A Convex Polynomial that is not SOS-Convex

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A Convex Polynomial that is not SOS-Convex

• Amir Ali Ahmadi
• Pablo A. Parrilo
• Laboratory for Information and Decision
  SystemsMassachusetts Institute of Technology
• FRG Semidefinite Optimization and Convex
  Algebraic Geometry
• May 2009 - MIT

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Deciding Convexity
Given a multivariate polynomial p(x)p(x1,, xn
) of even degree, how to decide if it is
convex? A concrete example
Most direct application global optimization

• Global minimization of polynomials is NP-hard
  even when the degree is 4
• But in presence of convexity, no local minima
  exist, and simple gradient methods can find a
  global min

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Other Applications

• In many problems, we would like to parameterize a
  family of convex polynomials that perhaps
• serve as a convex envelope to a non-convex
  function
• approximate a more complicated function
• fit data samples with small error

To address these questions, we need an
understanding of the algebraic structure of the
set of convex polynomials
Magnani, Lall, Boyd
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Convexity and the Second Derivative
Fact a polynomial p(x) is convex if and only if
its Hessian H(x) is positive semidefinite (PSD)
Equivalently, H(x) is PSD if and only if the
scalar polynomial yTH(x)y in 2n variables xy
is positive semidefinite (psd) Back to our
example
But can we efficiently check if H(x) is PSD for
all x?
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Complexity of Deciding Convexity

• Checking polynomial nonnegativity is NP-hard for
  degree 4 or larger
• However, there is additional structure in the
  polynomial yTH(x)y
• Quadratic in y (a biform)
• H(x) is a matrix of second derivatives ?
  partial derivatives commute

Pardalos and Vavasis (92) included the
following question proposed by Shor on a list of
the seven most important open problems in
complexity theory for numerical
optimization What is the complexity of deciding
convexity of a multivariate polynomial of degree
four? To the best of our knowledge still open
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SOS-Convexity
Defn. (Helton, Nie) a polynomial p(x) is
sos-convex if its Hessian factors as for a
possibly nonsquare polynomial matrix M(x).
As we will see, checking sos-convexity can be
cast as the feasibility of a semidefinite program
(SDP), which can be solved in polynomial time
using interior-point methods.
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SOS-convexity (Ctnd.)

• sos-convexity in the literature
• Semidefinite representability of semialgebraic
  sets Helton, Nie
• Generalization of Jensens inequality Lasserre
• Polynomial fitting, minimum volume convex sets
  Magnani, Lall, Boyd

Question that has been raised Q must every
convex polynomial be sos-convex?
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Agenda

• Nonnegativity and sum of squares
• A bit of history
• Connection to semidefinite programming
• SOS-matrices
• Other (equivalent?) notions for sos-convexity
• Our counterexample (convex but not sos-convex)
• Ideas behind the proof
• Several remarks
• How did we find it?
• Conclusions

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Nonnegative and Sum of Squares Polynomials
Defn. A polynomial p(x) is nonnegative or
positive semidefinite (psd) if Defn. A
polynomial p(x) is a sum of squares (sos) if
there exist some other polynomials q1(x),, qm(x)
such that

• p(x) sos ? p(x) psd (obvious)
• When is the converse true?

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Hilberts 1888 Paper
In 1888, Hilbert proved that a nonnegative
polynomial p(x) of degree d in n variables must
be sos only in the following cases

• n1 (univariate polynomials of any degree)
• d2 (quadratic polynomials in any number of
  variables)
• n2 and d4 (bivariate quartics)

In all other cases, there are polynomials that
are psd but not sos
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The Celebrated Example of Motzkin
The first concrete counterexample was found about
80 years later!
This polynomial is psd but not sos
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Sum of Squares and Semidefinite Programming
Unlike nonnegativity, checking whether a
polynomial is SOS is a tractable problem
Thm A polynomial p(x) of degree 2d is SOS if
and only if there exists a PSD matrix Q such that
where z is the vector of monomials of degree up
to d
Feasible set is the intersection of an affine
subspace and the PSD cone, and thus is a
semidefinite program.
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SOS matrices
Defn. (Kojima,Gatermann-Parrilo) A
symmetric polynomial matrix P(x) is an sos-matrix
if for a possibly nonsquare polynomial matrix
M(x).
Lemma P(x) is an sos-matrix if and only if the
scalar polynomial yTP(x)y in xy is sos.
Therefore, can solve an SDP to check if P(x) is
an sos-matrix.
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PSD matrices may not be SOS
Explicit biform examples of Choi, Reznick (and
others), yield PSD matrices that are not SOS.
For instance, the biquadratic Choi form can be
rewritten as
However this example (and all others weve
found), is not a valid Hessian
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Equivalent notions for convexity

• Basic definition

• First order condition

• Second order condition

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Each condition can be SOS-ified
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A convex polynomial that is not sos-convex
Without further ado...
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Our Counterexample
A homogeneous polynomial in three variables, of
degree 8.

• Claim
• p(x) is convex H(x) is PSD
• p(x) is not sos-convex H(x) ? MT(x)M(x)

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Proof H(x) is PSD
Claim
Or equivalently the scalar polynomial
is sos.
Proof Exact sos decomposition, with rational
coefficients. Exploiting symmetries of this
polynomial, we solve SDPs of significantly
reduced size
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Rational SOS Decomposition
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Rational SOS Decomposition
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Proof H(x)?MT(x)M(x)
Lemma if H(x) is an sos-matrix, then all its
2n-1 principal minors are sos polynomials. In
particular, all diagonal elements are sos.
Proof follows from the Cauchy-Binet formula.
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Separating Hyperplane
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A few remarks

• Our counterexample is robust to small
  perturbations
• Follows from inequalities being strict
• A dehomogenized version is still convex but
  not sos-convex
• Minimal in the number of variables
• Almost minimal in the degree

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How did we find this polynomial?
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Messages to take home

• SOS-relaxation is a tractable technique for
  certifying positive semidefiniteness of scalar or
  matrix polynomials
• We specialized to convexity and sos-convexity
• Three natural notions for sos-convexity are
  equivalent
• Not always exact
• But very powerful (at least for low degrees and
  dimensions)
• Proposed a convex relaxation to search over a
  restricted family of psd polynomials that are not
  sos
• Open whats the complexity of deciding
  convexity?
• Our result further supports the hypothesis that
  it must be a hard problem

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• Want to know more?
• Preprint at http//arxiv.org/abs/0903.1287
• Questions?