthur Van Camp Jasper De Bock Stavros Lopatatzidis Decision Support Systems Utrecht University Linda van der Gaag Silja Renooij Marjan van den Akker Steven Woudenberg Merel Rietbergen Algorithms & Complexity Centrum Wiskunde & Informatica Peter Grünwald Erik Quaeghebeur

SLIDE 4

SYSTeMS Ghent University Gert de Cooman Arthur Van Camp Jasper De Bock Stavros Lopatatzidis Decision Support Systems Utrecht University Linda van der Gaag Silja Renooij Marjan van den Akker Steven Woudenberg Merel Rietbergen Algorithms & Complexity Centrum Wiskunde & Informatica Peter Grünwald Erik Quaeghebeur

SLIDE 5

SYSTeMS Ghent University Gert de Cooman Arthur Van Camp Jasper De Bock Stavros Lopatatzidis Decision Support Systems Utrecht University Linda van der Gaag Silja Renooij Marjan van den Akker Steven Woudenberg Merel Rietbergen Algorithms & Complexity Centrum Wiskunde & Informatica Peter Grünwald Erik Quaeghebeur

SLIDE 6

SYSTeMS Ghent University Gert de Cooman Arthur Van Camp Jasper De Bock Stavros Lopatatzidis Decision Support Systems Utrecht University Linda van der Gaag Silja Renooij Marjan van den Akker Steven Woudenberg Merel Rietbergen Algorithms & Complexity Centrum Wiskunde & Informatica Peter Grünwald Erik Quaeghebeur

SLIDE 7

SYSTeMS Ghent University Gert de Cooman Arthur Van Camp Jasper De Bock Stavros Lopatatzidis Decision Support Systems Utrecht University Linda van der Gaag Silja Renooij Marjan van den Akker Steven Woudenberg Merel Rietbergen Algorithms & Complexity Centrum Wiskunde & Informatica Peter Grünwald Erik Quaeghebeur

SLIDE 8

Characterizing Coherence, Correcting Incoherence

Erik Quaeghebeur

SLIDE 9

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 10

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 11

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 12

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 13

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 14

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 15

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 16

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 17

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 18

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 19

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 20

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 21

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 22

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 23

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 24

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 25

g1 g2 P Q Pg1 Pg2 g1 g2 a 1 b

1 2

1 c 0

1 2

K Ω Pb Pa Pc

1 2

1

1 2

1 EP DP DQ min

SLIDE 26

Characterizing Coherence, Correcting Incoherence

I WANT YOU to crank out COHERENCE CHARACTERIZATIONS

• 1. Context

Basic setup:

• Finite possibility space Ω
• Finite set of gambles 𝒧 on Ω
• Lower previsions P on 𝒧

Matrix notation:

• ⋃︁Ω⋃︁-by-⋃︁𝒧⋃︁ matrix K with

gambles as columns

• the rows of K (columns of K⊺)

are the degenerate previsions

• the set 𝒯 of matrices S obtained

from the identity matrix I by changing at most one 1 to −1

• all-one (zero) column vector 1 (0)
• 2. Goals

Given K, find a non-redundant H- representations for the set of all P

• A. that avoid sure loss ()︁ΛA αA⌈︁),
• B. that avoid sure loss and for

which P ≥ min ()︁ΛB αB⌈︁),

• C. that are coherent ()︁ΛC αC⌈︁).
• 7. Experiments

The sparsity σ is the fraction of zero components in K. Procedure C1 is exponential in 1 − σ and ∼linear in ⋃︁Ω⋃︁: 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 10−2 10−1 100 101 102 ⋃︁Ω⋃︁ = 4 8 16 32 64 128 256 512 1024 2048 4096 ⋃︁Ω⋃︁ = 8192 σ [s] ⋃︁𝒧⋃︁ = 5 . . . and (at least) exponential in ⋃︁𝒧⋃︁: 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 10−3 10−2 10−1 100 101 102 103 ⋃︁𝒧⋃︁ = 3 ⋃︁𝒧⋃︁ = 4 ⋃︁𝒧⋃︁ = 6 ⋃︁𝒧⋃︁ = 8 ⋃︁𝒧⋃︁ = 9 ⋃︁𝒧⋃︁ = 12 σ [s] ⋃︁Ω⋃︁ = 6

• 3. Goal A: Characterizing ASL

Based on the existence of a dominating linear prevision:

• A1. ∃µI,νI ≥ 0 ∶

P = K⊺µI−IνI ∧ 1⊺µI = 1 ⌊︁K⊺ −I 1⊺ 0⊺}︁ )︁ΛA αA⌈︁ EN, RR

• A2. ∃µI ≥ 0 ∶

P ≤ K⊺µI ∧ 1⊺µI = 1 ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ I −K⊺ −I 1⊺ 1 −1⊺ −1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛA αA⌈︁ PJP, RR

• 4. Goal B: Characterizing ASL ≥

≥ ≥ min

• B1. Starting from )︁ΛA αA⌈︁:

⌊︁ ΛA αA −I −min}︁ )︁ΛB αB⌈︁ RR

• 5. Goal C: Characterizing coherence

Based on the existence of S-dominating linear previsions:

• C1. Analogous to A1 & intersection over all S in 𝒯:

∀S ∈ 𝒯 ∶ ∃µS,νS ≥ 0 ∶ P = K⊺µS−SνS ∧ 1⊺µS = 1 ⌊︁K⊺ −S 1⊺ 0⊺}︁ )︁ΛC αC⌈︁ EN, ISS∈𝒯, RR

• C2. Analogous to A2 & intersection over all S in 𝒯:

∀S ∈ 𝒯 ∶ ∃µS ≥ 0 ∶ SP ≤ SK⊺µS ∧ 1⊺µS = 1 ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ S −SK⊺ −I 1⊺ 1 −1⊺ −1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛC αC⌈︁ PJP, ISS∈𝒯, RR =∶ )︁AS,P AS,µS b0⌈︁

• C3. Block matrix form of C2:

)︁AP Aµ b⌈︁ ∶= ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ AI,P AI,µI b0 ⋮ ⋱ ⋮ AS,P AS,µS b0 ⋮ ⋱ ⋮ ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛC αC⌈︁ PJP, RR

• 6. Illustrations of Procedure C1

Pg1 1 2 1 Pg2 1 2 1 P b P a P c Ω = {a,b,c} K = ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ 1 0 1 2 1 0 1 2 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ ⋃︂1 1⨄︂ ⋃︂−1 1⨄︂ ⋃︂1 −1⨄︂ P b P a P c min facet enumerate the V-representa- tion for avoiding sure S-loss for each S in 𝒯 intersect and remove redundancy Pg1 Pg2 Pg3 P b P a P c Ω = {a,b,c} K = ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ 1 0 1 2 1 2 1 0 0 1 2 1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ min P b P a P c intersect and remove redundancy I WANT YOU to ERADICATE INCOHERENCE utterly

• 1. Context & Goal

Given: incoherent lower prevision P. Goal: Find a coherent correction to it.

• 2. Bring within bounds

If Pf ∉ ⋃︂min f,max f⨄︂ for some f in 𝒧, it is

• ut of bounds. To bring it within bounds:

BP f ∶= )︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌋︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ]︁ min f Pf ≤ min f, max f P f ≥ max f, P f

• therwise.

P BP Q BQ lower previsions

• ut of bounds
• 3. Downward correction

As the downward correction of P we take the lower envelope of the maximal coherent dominated lower previsions (proposed earlier by Pelessoni & Vicig, following Weichselberger), so the nadir point DP of the MOLP (cf. C) (†) maximize Q, subject to ΛCQ ≤ αC

Q ≤ P

• r the MOLP (cf. C3)

(‡) maximize Q, subject to AQQ+Aµµ ≤ b Q ≤ P. Some desirable properties:

• It is the maximal neutral correction

(‘no component tradeoffs’).

• The imprecision of the correction is

nondecreasing with incoherence. P DP Q DQ DP P dominated lower previsions extreme coherent dominated lower prevision For the future: Can the computation be simplified for special classes of P?

• 4. Experiments

With the M3-solver we used, computa- tion appears exponential in ⋃︁𝒧⋃︁; using pre-computed constraints (†) is more efficient than not (‡): 2 3 4 5 6 7 8 9 10 10−2 10−1 100 101 102 103 104

3 8 17 24 39 53 112 228 247 2 1 2 1 4 1 4 1 17 1 17 1 24 3 24 3 3 2 8 2 7 2 29 2 8 3 26 5 206 2 206 2 16 16 16 16

⋃︁𝒧⋃︁ [s] DP via (†) DP via (‡) ⋃︁Ω⋃︁ = 5, σ ≈ 1⇑2 We expect other solvers and certainly direct M2-solvers to perform more efficiently, but could not test any yet.

• 5. Upward correction

The standard upward correction of P is its natural extension EP, the unique minimal pointwise dominating co- herent lower prevision, so the the solution to the MOLP (cf. C) minimize EP, subject to ΛCEP ≤ αC

EP ≥ P

• r the MOLP (cf. C3)

(*) minimize EP, subject to AEPEP+Aµµ ≤ b EP ≥ P.

• The problem becomes a plain LP by

using the objective ∑g∈𝒧EPg.

• (*) decomposes into a classical for-

mulation of natural extension. P EP Q dominating lower previsions no natural extension in case of sure loss

• 1. Representations

Any convex polyhedron in Rn can be described in two ways: H-representation (intersection of half-spaces)

)︁A b⌈︁ ∶= {x ∈ Rn ∶ Ax ≤ b}

constraint matrix in Rk×n constraint vector in Rk V-representation (convex hull of points and rays)

⌊︁V w}︁ ∶= {x ∈ Rn ∶ x =V µ ∧ µ ≥ 0 ∧ w⊺µ = 1}

vector matrix in Rn×ℓ vector in Rℓ vector in (Rℓ)≥0 with components defining points (≠ 0) and rays (= 0)

• 2. Illustration

Here n = 2, k = 3, and ℓ = 4. constraint redundant constraint redundant point extreme ray vertex I WANT YOU to juggle POL YHEDRA like there’s no tomorrow

• 3. Tasks
• RR. Removing redundancy: if j is the

numberof non-redundant con- straints (or vectors), this requires solving k (or ℓ) linear programming problems of size n× j

• EN. Moving between H- and V-represent-

ations: done using vertex/facet enu- meration algorithms; polynomial in n, k, and ℓ.

• PJ. Projection on a lower-dimensional

space: easy with V-representations, hard with H-representations.

• IS. Intersection: easy with H-represent-

ations, hard with V-representations.

• 1. Formalization

Any multi-objective linear program (MOLP) can be put in the following form: maximize y =Cx, subject to Ax ≤ b and x ≥ 0

• bjective

vector in Rm

• bjective

matrix in Rm×n

• ptimization

vector in Rn constraint matrix in Rk×n constraint vector in Rk

• 3. Tasks

Main computational tasks in non- decreasing order of complexity:

• M1. Finding ˆ

y.

• M2. Finding ˇ

y.

• M3. Finding ext𝒵∗

and characterizing 𝒵∗.

• M4. Finding ext𝒴 ∗.
• M5. Characterizing 𝒴 ∗.
• 2. Illustration

Here m = n = 2 and k = 4. x1 x2 𝒴 𝒴 ∗ C1 C2 y1 y2 𝒵 𝒵∗ ˆ y ˇ y feasible optimization vectors {x ∈ Rn ∶ Ax ≤ b ∧ x ≥ 0} C-undominated optimization vectors {x ∈ 𝒴 ∶ (∀z ∈ 𝒴 ∶Cx ⇑ <Cz)} with vertices ext𝒴 ∗ undominated objective vectors {Cx ∶ x ∈ 𝒴 ∗} with vertices ext𝒵∗ ideal point, with ˆ yi = max{yi ∶ y ∈ 𝒵} nadir point, with ˇ yi = min{yi ∶ y ∈ 𝒵∗} feasible objective vectors {Cx ∶ x ∈ 𝒴} I WANT YOU to grok MUL TI-OBJECTIVE LINEAR PROGRAMMING

SYSTeMS Research Group Ghent University Erik Quaeghebeur Decision Support Systems Group Utrecht University

Coherence characterization procedures Incoherence correction procedures Polytope theory Multi-

• bjective

linear programming

SLIDE 27

Characterizing Coherence, Correcting Incoherence

I WANT YOU to crank out COHERENCE CHARACTERIZATIONS

• 1. Context

Basic setup:

• Finite possibility space Ω
• Finite set of gambles 𝒧 on Ω
• Lower previsions P on 𝒧

Matrix notation:

• ⋃︁Ω⋃︁-by-⋃︁𝒧⋃︁ matrix K with

gambles as columns

• the rows of K (columns of K⊺)

are the degenerate previsions

• the set 𝒯 of matrices S obtained

from the identity matrix I by changing at most one 1 to −1

• all-one (zero) column vector 1 (0)
• 2. Goals

Given K, find a non-redundant H- representations for the set of all P

• A. that avoid sure loss ()︁ΛA αA⌈︁),
• B. that avoid sure loss and for

which P ≥ min ()︁ΛB αB⌈︁),

• C. that are coherent ()︁ΛC αC⌈︁).
• 7. Experiments

The sparsity σ is the fraction of zero components in K. Procedure C1 is exponential in 1 − σ and ∼linear in ⋃︁Ω⋃︁: 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 10−2 10−1 100 101 102 ⋃︁Ω⋃︁ = 4 8 16 32 64 128 256 512 1024 2048 4096 ⋃︁Ω⋃︁ = 8192 σ [s] ⋃︁𝒧⋃︁ = 5 . . . and (at least) exponential in ⋃︁𝒧⋃︁: 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 10−3 10−2 10−1 100 101 102 103 ⋃︁𝒧⋃︁ = 3 ⋃︁𝒧⋃︁ = 4 ⋃︁𝒧⋃︁ = 6 ⋃︁𝒧⋃︁ = 8 ⋃︁𝒧⋃︁ = 9 ⋃︁𝒧⋃︁ = 12 σ [s] ⋃︁Ω⋃︁ = 6

• 3. Goal A: Characterizing ASL

Based on the existence of a dominating linear prevision:

• A1. ∃µI,νI ≥ 0 ∶

P = K⊺µI−IνI ∧ 1⊺µI = 1 ⌊︁K⊺ −I 1⊺ 0⊺}︁ )︁ΛA αA⌈︁ EN, RR

• A2. ∃µI ≥ 0 ∶

P ≤ K⊺µI ∧ 1⊺µI = 1 ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ I −K⊺ −I 1⊺ 1 −1⊺ −1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛA αA⌈︁ PJP, RR

• 4. Goal B: Characterizing ASL ≥

≥ ≥ min

• B1. Starting from )︁ΛA αA⌈︁:

⌊︁ ΛA αA −I −min}︁ )︁ΛB αB⌈︁ RR

• 5. Goal C: Characterizing coherence

Based on the existence of S-dominating linear previsions:

• C1. Analogous to A1 & intersection over all S in 𝒯:

∀S ∈ 𝒯 ∶ ∃µS,νS ≥ 0 ∶ P = K⊺µS−SνS ∧ 1⊺µS = 1 ⌊︁K⊺ −S 1⊺ 0⊺}︁ )︁ΛC αC⌈︁ EN, ISS∈𝒯, RR

• C2. Analogous to A2 & intersection over all S in 𝒯:

∀S ∈ 𝒯 ∶ ∃µS ≥ 0 ∶ SP ≤ SK⊺µS ∧ 1⊺µS = 1 ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ S −SK⊺ −I 1⊺ 1 −1⊺ −1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛC αC⌈︁ PJP, ISS∈𝒯, RR =∶ )︁AS,P AS,µS b0⌈︁

• C3. Block matrix form of C2:

)︁AP Aµ b⌈︁ ∶= ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ AI,P AI,µI b0 ⋮ ⋱ ⋮ AS,P AS,µS b0 ⋮ ⋱ ⋮ ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛC αC⌈︁ PJP, RR

• 6. Illustrations of Procedure C1

Pg1 1 2 1 Pg2 1 2 1 P b P a P c Ω = {a,b,c} K = ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ 1 0 1 2 1 0 1 2 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ ⋃︂1 1⨄︂ ⋃︂−1 1⨄︂ ⋃︂1 −1⨄︂ P b P a P c min facet enumerate the V-representa- tion for avoiding sure S-loss for each S in 𝒯 intersect and remove redundancy Pg1 Pg2 Pg3 P b P a P c Ω = {a,b,c} K = ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ 1 0 1 2 1 2 1 0 0 1 2 1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ min P b P a P c intersect and remove redundancy I WANT YOU to ERADICATE INCOHERENCE utterly

• 1. Context & Goal

Given: incoherent lower prevision P. Goal: Find a coherent correction to it.

• 2. Bring within bounds

If Pf ∉ ⋃︂min f,max f⨄︂ for some f in 𝒧, it is

• ut of bounds. To bring it within bounds:

BP f ∶= )︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌋︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ]︁ min f Pf ≤ min f, max f P f ≥ max f, P f

• therwise.

P BP Q BQ lower previsions

• ut of bounds
• 3. Downward correction

As the downward correction of P we take the lower envelope of the maximal coherent dominated lower previsions (proposed earlier by Pelessoni & Vicig, following Weichselberger), so the nadir point DP of the MOLP (cf. C) (†) maximize Q, subject to ΛCQ ≤ αC

Q ≤ P

• r the MOLP (cf. C3)

(‡) maximize Q, subject to AQQ+Aµµ ≤ b Q ≤ P. Some desirable properties:

• It is the maximal neutral correction

(‘no component tradeoffs’).

• The imprecision of the correction is

nondecreasing with incoherence. P DP Q DQ DP P dominated lower previsions extreme coherent dominated lower prevision For the future: Can the computation be simplified for special classes of P?

• 4. Experiments

With the M3-solver we used, computa- tion appears exponential in ⋃︁𝒧⋃︁; using pre-computed constraints (†) is more efficient than not (‡): 2 3 4 5 6 7 8 9 10 10−2 10−1 100 101 102 103 104

3 8 17 24 39 53 112 228 247 2 1 2 1 4 1 4 1 17 1 17 1 24 3 24 3 3 2 8 2 7 2 29 2 8 3 26 5 206 2 206 2 16 16 16 16

⋃︁𝒧⋃︁ [s] DP via (†) DP via (‡) ⋃︁Ω⋃︁ = 5, σ ≈ 1⇑2 We expect other solvers and certainly direct M2-solvers to perform more efficiently, but could not test any yet.

• 5. Upward correction

The standard upward correction of P is its natural extension EP, the unique minimal pointwise dominating co- herent lower prevision, so the the solution to the MOLP (cf. C) minimize EP, subject to ΛCEP ≤ αC

EP ≥ P

• r the MOLP (cf. C3)

(*) minimize EP, subject to AEPEP+Aµµ ≤ b EP ≥ P.

• The problem becomes a plain LP by

using the objective ∑g∈𝒧EPg.

• (*) decomposes into a classical for-

mulation of natural extension. P EP Q dominating lower previsions no natural extension in case of sure loss

• 1. Representations

Any convex polyhedron in Rn can be described in two ways: H-representation (intersection of half-spaces)

)︁A b⌈︁ ∶= {x ∈ Rn ∶ Ax ≤ b}

constraint matrix in Rk×n constraint vector in Rk V-representation (convex hull of points and rays)

⌊︁V w}︁ ∶= {x ∈ Rn ∶ x =V µ ∧ µ ≥ 0 ∧ w⊺µ = 1}

vector matrix in Rn×ℓ vector in Rℓ vector in (Rℓ)≥0 with components defining points (≠ 0) and rays (= 0)

• 2. Illustration

Here n = 2, k = 3, and ℓ = 4. constraint redundant constraint redundant point extreme ray vertex I WANT YOU to juggle POL YHEDRA like there’s no tomorrow

• 3. Tasks
• RR. Removing redundancy: if j is the

numberof non-redundant con- straints (or vectors), this requires solving k (or ℓ) linear programming problems of size n× j

• EN. Moving between H- and V-represent-

ations: done using vertex/facet enu- meration algorithms; polynomial in n, k, and ℓ.

• PJ. Projection on a lower-dimensional

space: easy with V-representations, hard with H-representations.

• IS. Intersection: easy with H-represent-

ations, hard with V-representations.

• 1. Formalization

Any multi-objective linear program (MOLP) can be put in the following form: maximize y =Cx, subject to Ax ≤ b and x ≥ 0

• bjective

vector in Rm

• bjective

matrix in Rm×n

• ptimization

vector in Rn constraint matrix in Rk×n constraint vector in Rk

• 3. Tasks

Main computational tasks in non- decreasing order of complexity:

• M1. Finding ˆ

y.

• M2. Finding ˇ

y.

• M3. Finding ext𝒵∗

and characterizing 𝒵∗.

• M4. Finding ext𝒴 ∗.
• M5. Characterizing 𝒴 ∗.
• 2. Illustration

Here m = n = 2 and k = 4. x1 x2 𝒴 𝒴 ∗ C1 C2 y1 y2 𝒵 𝒵∗ ˆ y ˇ y feasible optimization vectors {x ∈ Rn ∶ Ax ≤ b ∧ x ≥ 0} C-undominated optimization vectors {x ∈ 𝒴 ∶ (∀z ∈ 𝒴 ∶Cx ⇑ <Cz)} with vertices ext𝒴 ∗ undominated objective vectors {Cx ∶ x ∈ 𝒴 ∗} with vertices ext𝒵∗ ideal point, with ˆ yi = max{yi ∶ y ∈ 𝒵} nadir point, with ˇ yi = min{yi ∶ y ∈ 𝒵∗} feasible objective vectors {Cx ∶ x ∈ 𝒴} I WANT YOU to grok MUL TI-OBJECTIVE LINEAR PROGRAMMING

SYSTeMS Research Group Ghent University Erik Quaeghebeur Decision Support Systems Group Utrecht University

Coherence characterization procedures Incoherence correction procedures Polytope theory Multi-

• bjective

linear programming

SLIDE 28

Characterizing Coherence, Correcting Incoherence

I WANT YOU to crank out COHERENCE CHARACTERIZATIONS

• 1. Context

Basic setup:

• Finite possibility space Ω
• Finite set of gambles 𝒧 on Ω
• Lower previsions P on 𝒧

Matrix notation:

• ⋃︁Ω⋃︁-by-⋃︁𝒧⋃︁ matrix K with

gambles as columns

• the rows of K (columns of K⊺)

are the degenerate previsions

• the set 𝒯 of matrices S obtained

from the identity matrix I by changing at most one 1 to −1

• all-one (zero) column vector 1 (0)
• 2. Goals

Given K, find a non-redundant H- representations for the set of all P

• A. that avoid sure loss ()︁ΛA αA⌈︁),
• B. that avoid sure loss and for

which P ≥ min ()︁ΛB αB⌈︁),

• C. that are coherent ()︁ΛC αC⌈︁).
• 7. Experiments

The sparsity σ is the fraction of zero components in K. Procedure C1 is exponential in 1 − σ and ∼linear in ⋃︁Ω⋃︁: 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 10−2 10−1 100 101 102 ⋃︁Ω⋃︁ = 4 8 16 32 64 128 256 512 1024 2048 4096 ⋃︁Ω⋃︁ = 8192 σ [s] ⋃︁𝒧⋃︁ = 5 . . . and (at least) exponential in ⋃︁𝒧⋃︁: 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 10−3 10−2 10−1 100 101 102 103 ⋃︁𝒧⋃︁ = 3 ⋃︁𝒧⋃︁ = 4 ⋃︁𝒧⋃︁ = 6 ⋃︁𝒧⋃︁ = 8 ⋃︁𝒧⋃︁ = 9 ⋃︁𝒧⋃︁ = 12 σ [s] ⋃︁Ω⋃︁ = 6

• 3. Goal A: Characterizing ASL

Based on the existence of a dominating linear prevision:

• A1. ∃µI,νI ≥ 0 ∶

P = K⊺µI−IνI ∧ 1⊺µI = 1 ⌊︁K⊺ −I 1⊺ 0⊺}︁ )︁ΛA αA⌈︁ EN, RR

• A2. ∃µI ≥ 0 ∶

P ≤ K⊺µI ∧ 1⊺µI = 1 ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ I −K⊺ −I 1⊺ 1 −1⊺ −1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛA αA⌈︁ PJP, RR

• 4. Goal B: Characterizing ASL ≥

≥ ≥ min

• B1. Starting from )︁ΛA αA⌈︁:

⌊︁ ΛA αA −I −min}︁ )︁ΛB αB⌈︁ RR

• 5. Goal C: Characterizing coherence

Based on the existence of S-dominating linear previsions:

• C1. Analogous to A1 & intersection over all S in 𝒯:

∀S ∈ 𝒯 ∶ ∃µS,νS ≥ 0 ∶ P = K⊺µS−SνS ∧ 1⊺µS = 1 ⌊︁K⊺ −S 1⊺ 0⊺}︁ )︁ΛC αC⌈︁ EN, ISS∈𝒯, RR

• C2. Analogous to A2 & intersection over all S in 𝒯:

∀S ∈ 𝒯 ∶ ∃µS ≥ 0 ∶ SP ≤ SK⊺µS ∧ 1⊺µS = 1 ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ S −SK⊺ −I 1⊺ 1 −1⊺ −1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛC αC⌈︁ PJP, ISS∈𝒯, RR =∶ )︁AS,P AS,µS b0⌈︁

• C3. Block matrix form of C2:

)︁AP Aµ b⌈︁ ∶= ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ AI,P AI,µI b0 ⋮ ⋱ ⋮ AS,P AS,µS b0 ⋮ ⋱ ⋮ ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ )︁ΛC αC⌈︁ PJP, RR

• 6. Illustrations of Procedure C1

Pg1 1 2 1 Pg2 1 2 1 P b P a P c Ω = {a,b,c} K = ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ 1 0 1 2 1 0 1 2 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ ⋃︂1 1⨄︂ ⋃︂−1 1⨄︂ ⋃︂1 −1⨄︂ P b P a P c min facet enumerate the V-representa- tion for avoiding sure S-loss for each S in 𝒯 intersect and remove redundancy Pg1 Pg2 Pg3 P b P a P c Ω = {a,b,c} K = ⎨ ⎝ ⎝ ⎝ ⎝ ⎝ ⎪ 1 0 1 2 1 2 1 0 0 1 2 1 ⎬ ⎠ ⎠ ⎠ ⎠ ⎠ ⎮ min P b P a P c intersect and remove redundancy I WANT YOU to ERADICATE INCOHERENCE utterly

• 1. Context & Goal

Given: incoherent lower prevision P. Goal: Find a coherent correction to it.

• 2. Bring within bounds

If Pf ∉ ⋃︂min f,max f⨄︂ for some f in 𝒧, it is

• ut of bounds. To bring it within bounds:

BP f ∶= )︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌋︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ⌉︁ ]︁ min f Pf ≤ min f, max f P f ≥ max f, P f

• therwise.

P BP Q BQ lower previsions

• ut of bounds
• 3. Downward correction

As the downward correction of P we take the lower envelope of the maximal coherent dominated lower previsions (proposed earlier by Pelessoni & Vicig, following Weichselberger), so the nadir point DP of the MOLP (cf. C) (†) maximize Q, subject to ΛCQ ≤ αC

Q ≤ P

• r the MOLP (cf. C3)

(‡) maximize Q, subject to AQQ+Aµµ ≤ b Q ≤ P. Some desirable properties:

• It is the maximal neutral correction

(‘no component tradeoffs’).

• The imprecision of the correction is

nondecreasing with incoherence. P DP Q DQ DP P dominated lower previsions extreme coherent dominated lower prevision For the future: Can the computation be simplified for special classes of P?

• 4. Experiments

With the M3-solver we used, computa- tion appears exponential in ⋃︁𝒧⋃︁; using pre-computed constraints (†) is more efficient than not (‡): 2 3 4 5 6 7 8 9 10 10−2 10−1 100 101 102 103 104

3 8 17 24 39 53 112 228 247 2 1 2 1 4 1 4 1 17 1 17 1 24 3 24 3 3 2 8 2 7 2 29 2 8 3 26 5 206 2 206 2 16 16 16 16

⋃︁𝒧⋃︁ [s] DP via (†) DP via (‡) ⋃︁Ω⋃︁ = 5, σ ≈ 1⇑2 We expect other solvers and certainly direct M2-solvers to perform more efficiently, but could not test any yet.

• 5. Upward correction

The standard upward correction of P is its natural extension EP, the unique minimal pointwise dominating co- herent lower prevision, so the the solution to the MOLP (cf. C) minimize EP, subject to ΛCEP ≤ αC

EP ≥ P

• r the MOLP (cf. C3)

(*) minimize EP, subject to AEPEP+Aµµ ≤ b EP ≥ P.

• The problem becomes a plain LP by

using the objective ∑g∈𝒧EPg.

• (*) decomposes into a classical for-

mulation of natural extension. P EP Q dominating lower previsions no natural extension in case of sure loss

• 1. Representations

Any convex polyhedron in Rn can be described in two ways: H-representation (intersection of half-spaces)

)︁A b⌈︁ ∶= {x ∈ Rn ∶ Ax ≤ b}

constraint matrix in Rk×n constraint vector in Rk V-representation (convex hull of points and rays)

⌊︁V w}︁ ∶= {x ∈ Rn ∶ x =V µ ∧ µ ≥ 0 ∧ w⊺µ = 1}

vector matrix in Rn×ℓ vector in Rℓ vector in (Rℓ)≥0 with components defining points (≠ 0) and rays (= 0)

• 2. Illustration

Here n = 2, k = 3, and ℓ = 4. constraint redundant constraint redundant point extreme ray vertex I WANT YOU to juggle POL YHEDRA like there’s no tomorrow

• 3. Tasks
• RR. Removing redundancy: if j is the

numberof non-redundant con- straints (or vectors), this requires solving k (or ℓ) linear programming problems of size n× j

• EN. Moving between H- and V-represent-

ations: done using vertex/facet enu- meration algorithms; polynomial in n, k, and ℓ.

• PJ. Projection on a lower-dimensional

space: easy with V-representations, hard with H-representations.

• IS. Intersection: easy with H-represent-

ations, hard with V-representations.

• 1. Formalization

Any multi-objective linear program (MOLP) can be put in the following form: maximize y =Cx, subject to Ax ≤ b and x ≥ 0

• bjective

vector in Rm

• bjective

matrix in Rm×n

• ptimization

vector in Rn constraint matrix in Rk×n constraint vector in Rk

• 3. Tasks

Main computational tasks in non- decreasing order of complexity:

• M1. Finding ˆ

y.

• M2. Finding ˇ

y.

• M3. Finding ext𝒵∗

and characterizing 𝒵∗.

• M4. Finding ext𝒴 ∗.
• M5. Characterizing 𝒴 ∗.
• 2. Illustration

Here m = n = 2 and k = 4. x1 x2 𝒴 𝒴 ∗ C1 C2 y1 y2 𝒵 𝒵∗ ˆ y ˇ y feasible optimization vectors {x ∈ Rn ∶ Ax ≤ b ∧ x ≥ 0} C-undominated optimization vectors {x ∈ 𝒴 ∶ (∀z ∈ 𝒴 ∶Cx ⇑ <Cz)} with vertices ext𝒴 ∗ undominated objective vectors {Cx ∶ x ∈ 𝒴 ∗} with vertices ext𝒵∗ ideal point, with ˆ yi = max{yi ∶ y ∈ 𝒵} nadir point, with ˇ yi = min{yi ∶ y ∈ 𝒵∗} feasible objective vectors {Cx ∶ x ∈ 𝒴} I WANT YOU to grok MUL TI-OBJECTIVE LINEAR PROGRAMMING

SYSTeMS Research Group Ghent University Erik Quaeghebeur Decision Support Systems Group Utrecht University

Coherence characterization procedures Incoherence correction procedures Polytope theory Multi-

• bjective

linear programming

SLIDE 29

I WANT YOU to COME TO MY POSTER