Certificates of positivity and polynomial minimization in the multivariate Bernstein

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Bernstein basis Standard triangulation Control polytope Certificates of positivity Polynomial minimization

Certificates of positivity and polynomial minimization in the multivariate Bernstein

basis

Richard Leroy

IRMAR - Universit´ e de Rennes 1

R. Leroy — Certificates of positivity and polynomial minimization - 1 -

(2)

Motivation

Notations

f ∈ R [X ] = R [X ₁ , . . . , X _k ] of degree d

non degenerate simplex V = Conv [V ₀ , . . . , V _k ] ⊂ R ^k barycentric coordinates λ i (i = 0, . . . , k) :

polynomials of degree 1 P λ _i = 1

x ∈ V ⇔ ∀ i, λ i (x) ≥ 0

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Motivation

Notations

f ∈ R [X ] = R [X ₁ , . . . , X _k ] of degree d

non degenerate simplex V = Conv [V ₀ , . . . , V _k ] ⊂ R ^k barycentric coordinates λ i (i = 0, . . . , k) :

polynomials of degree 1 P λ _i = 1

x ∈ V ⇔ ∀ i, λ i (x) ≥ 0

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Motivation

Notations

f ∈ R [X ] = R [X ₁ , . . . , X _k ] of degree d

non degenerate simplex V = Conv [V ₀ , . . . , V _k ] ⊂ R ^k

barycentric coordinates λ i (i = 0, . . . , k) :

polynomials of degree 1 P λ _i = 1

x ∈ V ⇔ ∀ i, λ i (x) ≥ 0

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Motivation

Notations

f ∈ R [X ] = R [X ₁ , . . . , X _k ] of degree d

non degenerate simplex V = Conv [V ₀ , . . . , V _k ] ⊂ R ^k barycentric coordinates λ i (i = 0, . . . , k) :

polynomials of degree 1 P λ i = 1

x ∈ V ⇔ ∀ i, λ _i (x) ≥ 0

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Motivation

Example : standard simplex

∆ = { x ∈ R ^k | ∀ i , x i ≥ 0 et X

x i = 1 }

x ≥ 0 1 − x ≥ 0

x ≥ 0 y ≥ 0 1 − x − y ≥ 0

x ≥ 0 y ≥ 0 z ≥ 0

1 − x − y − z ≥ 0

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Motivation

Questions

Decide if f is positive on V (or not) Obtain a simple proof

, → certificate of positivity

, → formal proof checker (COQ)

Compute the minimum of f over V (and localize the minimizers)

, → epidemiology problems

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Motivation

Questions

Decide if f is positive on V (or not)

Obtain a simple proof

, → certificate of positivity

, → formal proof checker (COQ)

Compute the minimum of f over V (and localize the minimizers)

, → epidemiology problems

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Motivation

Questions

Decide if f is positive on V (or not) Obtain a simple proof

, → certificate of positivity

, → formal proof checker (COQ) Compute the minimum of f over V (and localize the minimizers)

, → epidemiology problems

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Motivation

Questions

Decide if f is positive on V (or not) Obtain a simple proof

, → certificate of positivity

, → formal proof checker (COQ) Compute the minimum of f over V (and localize the minimizers)

, → epidemiology problems

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Motivation

Questions

Decide if f is positive on V (or not) Obtain a simple proof

, → certificate of positivity

, → formal proof checker (COQ)

Compute the minimum of f over V (and localize the minimizers)

, → epidemiology problems

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Motivation

Questions

Decide if f is positive on V (or not) Obtain a simple proof

, → certificate of positivity

, → formal proof checker (COQ) Compute the minimum of f over V (and localize the minimizers)

, → epidemiology problems

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Motivation

Questions

Decide if f is positive on V (or not) Obtain a simple proof

, → certificate of positivity

, → formal proof checker (COQ) Compute the minimum of f over V (and localize the minimizers)

, → epidemiology problems

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Outline

1 Multivariate Bernstein basis 2 Standard triangulation

3 Control polytope : approximation and convergence 4 Certificates of positivity

5 Polynomial minimization

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1 Multivariate Bernstein basis 2 Standard triangulation

3 Control polytope : approximation and convergence 4 Certificates of positivity

5 Polynomial minimization

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Bernstein polynomials

Notations

multi-index α = (α 0 , . . . , α k ) ∈ N ^k+1

| α | = α 0 + · · · + α k = d multinomial coefficient ^d _α

= d !

α 0 ! . . . α k !

Bernstein polynomials of degree d with respect to V B _α ^d =

d α

λ ^α =

d α

λ 0 α

0

. . . λ k α

k

.

Appear naturally in the expansion 1 = 1 ^d = (λ 0 + · · · + λ k ) ^d = X

|α|=d

d α

λ ^α = X

|α|=d

B _α ^d .

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Bernstein polynomials

Notations

multi-index α = (α 0 , . . . , α k ) ∈ N ^k+1

| α | = α 0 + · · · + α k = d multinomial coefficient ^d _α

= d !

α 0 ! . . . α k !

Bernstein polynomials of degree d with respect to V B _α ^d =

d α

λ ^α =

d α

λ 0 α

0

. . . λ k α

k

.

Appear naturally in the expansion 1 = 1 ^d = (λ 0 + · · · + λ k ) ^d = X

|α|=d

d α

λ ^α = X

|α|=d

B _α ^d .

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Bernstein polynomials

Notations

multi-index α = (α 0 , . . . , α k ) ∈ N ^k+1

| α | = α 0 + · · · + α k = d multinomial coefficient ^d _α

= d !

α 0 ! . . . α k !

Bernstein polynomials of degree d with respect to V B _α ^d =

d α

λ ^α =

d α

λ 0 α

0

. . . λ k α

k

.

Appear naturally in the expansion 1 = 1 ^d = (λ + · · · + λ ) ^d = X

d

λ ^α = X

B ^d .

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Properties

nonnegative on V basis of R ^≤d [X ]

, → Bernstein coefficients : f = P

|α|=d

b _α (f , d, V )B _α ^d .

b(f , d , V ) : list of coefficients b α = b α (f , d , V )

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Properties

nonnegative on V

basis of R ^≤d [X ]

, → Bernstein coefficients : f = P

|α|=d

b _α (f , d, V )B _α ^d .

b(f , d , V ) : list of coefficients b α = b α (f , d , V )

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Properties

nonnegative on V basis of R ^≤d [X ]

, → Bernstein coefficients : f = P

|α|=d

b _α (f , d, V )B _α ^d .

b(f , d , V ) : list of coefficients b α = b α (f , d , V )

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Properties

nonnegative on V basis of R ^≤d [X ]

, → Bernstein coefficients : f = P

|α|=d

b _α (f , d , V )B _α ^d .

b(f , d , V ) : list of coefficients b α = b α (f , d , V )

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Control net

Gr´ eville grid : points N α = α 0 V ₀ + · · · + α k V _k Control net : points (N _α , b _α ) d

Discrete graph of f : points (N _α , f (N _α ))

Graph of f

Control points

Gr´eville points

0 1

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Control net

Gr´ eville grid : points N α = α 0 V ₀ + · · · + α k V _k d

Control net : points (N _α , b _α )

Discrete graph of f : points (N _α , f (N _α ))

Graph of f

Control points

Gr´eville points

0 1

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Control net

Gr´ eville grid : points N α = α 0 V ₀ + · · · + α k V _k Control net : points (N _α , b _α ) d

Discrete graph of f : points (N _α , f (N _α ))

Graph of f

Control points

Gr´eville points

0 1

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Control net

Gr´ eville grid : points N α = α 0 V ₀ + · · · + α k V _k Control net : points (N _α , b _α ) d

Discrete graph of f : points (N α , f (N α ))

Graph of f

Control points

Gr´eville points

0 1

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Control net

Gr´ eville grid : points N α = α 0 V ₀ + · · · + α k V _k Control net : points (N _α , b _α ) d

Discrete graph of f : points (N α , f (N α ))

Graph of f

Control points

Gr´eville points

0 1

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Interpolation properties

Linear precision

If d ≤ 1 : b _α = f (N _α ) Interpolation at vertices

b _de

_i

= f (V _i )

What about the other coefficients when d ≥ 2 ?

, → bound on the gap between f (N α ) and b α

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Interpolation properties

Linear precision

If d ≤ 1 : b _α = f (N _α )

Interpolation at vertices

b _de

_i

= f (V _i )

What about the other coefficients when d ≥ 2 ?

, → bound on the gap between f (N α ) and b α

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Interpolation properties

Linear precision

If d ≤ 1 : b _α = f (N _α ) Interpolation at vertices

b _de

_i

= f (V _i )

What about the other coefficients when d ≥ 2 ?

, → bound on the gap between f (N α ) and b α

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Interpolation properties

Linear precision

If d ≤ 1 : b _α = f (N _α ) Interpolation at vertices

b _de

_i

= f (V _i )

What about the other coefficients when d ≥ 2 ?

, → bound on the gap between f (N α ) and b α

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Interpolation properties

Linear precision

If d ≤ 1 : b _α = f (N _α ) Interpolation at vertices

b _de

_i

= f (V _i )

What about the other coefficients when d ≥ 2 ?

, → bound on the gap between f (N α ) and b α

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1 Multivariate Bernstein basis 2 Standard triangulation

3 Control polytope : approximation and convergence 4 Certificates of positivity

5 Polynomial minimization

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Kuhn’s triangulation of the cube

Goal : Triangulate the unit cube C = [0, 1] ^k .

Idea : ∀ σ ∈ S _k , consider the simplex V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] defined as follows :

V ₀ ^σ = (0, . . . , 0)

V _i ^σ = e _σ(1) + · · · + e _σ(i ₎ (1 ≤ i ≤ k).

Result : These simplices form a triangulation of C .

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Kuhn’s triangulation of the cube

Goal : Triangulate the unit cube C = [0, 1] ^k .

Idea : ∀ σ ∈ S _k , consider the simplex V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] defined as follows :

V ₀ ^σ = (0, . . . , 0)

V _i ^σ = e _σ(1) + · · · + e _σ(i ₎ (1 ≤ i ≤ k). Result : These simplices form a triangulation of C .

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Kuhn’s triangulation of the cube

Goal : Triangulate the unit cube C = [0, 1] ^k .

Idea : ∀ σ ∈ S _k , consider the simplex V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] defined as follows :

V ₀ ^σ = (0, . . . , 0)

V _i ^σ = e _σ(1) + · · · + e _σ(i ₎ (1 ≤ i ≤ k).

Result : These simplices form a triangulation of C .

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Kuhn’s triangulation of the cube

Goal : Triangulate the unit cube C = [0, 1] ^k .

Idea : ∀ σ ∈ S _k , consider the simplex V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] defined as follows :

V ₀ ^σ = (0, . . . , 0)

V _i ^σ = e _σ(1) + · · · + e _σ(i ₎ (1 ≤ i ≤ k).

Result : These simplices form a triangulation of C .

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Kuhn’s triangulation of the cube

In dimension 2 and 3

(1 2) (2 1)

Figure : Kuhn’s triangulation in dimension 2

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Kuhn’s triangulation of the cube

In dimension 2 and 3

x

1

x

3

x

2

(3 2 1) (3 1 2) (1 2 3)

(2 3 1) (1 3 2) (2 1 3)

Figure : Kuhn’s triangulation in dimension 3

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Kuhn’s triangulation of the cube

Adjacencies

V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] and V ^σ

⁰

=

V ₀ ^σ

⁰

, . . . , V _k ^σ

⁰

are adjacent m

∃ p, σ ⁰ (p) = σ(p + 1) and σ ⁰ (p + 1) = σ(p)

⇓

V p−1 , V _p ^σ , V _p ^σ

⁰

, V p+1 form a parallelogram.

(3 2 1) (3 1 2)

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Kuhn’s triangulation of the cube

Adjacencies

V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] and V ^σ

⁰

=

V ₀ ^σ

⁰

, . . . , V _k ^σ

⁰

are adjacent

m

∃ p, σ ⁰ (p) = σ(p + 1) and σ ⁰ (p + 1) = σ(p)

⇓

V p−1 , V _p ^σ , V _p ^σ

⁰

, V p+1 form a parallelogram.

(3 2 1) (3 1 2)

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Kuhn’s triangulation of the cube

Adjacencies

V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] and V ^σ

⁰

=

V ₀ ^σ

⁰

, . . . , V _k ^σ

⁰

are adjacent

m

∃ p, σ ⁰ (p) = σ(p + 1) and σ ⁰ (p + 1) = σ(p)

⇓

V p−1 , V _p ^σ , V _p ^σ

⁰

, V p+1 form a parallelogram.

(3 2 1) (3 1 2)

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Kuhn’s triangulation of the cube

Adjacencies

V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] and V ^σ

⁰

=

V ₀ ^σ

⁰

, . . . , V _k ^σ

⁰

are adjacent m

∃ p, σ ⁰ (p) = σ(p + 1) and σ ⁰ (p + 1) = σ(p)

⇓

V p−1 , V _p ^σ , V _p ^σ

⁰

, V p+1 form a parallelogram.

(3 2 1) (3 1 2)

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Kuhn’s triangulation of the cube

Adjacencies

V ^σ = [V ₀ ^σ , . . . , V _k ^σ ] and V ^σ

⁰

=

V ₀ ^σ

⁰

, . . . , V _k ^σ

⁰

are adjacent m

∃ p, σ ⁰ (p) = σ(p + 1) and σ ⁰ (p + 1) = σ(p)

⇓

V p−1 , V _p ^σ , V _p ^σ

⁰

, V p+1 form a parallelogram.

(3 2 1) (3 1 2)

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k} Reorder the images of F into f ₁ , . . . , f _k Define the vertex V ₀ ^F = 1

d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` ) Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } . Then define the simplex V ^F = V ₀ ^F + 1

d V ^σ

^F

= h

V ₀ ^F , V ₀ ^F + e _σ

_F

₍₁₎

d , . . . , V ₀ ^F + e _σ

_F

₍₁₎

d + · · · + e _σ

_F

_(k ₎ d

i .

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i .

Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k} Reorder the images of F into f ₁ , . . . , f _k Define the vertex V ₀ ^F = 1

d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` ) Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } . Then define the simplex V ^F = V ₀ ^F + 1

d V ^σ

^F

= h

V ₀ ^F , V ₀ ^F + e _σ

_F

₍₁₎

d , . . . , V ₀ ^F + e _σ

_F

₍₁₎

d + · · · + e _σ

_F

_(k ₎ d

i

.

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k ^} Reorder the images of F into f ₁ , . . . , f _k Define the vertex V ₀ ^F = 1

d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` ) Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } . Then define the simplex V ^F = V ₀ ^F + 1

d V ^σ

^F

= h

V ₀ ^F , V ₀ ^F + e _σ

_F

₍₁₎

d , . . . , V ₀ ^F + e _σ

_F

₍₁₎

d + · · · + e _σ

_F

_(k ₎ d

i .

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k ^}

Reorder the images of F into f ₁ , . . . , f _k Define the vertex V ₀ ^F = 1

d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` ) Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } . Then define the simplex V ^F = V ₀ ^F + 1

d V ^σ

^F

= h

V ₀ ^F , V ₀ ^F + e _σ

_F

₍₁₎

d , . . . , V ₀ ^F + e _σ

_F

₍₁₎

d + · · · + e _σ

_F

_(k ₎ d

i

.

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k ^} Reorder the images of F into f ₁ , . . . , f _k

Define the vertex V ₀ ^F = 1 d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` ) Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } . Then define the simplex V ^F = V ₀ ^F + 1

d V ^σ

^F

= h

V ₀ ^F , V ₀ ^F + e _σ

_F

₍₁₎

d , . . . , V ₀ ^F + e _σ

_F

₍₁₎

d + · · · + e _σ

_F

_(k ₎ d

i .

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k ^} Reorder the images of F into f ₁ , . . . , f _k Define the vertex V ₀ ^F = 1

d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` )

Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } . Then define the simplex V ^F = V ₀ ^F + 1

d V ^σ

^F

= h

V ₀ ^F , V ₀ ^F + e _σ

_F

₍₁₎

d , . . . , V ₀ ^F + e _σ

_F

₍₁₎

d + · · · + e _σ

_F

_(k ₎ d

i

.

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k ^} Reorder the images of F into f ₁ , . . . , f _k Define the vertex V ₀ ^F = 1

d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` ) Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } .

Then define the simplex V ^F = V ₀ ^F + 1 d V ^σ

^F

= h

V ₀ ^F , V ₀ ^F + e _σ

_F

₍₁₎

d , . . . , V ₀ ^F + e _σ

_F

₍₁₎

d + · · · + e _σ

_F

_(k ₎ d

i .

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Standard triangulation of a simplex

Goal : Triangulate the simplex V = h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . Idea :

Fix d ≥ 1 and F ∈ { 1, . . . , d } ^{1,...,k ^} Reorder the images of F into f ₁ , . . . , f _k Define the vertex V ₀ ^F = 1

d

k

P

`=1

(f _`+1 − f _` ) (e ₁ + . . . + e _` ) Define a permutation σ ∈ S _k as follows :

∀ j ∈ { 1, . . . , k } , σ F (j ) = # { ` ∈ { 1, . . . , k } | F (`) < F (j ) } + # { ` ∈ { 1, . . . , j } | F (`) = F (j ) } . Then define the simplex V ^F = V ₀ ^F + 1

d V ^σ

^F

=

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Standard triangulation of a simplex

Standard triangulation of degree d

The collection (V ^F ), for all F ∈ { 1, . . . , d } ^{1,...,k} , is a triangulation of h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . The standard triangulation T _d (V ) of degree d of any simplex V ⊂ R ^k is then obtained by affine transformation.

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Standard triangulation of a simplex

Standard triangulation of degree d

The collection (V ^F ), for all F ∈ { 1, . . . , d } ^{1,...,k} , is a triangulation of h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i .

The standard triangulation T _d (V ) of degree d of any

simplex V ⊂ R ^k is then obtained by affine transformation.

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Standard triangulation of a simplex

Standard triangulation of degree d

The collection (V ^F ), for all F ∈ { 1, . . . , d } ^{1,...,k} , is a triangulation of h

~ 0, e ₁ , e ₁ + e ₂ , . . . , e ₁ + e ₂ + · · · + e _k i . The standard triangulation T _d (V ) of degree d of any simplex V ⊂ R ^k is then obtained by affine transformation.

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Standard triangulation of a simplex

In dimension 2, degree 2

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Standard triangulation of a simplex

In dimension 3, degree 2

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Standard triangulation of a simplex

In dimension 3, degree 2

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Standard triangulation of a simplex

In dimension 3, degree 2

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Standard triangulation of a simplex

of degree 2

^N

Important property :

T d (T ` (V )) = T d` (V )

Here, we will consider standard triangulations of degree

2 ^N as consecutive standard triangulations of degree 2.

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Standard triangulation of a simplex

of degree 2

^N

Important property :

T d (T ` (V )) = T d` (V )

Here, we will consider standard triangulations of degree 2 ^N as consecutive standard triangulations of degree 2.

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Standard triangulation of a simplex

of degree 2

^N

Important property :

T d (T ` (V )) = T d` (V )

Here, we will consider standard triangulations of degree

2 ^N as consecutive standard triangulations of degree 2.

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1 Multivariate Bernstein basis 2 Standard triangulation

3 Control polytope : approximation and convergence 4 Certificates of positivity

5 Polynomial minimization

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Control polytope

Definition

Let f be a polynomial of degree d , and V a simplex.

The control polytope associated to f and V is the unique continuous function ˆ f , piecewise linear over each simplex of the standard triangulation of degree d of V and satisfaying the following interpolation property :

∀| α | = d , ˆ f (N _α ) = b _α .

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Control polytope

Convexity property

Theorem

The following are equivalent :

(i ) The control polytope ˆ f is convex. (ii ) ∀| γ | = d − 2, ∀ 0 ≤ i < j ≤ k,

b γ+e

_i

+e

j−1

+ b γ+e

i−1

+e

_j

− b γ+e

_i

+e

_j

− b γ+e

i−1

+e

j−1

| {z }

∆

2

b

γ,i,j

(f ,d,V )

≥ 0.

∆ 2 b γ,i,j (f , d , V ) : second differences k ∆ ₂ b(f , d , V ) k ∞ = max

|γ|=d−2 0≤i<j≤k

| ∆ ₂ b _γ,i,j (f , d, V ) | .

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Control polytope

Convexity property

Theorem

The following are equivalent :

(i ) The control polytope ˆ f is convex.

(ii ) ∀| γ | = d − 2, ∀ 0 ≤ i < j ≤ k,

b _γ+e

_i

_+e

_j−1

+ b _γ+e

_i−1

_+e

_j

− b _γ+e

_i

_+e

_j

− b _γ+e

_i−1

_+e

_j−1

| {z }

∆

2

b

γ,i,j

(f ,d,V )

≥ 0.

∆ 2 b γ,i,j (f , d , V ) : second differences k ∆ ₂ b(f , d , V ) k ∞ = max

|γ|=d−2 0≤i<j≤k

| ∆ ₂ b _γ,i,j (f , d, V ) | .

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Control polytope

Convexity property

Theorem

The following are equivalent :

(i ) The control polytope ˆ f is convex.

(ii ) ∀| γ | = d − 2, ∀ 0 ≤ i < j ≤ k,

b _γ+e

_i

_+e

_j−1

+ b _γ+e

_i−1

_+e

_j

− b _γ+e

_i

_+e

_j

− b _γ+e

_i−1

_+e

_j−1

| {z }

∆

2

b

γ,i,j

(f ,d,V )

≥ 0.

∆ 2 b γ,i,j (f , d , V ) : second differences

k ∆ ₂ b(f , d , V ) k ∞ = max

|γ|=d−2 0≤i<j≤k

| ∆ ₂ b _γ,i,j (f , d, V ) | .

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Control polytope

Convexity property

Theorem

The following are equivalent :

(i ) The control polytope ˆ f is convex.

(ii ) ∀| γ | = d − 2, ∀ 0 ≤ i < j ≤ k,

b _γ+e

_i

_+e

_j−1

+ b _γ+e

_i−1

_+e

_j

− b _γ+e

_i

_+e

_j

− b _γ+e

_i−1

_+e

_j−1

| {z }

∆

2

b

γ,i,j

(f ,d,V )

≥ 0.

∆ 2 b γ,i,j (f , d , V ) : second differences k ∆ ₂ b(f , d , V ) k ∞ = max

|γ|=d−2 | ∆ ₂ b _γ,i,j (f , d, V ) | .

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Control polytope

Approximation

Theorem (’08)

|α|=d max | f (N _α ) − b _α | ≤ k(k + 2)

24 d k ∆ ₂ b(f , d, V ) k ∞ . The bound is sharp.

Nairn et al. (’99) : bound (d /8) k ∆ ₂ b k ∞ when k = 1. Reif (’00) : bound (d /3) k ∆ ₂ b k ∞ when k = 2.

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Control polytope

Approximation

Theorem (’08)

|α|=d max | f (N _α ) − b _α | ≤ k(k + 2)

24 d k ∆ ₂ b(f , d, V ) k ∞ .

The bound is sharp.

Nairn et al. (’99) : bound (d /8) k ∆ ₂ b k ∞ when k = 1.

Reif (’00) : bound (d /3) k ∆ ₂ b k ∞ when k = 2.

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Control polytope

Approximation

Theorem (’08)

|α|=d max | f (N _α ) − b _α | ≤ k(k + 2)

24 d k ∆ ₂ b(f , d, V ) k ∞ . The bound is sharp.

Nairn et al. (’99) : bound (d /8) k ∆ ₂ b k ∞ when k = 1. Reif (’00) : bound (d /3) k ∆ ₂ b k ∞ when k = 2.

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Control polytope

Approximation

Theorem (’08)

|α|=d max | f (N _α ) − b _α | ≤ k(k + 2)

24 d k ∆ ₂ b(f , d, V ) k ∞ . The bound is sharp.

Nairn et al. (’99) : bound (d /8) k ∆ ₂ b k ∞ when k = 1.

Reif (’00) : bound (d /3) k ∆ ₂ b k ∞ when k = 2.

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Control polytope

Approximation

Theorem (’08)

|α|=d max | f (N _α ) − b _α | ≤ k(k + 2)

24 d k ∆ ₂ b(f , d, V ) k ∞ . The bound is sharp.

Nairn et al. (’99) : bound (d /8) k ∆ ₂ b k ∞ when k = 1.

Reif (’00) : bound (d /3) k ∆ ₂ b k ∞ when k = 2.

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Control polytope

Convergence under degree elevation

Theorem (’08)

Let f ∈ R [X ] of degree d , over the standard simplex ∆. Let N ≥ 1. Then :

max

|α|=2

^N

d

f α 1

2 ^N d , . . . , α k

2 ^N d

− b _α (f , 2 ^N d, ∆)

≤ k(k + 2) 24

d(d − 1)

2 ^N d − 1 k ∆ 2 b(f , d , ∆) k ∞ .

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Control polytope

Convergence under degree elevation

Theorem (’08)

Let f ∈ R [X ] of degree d , over the standard simplex ∆.

Let N ≥ 1. Then : max

|α|=2

^N

d

f α 1

2 ^N d , . . . , α k

2 ^N d

− b _α (f , 2 ^N d, ∆)

≤ k(k + 2) 24

d (d − 1)

2 ^N d − 1 k ∆ 2 b(f , d , ∆) k ∞ .

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Control polytope

Convergence under subdivision

Theorem (’08)

Let f ∈ R [X ] of degree d , over the standard simplex ∆. Let U ∈ T ₂

N

(∆) (N ≥ 1). Then :

|α|=d max | f (N α ) − b α (f , d , U) |

≤

k (k + 2) 24

2 dk(k + 1)(k + 3)

2 ^2N k ∆ ₂ b(f , d, ∆) k ∞ .

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Control polytope

Convergence under subdivision

Theorem (’08)

Let f ∈ R [X ] of degree d , over the standard simplex ∆.

Let U ∈ T ₂

N

(∆) (N ≥ 1). Then :

|α|=d max | f (N α ) − b α (f , d , U) |

≤

k(k + 2) 24

2 dk(k + 1)(k + 3)

2 ^2N k ∆ ₂ b(f , d, ∆) k ∞ .

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1 Multivariate Bernstein basis 2 Standard triangulation

3 Control polytope : approximation and convergence 4 Certificates of positivity

5 Polynomial minimization

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