Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Fraction of Nonnegative Polynomials which are
Sums of Squares
Choosing a
direction
Finding the
endpoints
Caitlin A. Lownes
Texas A & M University - REU Program
July 26, 2011
Caitlin A. Lownes
Sums of Squares
Introduction
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
A polynomial f ∈ R[x1 , . . . , xn ] is a sum of squares
polynomial (SOS) if f = Σki=1 pi2 for some polynomials pi .
Hit and Run
Choosing a
direction
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
Introduction
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
A polynomial f ∈ R[x1 , . . . , xn ] is a sum of squares
polynomial (SOS) if f = Σki=1 pi2 for some polynomials pi .
Parrilo created an algorithm to optimize SOS polynomials
in polynomial time via semidefinite programming.
Polynomial optimization has applications in many areas
such as electrical engineering.
Caitlin A. Lownes
Sums of Squares
Introduction
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
A polynomial f ∈ R[x1 , . . . , xn ] is a sum of squares
polynomial (SOS) if f = Σki=1 pi2 for some polynomials pi .
Parrilo created an algorithm to optimize SOS polynomials
in polynomial time via semidefinite programming.
Polynomial optimization has applications in many areas
such as electrical engineering.
All SOS polynomials are nonnegative. How many
nonnegative polynomials are SOS?
Caitlin A. Lownes
Sums of Squares
Previous work
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Hilbert showed that all nonnegative univariate
polynomials, quadratic forms, and ternary quartics are
sums of squares. For all other cases, there exist
nonnegative polynomials which are not SOS.
Choosing a
direction
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
Previous work
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Hilbert showed that all nonnegative univariate
polynomials, quadratic forms, and ternary quartics are
sums of squares. For all other cases, there exist
nonnegative polynomials which are not SOS.
For nonnegative polynomials of fixed degree, previous
results by Blekherman show that the fraction of
nonnegative polynomials that are SOS approaches zero as
the number of variables increases.
Caitlin A. Lownes
Sums of Squares
Previous work
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Hilbert showed that all nonnegative univariate
polynomials, quadratic forms, and ternary quartics are
sums of squares. For all other cases, there exist
nonnegative polynomials which are not SOS.
For nonnegative polynomials of fixed degree, previous
results by Blekherman show that the fraction of
nonnegative polynomials that are SOS approaches zero as
the number of variables increases.
What about polynomials in few variables of low degree?
Caitlin A. Lownes
Sums of Squares
Cone of Polynomials
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Focus on bivariate polynomials f (x, y ), degy (f ) and
degx (f ) at most 4:
f (x, y ) = c1 + c2 x + c3 x 2 + c4 x 3 + c5 x 4 + c6 y + c7 xy +
c8 x 2 y + c9 x 3 y + c10 x 4 y + c11 y 2 + c12 xy 2 + c13 x 2 y 2 +
c14 x 3 y 2 + c15 x 4 y 2 + c16 y 3 + c17 xy 3 + c18 x 2 y 3 + c19 x 3 y 3 +
c20 x 4 y 3 + c21 y 4 + c22 xy 4 + c23 x 2 y 4 + c24 x 3 y 4 + c25 x 4 y 4
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
Cone of Polynomials
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Focus on bivariate polynomials f (x, y ), degy (f ) and
degx (f ) at most 4:
f (x, y ) = c1 + c2 x + c3 x 2 + c4 x 3 + c5 x 4 + c6 y + c7 xy +
c8 x 2 y + c9 x 3 y + c10 x 4 y + c11 y 2 + c12 xy 2 + c13 x 2 y 2 +
c14 x 3 y 2 + c15 x 4 y 2 + c16 y 3 + c17 xy 3 + c18 x 2 y 3 + c19 x 3 y 3 +
c20 x 4 y 3 + c21 y 4 + c22 xy 4 + c23 x 2 y 4 + c24 x 3 y 4 + c25 x 4 y 4
The set of nonnegative polynomials of this type form a 25
dimensional cone, and the set of sums of squares of
polynomials form a cone inside.
Caitlin A. Lownes
Sums of Squares
Cone of Polynomials
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Focus on bivariate polynomials f (x, y ), degy (f ) and
degx (f ) at most 4:
f (x, y ) = c1 + c2 x + c3 x 2 + c4 x 3 + c5 x 4 + c6 y + c7 xy +
c8 x 2 y + c9 x 3 y + c10 x 4 y + c11 y 2 + c12 xy 2 + c13 x 2 y 2 +
c14 x 3 y 2 + c15 x 4 y 2 + c16 y 3 + c17 xy 3 + c18 x 2 y 3 + c19 x 3 y 3 +
c20 x 4 y 3 + c21 y 4 + c22 xy 4 + c23 x 2 y 4 + c24 x 3 y 4 + c25 x 4 y 4
The set of nonnegative polynomials of this type form a 25
dimensional cone, and the set of sums of squares of
polynomials form a cone inside.
Intersect the cones with the hyperplane of polynomials
R
S 1 ×S 1 f dµ = 1.
Caitlin A. Lownes
Sums of Squares
Main Idea
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
24 dimensional convex body of sum of squares polynomials
inside convex body of nonnegative polynomials.
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
Main Idea
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
24 dimensional convex body of sum of squares polynomials
inside convex body of nonnegative polynomials.
Find ratio of the volumes to find the fraction.
Caitlin A. Lownes
Sums of Squares
Hit and Run
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Figure: Hit and Run algorithm
Caitlin A. Lownes
Sums of Squares
Choosing a direction
Sums of
Squares
Caitlin A.
Lownes
Begin with a polynomial f in the convex body.
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
Choosing a direction
Sums of
Squares
Caitlin A.
Lownes
Begin with a polynomial f in the convex body.
Introduction
Main Idea
Hit and Run
Choosing a
direction
Choose a direction v uniformly from the space of
polynomials
R
S 1 ×S 1 g dµ = 0.
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
Choosing a direction
Sums of
Squares
Caitlin A.
Lownes
Begin with a polynomial f in the convex body.
Introduction
Choosing a
direction
Choose a direction v uniformly from the space of
polynomials
R
S 1 ×S 1 g dµ = 0.
Finding the
endpoints
Then,
Main Idea
Hit and Run
R
S 1 ×S 1(f
Caitlin A. Lownes
+ t · v ) dµ = 1.
Sums of Squares
Choosing a direction
Sums of
Squares
Caitlin A.
Lownes
Begin with a polynomial f in the convex body.
Introduction
Choosing a
direction
Choose a direction v uniformly from the space of
polynomials
R
S 1 ×S 1 g dµ = 0.
Finding the
endpoints
Then,
Main Idea
Hit and Run
R
S 1 ×S 1(f
+ t · v ) dµ = 1.
How do we find the values of t at the endpoints?
Caitlin A. Lownes
Sums of Squares
The support A of a polynomial
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
A-discriminant
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Given a polynomial h(x1 , . . . , xn ) with support A, the
A-discriminant ∆A (h) is an irreducible polynomial in the
coefficients of h which vanishes when h has a degenerate
∂h
root (i.e. ∂x
= 0 for all i).
i
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
A-discriminant
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Given a polynomial h(x1 , . . . , xn ) with support A, the
A-discriminant ∆A (h) is an irreducible polynomial in the
coefficients of h which vanishes when h has a degenerate
∂h
root (i.e. ∂x
= 0 for all i).
i
Simple example:
f (x) = ax 2 + bx + c, ∆A (f ) = b 2 − 4ac
Caitlin A. Lownes
Sums of Squares
A-discriminant
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Given a polynomial h(x1 , . . . , xn ) with support A, the
A-discriminant ∆A (h) is an irreducible polynomial in the
coefficients of h which vanishes when h has a degenerate
∂h
root (i.e. ∂x
= 0 for all i).
i
Simple example:
f (x) = ax 2 + bx + c, ∆A (f ) = b 2 − 4ac
A nonnegative polynomial h is on the boundary of our cone
when ∆A (h) = 0. However, ∆A is not easy to compute!
Caitlin A. Lownes
Sums of Squares
Finding the values of t
Sums of
Squares
Caitlin A.
Lownes
Introduction
The resultant of polynomials h1 , . . . , hk is an irreducible
polynomial in the coefficients of h1 , . . . , hk which vanishes
when h1 , . . . , hk have a common root.
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
Caitlin A. Lownes
Sums of Squares
Finding the values of t
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
The resultant of polynomials h1 , . . . , hk is an irreducible
polynomial in the coefficients of h1 , . . . , hk which vanishes
when h1 , . . . , hk have a common root.
The principal A-determinant EA is the following resultant:
∂h
∂h
, y ∂y
)
EA (h) = Res(A,A,A) (h, x ∂x
When h is bivariate, we know how to compute this
resultant.
Caitlin A. Lownes
Sums of Squares
Finding the values of t
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
The resultant of polynomials h1 , . . . , hk is an irreducible
polynomial in the coefficients of h1 , . . . , hk which vanishes
when h1 , . . . , hk have a common root.
The principal A-determinant EA is the following resultant:
∂h
∂h
, y ∂y
)
EA (h) = Res(A,A,A) (h, x ∂x
When h is bivariate, we know how to compute this
resultant.
EA is a multiple of the A-discriminant:
EA (h) = (∆A (h))(∆ ∆ ∆| ∆| ∆· ∆· ∆· ∆· )
Caitlin A. Lownes
Sums of Squares
Finding the values of t
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run
Choosing a
direction
Finding the
endpoints
The resultant of polynomials h1 , . . . , hk is an irreducible
polynomial in the coefficients of h1 , . . . , hk which vanishes
when h1 , . . . , hk have a common root.
The principal A-determinant EA is the following resultant:
∂h
∂h
, y ∂y
)
EA (h) = Res(A,A,A) (h, x ∂x
When h is bivariate, we know how to compute this
resultant.
EA is a multiple of the A-discriminant:
EA (h) = (∆A (h))(∆ ∆ ∆| ∆| ∆· ∆· ∆· ∆· )
To find the values of t at the endpoints, find the roots of
∆A (f + t · v ) closest to 0!
Caitlin A. Lownes
Sums of Squares