Nonnegative polynomials modulo their gradient ideal

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Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?

Nonnegative polynomials modulo their gradient ideal

About the article

"Minimizing Polynomials via Sum of Squares over the Gradient Ideal"

from Demmel, Nie and Sturmfels.

Richard Leroy

6th May 2005

Universität Konstanz

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Plan

Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?

Introduction

Polynomials over their gradient varieties

Unconstrained optimization

What’s next ?

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Introduction : motivation and notations

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Introduction : motivation and notations

◮ Goal : Use semidefinite programming (SDP) to solve the

problem of minimizing a real polynomial over R ⁿ .

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Introduction : motivation and notations

◮ Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over R ⁿ .

◮ Idea : If u ∈ R ⁿ is a minimizer of a polynomial f ∈ R [X ] := R [X 1 , . . . , X _n ], then ∇ f (u) = 0, i.e.

∀ i = 1, . . . , n, ∂f

∂x _i (u) = 0

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Introduction : motivation and notations

◮ Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over R ⁿ .

◮ Idea : If u ∈ R ⁿ is a minimizer of a polynomial f ∈ R [X ] := R [X 1 , . . . , X _n ], then ∇ f (u) = 0, i.e.

∀ i = 1, . . . , n, ∂f

∂x _i (u) = 0

◮ Tools :

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Introduction : motivation and notations

◮ Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over R ⁿ .

◮ Idea : If u ∈ R ⁿ is a minimizer of a polynomial f ∈ R [X ] := R [X 1 , . . . , X _n ], then ∇ f (u) = 0, i.e.

∀ i = 1, . . . , n, ∂f

∂x _i (u) = 0

◮ Tools :

◮

(Real) algebraic geometry : Sos representation of a

nonnegative polynomial modulo its gradient ideal

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Introduction : motivation and notations

◮ Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over R ⁿ .

◮ Idea : If u ∈ R ⁿ is a minimizer of a polynomial f ∈ R [X ] := R [X 1 , . . . , X _n ], then ∇ f (u) = 0, i.e.

∀ i = 1, . . . , n, ∂f

∂x _i (u) = 0

◮ Tools :

◮

(Real) algebraic geometry : Sos representation of a nonnegative polynomial modulo its gradient ideal

◮

SDP : duality theory (sos representation / moment

approach)

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Polynomials over their gradient varieties

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Polynomials over their gradient varieties

◮ Notations :

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Polynomials over their gradient varieties

◮ Notations :

◮

Gradient varieties :

V grad (f ) := {u ∈ C ⁿ : ∇f (u) = 0} ⊂ C ⁿ V _grad

^R

(f ) := {u ∈ R ⁿ : ∇f (u) = 0} ⊂ R ⁿ

◮

Gradient ideal :

I grad (f ) := h∇f (X)i = ∂f

∂x

1

, . . . , ∂f

∂x n

⊂ R [X ]

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Polynomials over their gradient varieties

◮ Notations :

◮

Gradient varieties :

V grad (f ) := {u ∈ C ⁿ : ∇f (u) = 0} ⊂ C ⁿ V _grad

^R

(f ) := {u ∈ R ⁿ : ∇f (u) = 0} ⊂ R ⁿ

◮

Gradient ideal :

I grad (f ) := h∇f (X)i = ∂f

∂x

1

, . . . , ∂f

∂x n

⊂ R [X ]

◮ Theorem 1

f ≥ 0 on V _grad ^R (f ) I _grad (f ) radical

⇒ f sos modulo I _grad (f ) :

∃q i , φ _j ∈ R [X ], f = X s

i=1

q _i ² + X n

j =1

φ _j ∂f

∂x _j

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Proof

The proof is based on the following two lemmas :

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Proof

The proof is based on the following two lemmas :

◮ lemma 1.1

V ₁ , . . . , V _r pairwise disjoint varieties in C ⁿ

⇓

∃ p ₁ , . . . , p _r ∈ R [X ], ∀ i, j , p _i (V j ) = δ _ij

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Proof

The proof is based on the following two lemmas :

◮ lemma 1.1

V ₁ , . . . , V _r pairwise disjoint varieties in C ⁿ

⇓

∃ p ₁ , . . . , p _r ∈ R [X ], ∀ i, j , p _i (V j ) = δ _ij

◮ lemma 1.2

W irreducible subvariety of V _grad (f ) s.t. W ∩ R ⁿ 6= ∅

⇓

f ≡ const on W

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In case I grad ( f ) is not radical

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In case I grad ( f ) is not radical

◮ Example :

f (x , y , z) := x ⁸ + y ⁸ + z ⁸ + x ⁴ y ² + x ² y ⁴ + z ⁶ − 3x ² y ² z ²

| {z }

Motzkin polynomial M(x,y,z)

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In case I grad ( f ) is not radical

◮ Example :

f (x , y , z) := x ⁸ + y ⁸ + z ⁸ + x ⁴ y ² + x ² y ⁴ + z ⁶ − 3x ² y ² z ²

| {z }

Motzkin polynomial M(x,y,z)

◮

Fact 1 : f ≡ 1

4 M (mod I grad (f ))

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In case I grad ( f ) is not radical

◮ Example :

f (x , y , z) := x ⁸ + y ⁸ + z ⁸ + x ⁴ y ² + x ² y ⁴ + z ⁶ − 3x ² y ² z ²

| {z }

Motzkin polynomial M(x,y,z)

◮

Fact 1 : f ≡ 1

4 M (mod I grad (f ))

◮

Fact 2 : M is not a sos in R [x, y , z ]/ I grad (f )

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In case I grad ( f ) is not radical

◮ Example :

f (x , y , z) := x ⁸ + y ⁸ + z ⁸ + x ⁴ y ² + x ² y ⁴ + z ⁶ − 3x ² y ² z ²

| {z }

Motzkin polynomial M(x,y,z)

◮

Fact 1 : f ≡ 1

4 M (mod I grad (f ))

◮

Fact 2 : M is not a sos in R [x, y , z ]/ I grad (f )

◮

Fact 3 : Ask Claus Scheiderer for more details

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In case I grad ( f ) is not radical

◮ Example :

f (x , y , z) := x ⁸ + y ⁸ + z ⁸ + x ⁴ y ² + x ² y ⁴ + z ⁶ − 3x ² y ² z ²

| {z }

Motzkin polynomial M(x,y,z)

◮

Fact 1 : f ≡ 1

4 M (mod I grad (f ))

◮

Fact 2 : M is not a sos in R [x, y , z ]/ I grad (f )

◮

Fact 3 : Ask Claus Scheiderer for more details

◮ Theorem 2

f > 0 on V _grad ^R (f ) ⇒ f sos modulo I _grad (f )

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Unconstrained optimization

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Unconstrained optimization

◮ Notations :

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Unconstrained optimization

◮ Notations :

◮

deg(f ) = d even

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Unconstrained optimization

◮ Notations :

◮

deg(f ) = d even

◮

f i := ∂f

∂x i

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Unconstrained optimization

◮ Notations :

◮

deg(f ) = d even

◮

f i := ∂f

∂x i

◮

∀k ≥ d , f ∈ R [X ] k : f = P

f

_α

x

^α

! f ∈ R

^νn,k

where ν n,k =

n + k k

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Unconstrained optimization

◮ Notations :

◮

deg(f ) = d even

◮

f i := ∂f

∂x i

◮

∀k ≥ d , f ∈ R [X ] k : f = P

f

_α

x

^α

! f ∈ R

^νn,k

where ν n,k =

n + k k

◮

∀N , mon

_N

( x ) =

^t

( 1 , x

₁

, . . . , x

_n

, x

₁²

, x

₁

x

₂

, . . . , x

_n^N

) ∈ R

^ν^n,N

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Unconstrained optimization

◮ Notations :

◮

deg(f ) = d even

◮

f i := ∂f

∂x i

◮

∀k ≥ d , f ∈ R [X ] k : f = P

f

_α

x

^α

! f ∈ R

^νn,k

where ν n,k =

n + k k

◮

∀N , mon

_N

( x ) =

^t

( 1 , x

₁

, . . . , x

_n

, x

₁²

, x

₁

x

₂

, . . . , x

_n^N

) ∈ R

^ν^n,N

◮ Restrictive hypothesis (H ) :

f attains its infimum f ^∗ over R ⁿ

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SDP relaxations

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SDP relaxations

◮ Primal SDP : moment formulation

(P ) :



 

 

 

 

f _N,mom ^∗ := inf

y

t fy = P f _α y _α

s .t.







∀ i, M _N ₋

d

2

(f i ∗ y ) = 0 M _N (y ) 0

y ₀ = 1

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SDP relaxations

◮ Primal SDP : moment formulation

(P ) :



 

 

 

 

f _N,mom ^∗ := inf

y

t fy = P f _α y _α

s .t.







∀ i, M _N ₋

d

2

(f i ∗ y ) = 0 M _N (y ) 0

y ₀ = 1

◮ Dual SDP : sos formulation

(D) :



 



 



f _N,grad ^∗ := sup

γ ∈R

γ

s .t.



 



 



f − γ = σ + P n j =1

φ j

∂f

∂x _j

σ ∈ P

( R [X ] N ) ²

φ _j ∈ R [X ] _2N−d+1

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SDP relaxations

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SDP relaxations

◮ Theorem 3

Under the assumption (H), the following holds :

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SDP relaxations

◮ Theorem 3

Under the assumption (H), the following holds :

◮

lim

N f _N,grad

^∗

= lim

N f _N,mom

^∗

= f

^∗

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SDP relaxations

◮ Theorem 3

Under the assumption (H), the following holds :

◮

lim

N f _N,grad

^∗

= lim

N f _N,mom

^∗

= f

^∗

◮

I grad (f ) radical ⇒ ∃N

⁰

, f _N

^∗₀_,grad

= f _N

^∗₀_,mom

= f

^∗

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SDP relaxations

◮ Theorem 3

Under the assumption (H), the following holds :

◮

lim

N f _N,grad

^∗

= lim

N f _N,mom

^∗

= f

^∗

◮

I grad (f ) radical ⇒ ∃N

⁰

, f _N

^∗₀_,grad

= f _N

^∗₀_,mom

= f

^∗

◮ Extracting solutions

In practice, Lasserre and Henrion’s technique :

If, for some N , and some optimal primal solution y ^∗ , we have

rank M _N (y ^∗ ) = rank M _N−d/2 (y ^∗ )

then we have reached the global minimum f ^∗ , and one can extract global minimizers (implemented in

Gloptipoly).

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Nonnegative polynomials modulo their gradient ideal