Sums of
Squares
Caitlin A.
Lownes
Introduction Fraction of Nonnegative Polynomials which are
Main Idea
Hit and Run
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
A polynomial f ∈ R[x1 , . . . , xn ] is a sum of squares
Main Idea
polynomial (SOS) if f = Σk pi2 for some polynomials pi .
i=1
Hit and Run
Choosing a
direction
Finding the
endpoints
Caitlin A. Lownes Sums of Squares
Introduction
Sums of
Squares
Caitlin A.
Lownes
Introduction
A polynomial f ∈ R[x1 , . . . , xn ] is a sum of squares
Main Idea
polynomial (SOS) if f = Σk pi2 for some polynomials pi .
i=1
Hit and Run Parrilo created an algorithm to optimize SOS polynomials
Choosing a
direction
in polynomial time via semidefinite programming.
Finding the
Polynomial optimization has applications in many areas
endpoints such as electrical engineering.
Caitlin A. Lownes Sums of Squares
Introduction
Sums of
Squares
Caitlin A.
Lownes
Introduction
A polynomial f ∈ R[x1 , . . . , xn ] is a sum of squares
Main Idea
polynomial (SOS) if f = Σk pi2 for some polynomials pi .
i=1
Hit and Run Parrilo created an algorithm to optimize SOS polynomials
Choosing a
direction
in polynomial time via semidefinite programming.
Finding the
Polynomial optimization has applications in many areas
endpoints 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
Hilbert showed that all nonnegative univariate
Introduction
polynomials, quadratic forms, and ternary quartics are
Main Idea
sums of squares. For all other cases, there exist
Hit and Run
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
Hilbert showed that all nonnegative univariate
Introduction
polynomials, quadratic forms, and ternary quartics are
Main Idea
sums of squares. For all other cases, there exist
Hit and Run
nonnegative polynomials which are not SOS.
Choosing a
direction For nonnegative polynomials of fixed degree, previous
Finding the
endpoints
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
Hilbert showed that all nonnegative univariate
Introduction
polynomials, quadratic forms, and ternary quartics are
Main Idea
sums of squares. For all other cases, there exist
Hit and Run
nonnegative polynomials which are not SOS.
Choosing a
direction For nonnegative polynomials of fixed degree, previous
Finding the
endpoints
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. Focus on bivariate polynomials f (x, y ), degy (f ) and
Lownes
degx (f ) at most 4:
Introduction
f (x, y ) = c1 + c2 x + c3 x 2 + c4 x 3 + c5 x 4 + c6 y + c7 xy +
Main Idea
c8 x 2 y + c9 x 3 y + c10 x 4 y + c11 y 2 + c12 xy 2 + c13 x 2 y 2 +
Hit and Run
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 +
Choosing a
direction 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. Focus on bivariate polynomials f (x, y ), degy (f ) and
Lownes
degx (f ) at most 4:
Introduction
f (x, y ) = c1 + c2 x + c3 x 2 + c4 x 3 + c5 x 4 + c6 y + c7 xy +
Main Idea
c8 x 2 y + c9 x 3 y + c10 x 4 y + c11 y 2 + c12 xy 2 + c13 x 2 y 2 +
Hit and Run
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 +
Choosing a
direction 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 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. Focus on bivariate polynomials f (x, y ), degy (f ) and
Lownes
degx (f ) at most 4:
Introduction
f (x, y ) = c1 + c2 x + c3 x 2 + c4 x 3 + c5 x 4 + c6 y + c7 xy +
Main Idea
c8 x 2 y + c9 x 3 y + c10 x 4 y + c11 y 2 + c12 xy 2 + c13 x 2 y 2 +
Hit and Run
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 +
Choosing a
direction 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 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
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 24 dimensional convex body of sum of squares polynomials
Choosing a inside convex body of nonnegative polynomials.
direction
Finding the
endpoints
Caitlin A. Lownes Sums of Squares
Main Idea
Sums of
Squares
Caitlin A.
Lownes
Introduction
Main Idea
Hit and Run 24 dimensional convex body of sum of squares polynomials
Choosing a inside convex body of nonnegative polynomials.
direction
Finding the Find ratio of the volumes to find the fraction.
endpoints
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
Choose a direction v uniformly from the space of
Hit and Run polynomials
Choosing a
direction 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
Main Idea
Choose a direction v uniformly from the space of
Hit and Run polynomials
Choosing a
direction S 1 ×S 1 g dµ = 0.
Finding the
endpoints Then,
S 1 ×S 1(f + t · v ) dµ = 1.
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
Choose a direction v uniformly from the space of
Hit and Run polynomials
Choosing a
direction S 1 ×S 1 g dµ = 0.
Finding the
endpoints Then,
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
Given a polynomial h(x1 , . . . , xn ) with support A, the
Introduction
A-discriminant ∆A (h) is an irreducible polynomial in the
Main Idea
coefficients of h which vanishes when h has a degenerate
Hit and Run
∂h
Choosing a
root (i.e. ∂xi = 0 for all i).
direction
Finding the
endpoints
Caitlin A. Lownes Sums of Squares
A-discriminant
Sums of
Squares
Caitlin A.
Lownes
Given a polynomial h(x1 , . . . , xn ) with support A, the
Introduction
A-discriminant ∆A (h) is an irreducible polynomial in the
Main Idea
coefficients of h which vanishes when h has a degenerate
Hit and Run
∂h
Choosing a
root (i.e. ∂xi = 0 for all i).
direction
Simple example:
Finding the
endpoints
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
Given a polynomial h(x1 , . . . , xn ) with support A, the
Introduction
A-discriminant ∆A (h) is an irreducible polynomial in the
Main Idea
coefficients of h which vanishes when h has a degenerate
Hit and Run
∂h
Choosing a
root (i.e. ∂xi = 0 for all i).
direction
Simple example:
Finding the
endpoints
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.
The resultant of polynomials h1 , . . . , hk is an irreducible
Lownes
polynomial in the coefficients of h1 , . . . , hk which vanishes
Introduction 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.
The resultant of polynomials h1 , . . . , hk is an irreducible
Lownes
polynomial in the coefficients of h1 , . . . , hk which vanishes
Introduction when h1 , . . . , hk have a common root.
Main Idea
The principal A-determinant EA is the following resultant:
Hit and Run ∂h ∂h
EA (h) = Res(A,A,A) (h, x ∂x , y ∂y )
Choosing a
direction
When h is bivariate, we know how to compute this
Finding the
endpoints resultant.
Caitlin A. Lownes Sums of Squares
Finding the values of t
Sums of
Squares
Caitlin A.
The resultant of polynomials h1 , . . . , hk is an irreducible
Lownes
polynomial in the coefficients of h1 , . . . , hk which vanishes
Introduction when h1 , . . . , hk have a common root.
Main Idea
The principal A-determinant EA is the following resultant:
Hit and Run ∂h ∂h
EA (h) = Res(A,A,A) (h, x ∂x , y ∂y )
Choosing a
direction
When h is bivariate, we know how to compute this
Finding the
endpoints 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.
The resultant of polynomials h1 , . . . , hk is an irreducible
Lownes
polynomial in the coefficients of h1 , . . . , hk which vanishes
Introduction when h1 , . . . , hk have a common root.
Main Idea
The principal A-determinant EA is the following resultant:
Hit and Run ∂h ∂h
EA (h) = Res(A,A,A) (h, x ∂x , y ∂y )
Choosing a
direction
When h is bivariate, we know how to compute this
Finding the
endpoints 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