1. 1 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

2. 2 Deciding Convexity Given a multivariate polynomial p(x):=p(x 1,…, x n ) 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

3. 3 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 [Magnani, Lall, Boyd] To address these questions, we need an understanding of the algebraic structure of the set of convex polynomials

4. 4 Convexity and the Second Derivative But can we efficiently check if H(x) is PSD for all x? Equivalently, H(x) is PSD if and only if the scalar polynomial y T H(x)y in 2n variables [x;y] is positive semidefinite (psd) Back to our example: Fact: a polynomial p(x) is convex if and only if its Hessian H(x) is positive semidefinite (PSD)

5. 5 Complexity of Deciding Convexity Checking polynomial nonnegativity is NP-hard for degree 4 or larger However, there is additional structure in the polynomial y T H(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

6. 6 SOS-Convexity p(x) sos-convexp(x) convex 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. Defn. ([Helton, Nie]): a polynomial p(x) is sos-convex if its Hessian factors as for a possibly nonsquare polynomial matrix M(x).

7. 7 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? No Our main contribution (via an explicit counterexample)

8. 8 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

9. 9 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 q 1 (x),…, q m (x) such that p(x) sos p(x) psd (obvious) When is the converse true?

10. 10 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: n=1 (univariate polynomials of any degree) d=2 (quadratic polynomials in any number of variables) n=2 and d=4 (bivariate quartics) In all other cases, there are polynomials that are psd but not sos

11. 11 The Celebrated Example of Motzkin The first concrete counterexample was found about 80 years later! This polynomial is psd but not sos

12. 12 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.

13. 13 SOS matrices Therefore, can solve an SDP to check if P(x) is an sos-matrix. 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 y T P(x)y in [x;y] is sos.

14. 14 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:

15. 15 Equivalent notions for convexity Basic definition: First order condition: Second order condition:

16. 16 Each condition can be SOS-ified Basic definition: First order condition: Second order condition: A B C A A Thm: CBA Proof: mimics the standard proof closely and uses closedness of the SOS cone

17. 17 A convex polynomial that is not sos-convex Without further ado... B C A Need a polynomial p(x) such that all the following polynomials are psd but not sos.

18. 18 Our Counterexample Claim: p(x) is convex: H(x) is PSD p(x) is not sos-convex: H(x) M T (x)M(x) A homogeneous polynomial in three variables, of degree 8.

19. 19 Proof: H(x) is PSD Or equivalently the scalar polynomial is sos. Claim: Proof: Exact sos decomposition, with rational coefficients. Exploiting symmetries of this polynomial, we solve SDPs of significantly reduced size

20. 20 Rational SOS Decomposition

21. 21 Rational SOS Decomposition

22. 22 Proof: H(x)M T (x)M(x) Lemma: if H(x) is an sos-matrix, then all its 2 n -1 principal minors are sos polynomials. In particular, all diagonal elements are sos. Proof: follows from the Cauchy-Binet formula. Therefore, it suffices to show that is not sos. We do this by a duality argument.

23. 23 Separating Hyperplane SOS PSD H(1,1 ) µ

24. 24 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

25. 25 How did we find this polynomial? parameterize H(x) add Hessian constraints (partial derivatives must commute) solve this sos-program SOS PSD H(1,1 ) µ M

26. 26 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

27. 27 Want to know more? Preprint at http://arxiv.org/abs/0903.1287http://arxiv.org/abs/0903.1287 Questions?