Nonnegative polynomials modulo their gradient ideal
Nonnegative polynomials modulo their gradient ideal About the article "Minimizing Polynomials via Sum of Squares over the Gradient Ideal" from Demmel, Nie and Sturmfels.
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Richard Leroy
6th May 2005 Universität Konstanz
Nonnegative polynomials modulo their gradient ideal
Introduction
Plan Introduction
Polynomials over their gradient varieties
Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
What’s next ?
Introduction : motivation and notations
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Introduction : motivation and notations ◮
Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over Rn .
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Introduction : motivation and notations ◮
◮
Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over Rn . Idea : If u ∈ Rn is a minimizer of a polynomial f ∈ R[X ] := R[X1 , . . . , Xn ], then ∇f (u) = 0, i.e. ∀i = 1, . . . , n,
∂f (u) = 0 ∂xi
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Introduction : motivation and notations ◮
◮
Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over Rn . Idea : If u ∈ Rn is a minimizer of a polynomial f ∈ R[X ] := R[X1 , . . . , Xn ], then ∇f (u) = 0, i.e. ∀i = 1, . . . , n,
◮
Tools :
∂f (u) = 0 ∂xi
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Introduction : motivation and notations ◮
◮
Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over Rn . Idea : If u ∈ Rn is a minimizer of a polynomial f ∈ R[X ] := R[X1 , . . . , Xn ], then ∇f (u) = 0, i.e. ∀i = 1, . . . , n,
◮
∂f (u) = 0 ∂xi
Tools : ◮
(Real) algebraic geometry : Sos representation of a nonnegative polynomial modulo its gradient ideal
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Introduction : motivation and notations ◮
◮
Goal : Use semidefinite programming (SDP) to solve the problem of minimizing a real polynomial over Rn . Idea : If u ∈ Rn is a minimizer of a polynomial f ∈ R[X ] := R[X1 , . . . , Xn ], then ∇f (u) = 0, i.e. ∀i = 1, . . . , n,
◮
∂f (u) = 0 ∂xi
Tools : ◮
◮
(Real) algebraic geometry : Sos representation of a nonnegative polynomial modulo its gradient ideal SDP : duality theory (sos representation / moment approach)
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Polynomials over their gradient varieties
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Polynomials over their gradient varieties ◮
Notations :
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Polynomials over their gradient varieties ◮
Nonnegative polynomials modulo their gradient ideal
Notations : ◮
Gradient varieties : Plan
n
◮
n
Vgrad (f ) := {u ∈ C : ∇f (u) = 0} ⊂ C
Introduction
R Vgrad (f ) := {u ∈ Rn : ∇f (u) = 0} ⊂ Rn
Polynomials over their gradient varieties
Gradient ideal : Igrad (f ) := h∇f (X )i =
∂f ∂f ,..., ∂x1 ∂xn
Unconstrained optimization
⊂ R[X ]
What’s next ?
Polynomials over their gradient varieties ◮
Nonnegative polynomials modulo their gradient ideal
Notations : ◮
Gradient varieties : Plan
n
◮
Vgrad (f ) := {u ∈ C : ∇f (u) = 0} ⊂ C
Introduction
R Vgrad (f ) := {u ∈ Rn : ∇f (u) = 0} ⊂ Rn
Polynomials over their gradient varieties
Gradient ideal : Igrad (f ) := h∇f (X )i =
◮
n
∂f ∂f ,..., ∂x1 ∂xn
Unconstrained optimization
⊂ R[X ]
Theorem 1 R (f ) f ≥ 0 on Vgrad Igrad (f ) radical
⇒ f sos modulo Igrad (f ) :
∃qi , φj ∈ R[X ], f =
s X i=1
qi2
+
n X j=1
φj
∂f ∂xj
What’s next ?
Proof The proof is based on the following two lemmas :
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Proof The proof is based on the following two lemmas : ◮
lemma 1.1
Nonnegative polynomials modulo their gradient ideal
Plan
V1 , . . . , Vr pairwise disjoint varieties in Cn ⇓ ∃p1 , . . . , pr ∈ R[X ], ∀i, j, pi (Vj ) = δij
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Proof The proof is based on the following two lemmas : ◮
lemma 1.1
Plan
V1 , . . . , Vr pairwise disjoint varieties in Cn ⇓ ∃p1 , . . . , pr ∈ R[X ], ∀i, j, pi (Vj ) = δij
◮
Nonnegative polynomials modulo their gradient ideal
lemma 1.2 W irreducible subvariety of Vgrad (f ) s.t. W ∩ Rn 6= ∅ ⇓ f ≡ const on W
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
In case Igrad (f ) is not radical
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
In case Igrad (f ) is not radical ◮
Nonnegative polynomials modulo their gradient ideal
Example :
Plan
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)
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
In case Igrad (f ) is not radical ◮
Nonnegative polynomials modulo their gradient ideal
Example :
Plan
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)
◮
1 Fact 1 : f ≡ M (mod Igrad (f )) 4
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
In case Igrad (f ) is not radical ◮
Nonnegative polynomials modulo their gradient ideal
Example :
Plan
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)
◮ ◮
1 Fact 1 : f ≡ M (mod Igrad (f )) 4 Fact 2 : M is not a sos in R [x, y , z]/Igrad (f )
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
In case Igrad (f ) is not radical ◮
Nonnegative polynomials modulo their gradient ideal
Example :
Plan
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)
◮ ◮ ◮
1 Fact 1 : f ≡ M (mod Igrad (f )) 4 Fact 2 : M is not a sos in R [x, y , z]/Igrad (f ) Fact 3 : Ask Claus Scheiderer for more details
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
In case Igrad (f ) is not radical ◮
Nonnegative polynomials modulo their gradient ideal
Example :
Plan
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)
◮ ◮ ◮
◮
1 Fact 1 : f ≡ M (mod Igrad (f )) 4 Fact 2 : M is not a sos in R [x, y , z]/Igrad (f ) Fact 3 : Ask Claus Scheiderer for more details
Theorem 2 R f > 0 on Vgrad (f ) ⇒ f sos modulo Igrad (f )
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
◮
Notations :
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
◮
Notations : ◮
deg(f ) = d even
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
◮
Notations : ◮ ◮
deg(f ) = d even ∂f fi := ∂xi
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
◮
Notations : ◮ ◮ ◮
deg(f ) = d even ∂f fi := ∂xi P ∀k ≥ d, f ∈ R[X ]k : f = fα x α ! f ∈ Rνn,k n+k where νn,k = k
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
◮
Plan
Notations : ◮ ◮ ◮
deg(f ) = d even ∂f fi := ∂xi P ∀k ≥ d, f ∈ R[X ]k : f = fα x α ! f ∈ Rνn,k n+k where νn,k = k t
◮
Nonnegative polynomials modulo their gradient ideal
∀N, monN (x) = (1, x1 , . . . , xn , x12 , x1 x2 , . . . , xnN ) ∈ Rνn,N
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
Unconstrained optimization
◮
Plan
Notations : ◮ ◮ ◮
deg(f ) = d even ∂f fi := ∂xi P ∀k ≥ d, f ∈ R[X ]k : f = fα x α ! f ∈ Rνn,k n+k where νn,k = k t
◮
◮
Nonnegative polynomials modulo their gradient ideal
∀N, monN (x) = (1, x1 , . . . , xn , x12 , x1 x2 , . . . , xnN ) ∈ Rνn,N
Restrictive hypothesis (H) : f attains its infimum f ∗ over Rn
Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations ◮
Primal SDP : moment formulation P ∗ t fN,mom := inf fy = fα yα y ∀i, MN− d2 (fi ∗ y ) = 0 (P) : s.t. M (y ) 0 N y0 = 1
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations ◮
◮
Primal SDP : moment formulation P ∗ t fN,mom := inf fy = fα yα y ∀i, MN− d2 (fi ∗ y ) = 0 (P) : s.t. M (y ) 0 N y0 = 1
Dual SDP : sos formulation ∗ fN,grad := sup γ γ∈R n P ∂f φj f −γ =σ+ (D) : ∂xj j=1 P s.t. 2 σ ∈ (R[X ]N ) φj ∈ R[X ]2N−d+1
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations ◮
Theorem 3 Under the assumption (H), the following holds :
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations ◮
Nonnegative polynomials modulo their gradient ideal
Theorem 3 Under the assumption (H), the following holds : ◮
∗ ∗ lim fN,grad = lim fN,mom =f∗ N
N
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations ◮
Nonnegative polynomials modulo their gradient ideal
Theorem 3 Under the assumption (H), the following holds : ◮
◮
Introduction
∗ ∗ lim fN,grad = lim fN,mom =f∗ N
N
Igrad (f ) radical ⇒ ∃N0 ,
fN∗0 ,grad
Plan
=
fN∗0 ,mom
=f
∗
Polynomials over their gradient varieties Unconstrained optimization What’s next ?
SDP relaxations ◮
Nonnegative polynomials modulo their gradient ideal
Theorem 3 Under the assumption (H), the following holds : ◮
◮
Introduction
∗ ∗ lim fN,grad = lim fN,mom =f∗ N
N
Igrad (f ) radical ⇒ ∃N0 ,
fN∗0 ,grad
Plan
=
fN∗0 ,mom
=f
∗
Polynomials over their gradient varieties Unconstrained optimization
◮
Extracting solutions In practice, Lasserre and Henrion’s technique : If, for some N, and some optimal primal solution y ∗ , we have rank MN (y ∗ ) = rank MN−d/2 (y ∗ ) then we have reached the global minimum f ∗ , and one can extract global minimizers (implemented in Gloptipoly).
What’s next ?
What’s next ?
Nonnegative polynomials modulo their gradient ideal
Plan Introduction Polynomials over their gradient varieties Unconstrained optimization What’s next ?
