The CONEstrip Algorithm Erik Quaeghebeur SYSTeMS Research Group, - PowerPoint PPT Presentation

The CONEstrip Algorithm Erik Quaeghebeur SYSTeMS Research Group, Ghent University, Belgium Erik.Quaeghebeur@UGent.be

Avoiding sure loss ▸ Finite possibility space Ω , ▸ Linear vector space L ∶= [ Ω → R ] , ▸ Finite set of gambles K ⋐ L , ▸ Lower prevision P ∈ [ K → R ] , ▸ Set of marginal gambles A ∶= { h − Ph ∶ h ∈ K } .

Avoiding sure loss ▸ Finite possibility space Ω , ▸ Linear vector space L ∶ = [ Ω → R ] , ▸ Finite set of gambles K ⋐ L , ▸ Lower prevision P ∈ [ K → R ] , ▸ Set of marginal gambles A ∶ = { h − Ph ∶ h ∈ K } . λ ∈ R A , find subject to ∑ g ∈A λ g ⋅ g ⋖ 0 and λ ≥ 0 .

Avoiding sure loss ▸ Finite possibility space Ω , ▸ Linear vector space L ∶ = [ Ω → R ] , ▸ Finite set of gambles K ⋐ L , ▸ Lower prevision P ∈ [ K → R ] , ▸ Set of marginal gambles A ∶ = { h − Ph ∶ h ∈ K } . λ ∈ R A , find subject to ∑ g ∈A λ g ⋅ g ⋖ 0 and λ ≥ 0 . ▸ Indicator function 1 B of an event B ⊆ Ω ; 1 ω ∶ = 1 { ω } for ω ∈ Ω . ( λ , µ ) ∈ R A × R Ω , find subject to ∑ g ∈A λ g ⋅ g + ∑ ω ∈ Ω µ ω ⋅ 1 ω = 0 and λ ≥ 0 and µ ≥ 1 .

Avoiding partial loss ▸ Set of (finite) events Ω ∗ , ▸ Finite set of (gamble, event)-pairs N ⋐ L × Ω ∗ , ▸ Conditional lower prevision P ∈ [ N → R ] , ▸ Set of (conditional marginal gamble, event)-pairs B ∶ = {([ h − P ( h ∣ B )] ⋅ 1 B , B ) ∶ ( h , B ) ∈ N } .

Avoiding partial loss ▸ Set of (finite) events Ω ∗ , ▸ Finite set of (gamble, event)-pairs N ⋐ L × Ω ∗ , ▸ Conditional lower prevision P ∈ [ N → R ] , ▸ Set of (conditional marginal gamble, event)-pairs B ∶ = {([ h − P ( h ∣ B )] ⋅ 1 B , B ) ∶ ( h , B ) ∈ N } . ( λ , ε ) ∈ R B × R B , find ∑ ( g , B )∈B λ g , B ⋅ [ g + ε g , B ⋅ 1 B ] ≤ 0 λ > 0 ε ⋗ 0 . subject to and and

Avoiding partial loss ▸ Set of (finite) events Ω ∗ , ▸ Finite set of (gamble, event)-pairs N ⋐ L × Ω ∗ , ▸ Conditional lower prevision P ∈ [ N → R ] , ▸ Set of (conditional marginal gamble, event)-pairs B ∶ = {([ h − P ( h ∣ B )] ⋅ 1 B , B ) ∶ ( h , B ) ∈ N } . ( λ , ε ) ∈ R B × R B , find ∑ ( g , B )∈B λ g , B ⋅ [ g + ε g , B ⋅ 1 B ] ≤ 0 λ > 0 ε ⋗ 0 . subject to and and ( λ , ν , µ ) ∈ R B × ( R B × R B ) × R Ω , find ∑ ( g , B )∈B λ g , B ⋅ [ ν g , B , g ⋅ g + ν g , B , B ⋅ 1 B ] + ∑ ω ∈ Ω µ ω ⋅ 1 ω = 0 subject to λ > 0 ν ⋗ 0 µ ≥ 0 . and to and and

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 g 3 g 1 g 4 g 10 R g 9 g 5 g 8 g 6 g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 { g k ∶ k = 1 .. 10 } g 3 g 1 g 4 { g 1 , g 2 } { g 2 , g 4 } { g 6 } { g 8 , g 9 } g 10 R g 9 g 5 g 8 g 6 { g 2 } { g 2 } { g 4 } g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} . Cone-in-facet representation: {{ g k ∶ k = 1 .. 10 } , { g 1 , g 2 } , { g 2 , g 4 } , { g 6 } , { g 8 , g 9 } , { g 2 } , { g 4 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 { g k ∶ k = 1 .. 10 } g 3 g 1 g 4 { g 1 , g 2 } { g 2 , g 4 } { g 6 } { g 8 , g 9 } g 10 R g 9 g 5 g 8 g 6 { g 2 } { g 2 } { g 4 } g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} . Cone-in-facet representation: {{ g k ∶ k = 1 .. 10 } , { g 1 , g 2 } , { g 2 , g 4 } , { g 6 } , { g 8 , g 9 } , { g 2 } , { g 4 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 { g k ∶ k = 1 .. 10 } g 3 g 1 g 4 { g 1 , g 2 } { g 2 , g 4 } { g 6 } { g 8 , g 9 } g 10 R g 9 g 5 g 8 g 6 { g 2 } { g 2 } { g 4 } g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} . Cone-in-facet representation: {{ g k ∶ k = 1 .. 10 } , { g 1 , g 2 } , { g 2 , g 4 } , { g 6 } , { g 8 , g 9 } , { g 2 } , { g 4 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 { g k ∶ k = 1 .. 10 } g 3 g 1 g 4 { g 1 , g 2 } { g 2 , g 4 } { g 6 } { g 8 , g 9 } g 10 R g 9 g 5 g 8 g 6 { g 2 } { g 2 } { g 4 } g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} . Cone-in-facet representation: {{ g k ∶ k = 1 .. 10 } , { g 1 , g 2 } , { g 2 , g 4 } , { g 6 } , { g 8 , g 9 } , { g 2 } , { g 4 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 { g k ∶ k = 1 .. 10 } g 3 g 1 g 4 { g 1 , g 2 } { g 2 , g 4 } { g 6 } { g 8 , g 9 } g 10 R g 9 g 5 g 8 g 6 { g 2 } { g 2 } { g 4 } g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} . Cone-in-facet representation: {{ g k ∶ k = 1 .. 10 } , { g 1 , g 2 } , { g 2 , g 4 } , { g 6 } , { g 8 , g 9 } , { g 2 } , { g 4 }} .

Representation of finitary general cones As a convex closure of a finite number of finitary open cones: R ∶ = {∑ D∈R λ D ⋅ ∑ g ∈D ν D , g ⋅ g ∶ λ > 0 , ν ⋗ 0 } R ⋐ L ∗ . for g 2 { g k ∶ k = 1 .. 10 } g 3 g 1 g 4 { g 1 , g 2 } { g 2 , g 4 } { g 6 } { g 8 , g 9 } g 10 R g 9 g 5 g 8 g 6 { g 2 } { g 2 } { g 4 } g 7 R ∶ = {{ g 3 , g 5 , g 10 } , { g 1 , g 2 } , { g 2 , g 7 } , { g 8 , g 9 } , { g 2 } , { g 4 } , { g 6 }} . Cone-in-facet representation: {{ g k ∶ k = 1 .. 10 } , { g 1 , g 2 } , { g 2 , g 4 } , { g 6 } , { g 8 , g 9 } , { g 2 } , { g 4 }} .