Sums of Squares Relaxations on The Hypercube
Greg Blekherman
Georgia Tech
MIP 2013
Joint Work with Joao Gouveia and James Pffeifer
Sums of Squares – High Level Overview
Main Problem: Minimize a quadratic function Q on the hypercube
H = {0, 1}n . Many problems in combinatorial optimization, e.g.
stable set and maxcut, can be stated in this way.
Given a nonnegative function Q : H → R sums of squares provide
a certificate of nonnegativity of Q. This certificate can be
computed via a semidefinite program.
Key Idea: Relax nonnegativity to sums of squares (in some way).
Nuts and Bolts
The hypercube H is defined by equations xi2 − xi = 0, i = 1, . . . , n.
A function Q is a sum of squares on H if we can write
Q=
X
qi2 +
X
(xi2 − xi )fi ,
with some polynomials qi and fi .
In order to compute need to truncate degrees of qi and fi . The
function Q is called k-sos if we can write Q as above and
deg qi ≤ k,
deg fi ≤ 2k − 2.
Why am I here?
Problem: Minimize Q on H. Compute best lower bound:
Q∗ =
min
Q−γ is nonnegative on H
γ.
Compute instead the k-sos minimum!
Qksos =
min
Q−γ is k-sos on H
γ.
This gives a hierarchy of semidefinite relaxations for Q∗ .
Key Question: For what values of k can we guarantee that Qksos
equals Q∗ or provides a good approximation of it?
How outlandish is this?
If we consider the {+1, −1} hypercube and let Q be the maxcut
quadratic function then Q1sos is in fact the Goemans-Williamson
semidefinite relaxation!
Lovász θ-function is also a 1-sos relaxation of a linear function on
the stable set polytope of a graph G .
Sums of squares can also be used to construct semidefinite lifts of
polytopes. Sums of squares (Lasserre) relaxations provide the best
known universal methods for constructing semidefinite relaxations.
More General Certificates
Can certify that Q is nonnegative on H by finding a sum of
squares multiplier h (of limited degree) such that
Q(1 + h) is k-sos.
Searching for h of fixed degree is still a semidefinite problem and
this is stronger than the Lasserre certificate!
Complication: Finding the best γ, such that
(Q − γ)(1 + h) is k-sos
is not a semidefinite program, due to the nonlinear term γh.
Can still approximate Q∗ by performing a binary search on the
values of γ.
Why Bother?
Sums of squares with multipliers are stronger. Also I believe that
they will be easier to analyze.
Our understanding of the quality of sums of squares relaxations is
very poor.
Degree Cancellation: Q = x1 + x2 . A 1-sos certificate for Q is
x1 + x2 = x12 + x22 − (x12 − x1 ) − (x22 − x2 ).
Degree Bounds
What is the least degree k such that any Q quadratic on H is
k-sos? Unknown. But k ≥ b n2 c + 1 (M. Laurent) and k ≤ n.
In degree n we can write interpolators (e.g. x1 · · · xn ) and thus any
nonnegative function is n-sos.
Theorem:(B, Gouveia, Pfeiffer) For any quadratic function Q
there exists a sum of squares multiplier h of degree at most
b n2 c + 2 such that Q(1 + h) is a sum of squares.
Moreover there exist nonnegative quadratic Q such that Qh is not
a sum of squares for any sum of squares h of degree at most b n2 c.
Are Multipliers Really Better?
Let S = {s1 , . . . , sn } be a finite set of distinct points in R. S is
defined by polynomial p = (x − s1 ) · · · (x − sn ). Given a quadratic
function Q nonnegative on S for what degree k is Q guaranteed to
be k-sos?
Interpolator Degree: in degree n − 1 can be interpolators, e.g.
(x − s1 ) · · · (x − sn−1 ), so Q is guaranteed to be (n − 1)-sos. Can
also build examples where the interpolators degree bound is sharp!
Can generalize this construction also to (special) solutions of n
quadratics in Rn . However the upper bounds of the previous
theorem still apply!
Multipliers guarantee certificates with degree roughly 1/2 of the
interpolator degree, and this is sharp (at least on H).
Some Convex Geometry
Suppose that Q has no k-sos multiplier h such that Qh is SOS.
k-sos functions on H form a a convex cone Σ2k .
Our assumption is:
Q · Σ2k ∩ Σ2k+2 = {0}.
Therefore there exists a linear functional ` separating the two
cones!
`(f ) ≥ 0, f ∈ Σ2k+2 and `(Qf ) ≤ 0, f ∈ Σ2k .
We can write ` as a linear combination of evaluations on the points
of H and then we do sign counting.
Next Steps
Would be great to obtain results on quality of approximation when
using multipliers.
The functions that need provably large multiplier degree seem to
be easy to approximate.
THANK YOU!