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A Convex Polynomial that is not SOS-Convex. Amir Ali Ahmadi Pablo A. Parrilo Laboratory for Information and Decision Systems Massachusetts Institute of Technology FRG: Semidefinite Optimization and Convex Algebraic Geometry May 2009 - MIT. Deciding Convexity.

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a convex polynomial that is not sos convex

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

deciding convexity
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
other applications
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]

convexity and the second derivative
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 [x;y] is positive semidefinite (psd)

Back to our example:

But can we efficiently check if H(x) is PSD for all x?

complexity of deciding convexity
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

sos convexity
SOS-Convexity

p(x)sos-convex

p(x)convex

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.

sos convexity ctnd
SOS-convexity (Ctnd.)

No

Our main contribution(via an explicit counterexample)

sos-convexity in the literature:

  • Semidefinite representability of semialgebraic sets [Helton, Nie]
  • Generalization of Jensen’s inequality [Lasserre]
  • Polynomial fitting, minimum volume convex sets [Magnani, Lall, Boyd]

Question that has been raised:

Q: must every convex polynomial be sos-convex?

agenda
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

nonnegative and sum of squares polynomials
Nonnegative and Sum of Squares Polynomials

Defn.Apolynomial p(x) is nonnegative or positive semidefinite (psd) if

Defn.Apolynomial 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?
hilbert s 1888 paper
Hilbert’s 1888 Paper

In 1888, Hilbert proved that a nonnegativepolynomial p(x) of degree d in n variables must be sosonly in the following cases:

  • n=1(univariate polynomials of any degree)
  • d=2(quadratic polynomials in any number of variables)
  • n=2 andd=4(bivariate quartics)

In all other cases, there are polynomials that are psd but not sos

the celebrated example of motzkin
The Celebrated Example of Motzkin

The first concrete counterexample was found about 80 years later!

This polynomial is psd but not sos

sum of squares and semidefinite programming
Sum of Squares and Semidefinite Programming

Unlike nonnegativity, checking whether a polynomial is SOS is a tractable problem

Thm:Apolynomial p(x) of degree 2d is SOS if and only if there exists a PSD matrixQ 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.

sos matrices
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 [x;y] is sos.

Therefore, can solve an SDP to check if P(x) is an sos-matrix.

psd matrices may not be sos
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 we’ve found), is not a valid Hessian:

equivalent notions for convexity
Equivalent notions for convexity
  • Basic definition:
  • First order condition:
  • Second order condition:
each condition can be sos ified
Each condition can be SOS-ified
  • Basic definition:
  • First order condition:
  • Second order condition:

Thm:

Proof:mimics the “standard” proof closely and uses closedness of the SOS cone

A’

B

C

A

A

C

B

A’

a convex polynomial that is not sos convex17
A convex polynomial that is not sos-convex

Need a polynomial p(x) such that all the following polynomials

are psd but not sos.

B

C

A

Without further ado...

our counterexample
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)
proof h x is psd
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

proof h x m t x m x
Proof: H(x)≠MT(x)M(x)

Therefore, it suffices to show that

is not sos.

We do this by a duality argument.

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.

separating hyperplane
Separating Hyperplane

PSD

H(1,1)

µ

SOS

a few remarks
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
how did we find this polynomial
How did we find this polynomial?

PSD

M

parameterize H(x)

H(1,1)

add Hessian constraints (partial derivatives must commute)

solve this sos-program

µ

SOS

messages to take home
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: what’s the complexity of deciding convexity?

Our result further supports the hypothesis that it must be a hard problem

slide27
Want to know more?

Preprint at http://arxiv.org/abs/0903.1287

Questions?