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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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
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
In many problems, we would like to parameterize a family of convex polynomials that perhaps:
To address these questions, we need an understanding of the algebraic structure of the set of convex polynomials
[Magnani, Lall, Boyd]
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?
Checking polynomial nonnegativity is NP-hard for degree 4 or larger
However, there is additional structure in the polynomial yTH(x)y:
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
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.
Our main contribution(via an explicit counterexample)
sos-convexity in the literature:
Question that has been raised:
Q: must every convex polynomial be sos-convex?
Other (equivalent?) notions for sos-convexity
Our counterexample (convex but not sos-convex)
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
In 1888, Hilbert proved that a nonnegativepolynomial p(x) of degree d in n variables must be sosonly in the following cases:
In all other cases, there are polynomials that are psd but not sos
The first concrete counterexample was found about 80 years later!
This polynomial is psd but not sos
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.
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.
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:
Proof:mimics the “standard” proof closely and uses closedness of the SOS cone
Need a polynomial p(x) such that all the following polynomials
are psd but not sos.
Without further ado...
A homogeneous polynomial in three variables, of degree 8.
Or equivalently the scalar polynomial
Proof:Exact sos decomposition, with rational coefficients.
Exploiting symmetries of this polynomial, we solve SDPs of significantly reduced size
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.
Our counterexample is robust to small perturbations
A dehomogenized version is still convex but not sos-convex
add Hessian constraints (partial derivatives must commute)
solve this sos-program
SOS-relaxation is a tractable technique for certifying positive semidefiniteness of scalar or matrix polynomials
We specialized to convexity and sos-convexity
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
Preprint at http://arxiv.org/abs/0903.1287