CONFIDENTIAL. Limited circulation. For review only. Sum of Squares and Polynomial Convexity Amir Ali Ahmadi and Pablo A. Parrilo Abstract— The notion of sos-convexity has recently been semidefiniteness of the Hessian matrix is replaced with proposed as a tractable sufficient condition for convexity of the existence of an appropriately defined sum of squares polynomials based on sum of squares decomposition. A multi- decomposition. As we will briefly review in this paper, by variate polynomial p(x) = p(x1; : : : ; xn) is said to be sos-convex if its Hessian H(x) can be factored as H(x) = M T (x) M (x) drawing some appealing connections between real algebra with a possibly nonsquare polynomial matrix M(x). It turns and numerical optimization, the latter problem can be re- out that one can reduce the problem of deciding sos-convexity duced to the feasibility of a semidefinite program. of a polynomial to the feasibility of a semidefinite program, Despite the relative recency of the concept of sos- which can be checked efficiently. Motivated by this computa- convexity, it has already appeared in a number of theo- tional tractability, it has been speculated whether every convex polynomial must necessarily be sos-convex. In this paper, we retical and practical settings. In [6], Helton and Nie use answer this question in the negative by presenting an explicit sos-convexity to give sufficient conditions for semidefinite example of a trivariate homogeneous polynomial of degree eight representability of semialgebraic sets. In [7], Lasserre uses that is convex but not sos-convex. sos-convexity to extend Jensen’s inequality in convex anal- ysis to linear functionals that are not necessarily probability I. INTRODUCTION measures. In a different work [8], Lasserre appeals to sos- In many problems in applied and computational math- convexity to give sufficient conditions for a polynomial ematics, we would like to decide whether a multivariate to belong to the quadratic module generated by a set of polynomial is convex or to parameterize a family of convex polynomials. More on the practical side, Magnani, Lall, and polynomials. Perhaps the most obvious instance appears Boyd [9] use sum of squares programming to find sos-convex in optimization. It is well known that in the absence of polynomials that best fit a set of data points or to find convexity, global minimization of polynomials is generally minimum volume convex sets, given by sub-level sets of sos- NP-hard [1], [2], [3]. However, if we somehow know a priori convex polynomials, that contain a set of points in space. that the polynomial is convex, nonexistence of local minima Even though it is well-known that sum of squares and is guaranteed and simple gradient descent methods can find a nonnegativity are not equivalent, because of the special global minimum. In many other practical settings, we might structure of the Hessian matrix, sos-convexity and convexity want to parameterize a family of convex polynomials that could potentially turn out to be equivalent. This speculation have certain properties, e.g., that serve as a convex envelope has been bolstered by the fact that finding a counterexample for a non-convex function, approximate a more complicated has shown to be difficult and attempts at giving a non- function, or fit some data points with minimum error. To constructive proof of its existence have seen no success address many questions of this type, we need to have an either. Our contribution in this paper is to give the first understanding of the algebraic structure of the set of convex such counterexample, i.e., the first example of a polynomial polynomials. that is convex but not sos-convex. This example is presented Over a decade ago, Pardalos and Vavasis [4] put the in Theorem 3.2. Our result further supports the hypothesis following question proposed by Shor on the list of seven that deciding convexity of polynomials should be a difficult most important open problems in complexity theory for problem. We hope that our counterexample, in a similar way numerical optimization: “Given a degree-4 polynomial in n to what other celebrated counterexamples [10]–[13] have variables, what is the complexity of determining whether this achieved, will help stimulate further research and clarify the polynomial describes a convex function?” To the best of our relationships between the geometric and algebraic aspects of knowledge, the question is still open but the general belief positivity and convexity. is that the problem should be hard (see the related work in The organization of the paper is as follows. Section II [5]). Not surprisingly, if testing membership to the set of is devoted to mathematical preliminaries required for under- convex polynomials is hard, searching and optimizing over standing the remainder of this paper. We begin this section them also turns out to be a hard problem. by introducing the cone of nonnegative and sum of squares The notion of sos-convexity has recently been proposed polynomials. We briefly discuss the connection between as a tractable relaxation for convexity based on semidefinite sum of squares decomposition and semidefinite programming programming. Broadly speaking, the requirement of positive highlighting also the dual problem. Formal definitions of sos- convex polynomials and sos-matrices are also given in this Amir Ali Ahmadi and Pablo A. Parrilo are with the Laboratory for section. In Section III, we present our main result, which is Information and Decision Systems, Department of Electrical Engineering and Computer Science at Massachusetts Institute of Technology. E-mail: an explicit example of a convex polynomial that is not sos- a a [email protected], [email protected]. convex. Some of the properties of this polynomial are also Preprint submitted to 48th IEEE Conference on Decision and Control. Received March 6, 2009. CONFIDENTIAL. Limited circulation. For review only. discussed at the end of this section. n variables can be homogenized into a form ph of degree d A more comprehensive version of this paper is presented in n + 1 variables, by adding a new variable y, and letting in [14]. Because our space is limited here, some of the x x p (x ; : : : ; x ; y) := ydp( 1 ;:::; n ): technical details and parts of the proofs have been omitted h 1 n y y and can be found in [14]. Furthermore, in [14], we explain The properties of being psd and sos are preserved under how we have found our counterexample using software and homogenization and dehomogenization. by utilizing techniques from sum of squares programming To make the ideas presented so far more concrete, we end and duality theory of semidefinite optimization. The method- this section by discussing the example of Motzkin. ology explained there is of independent interest since it can Example 2.1: The Motzkin polynomial be employed to search or optimize over a restricted family 4 2 2 4 2 2 of nonnegative polynomials that are not sums of squares. M(x1; x2) = x1x2 + x1x2 − 3x1x2 + 1 (2) II. MATHEMATICAL BACKGROUND is historically the first known example of a polynomial that A. Nonnegativity and sum of squares is psd but not sos. Positive semidefiniteness follows from the arithmetic-geometric inequality, and the nonexistence of an We denote by K[x] := K[x1; : : : ; xn] the ring of poly- sos decomposition can be shown by some clever algebraic nomials in n variables with coefficients in the field K. manipulations (see [16]) or by a duality argument. We can Throughout the paper, we will have K = R or K = Q.A homogenize this polynomial and obtain the Motzkin form polynomial p(x) 2 R[x] is said to be nonnegative or positive n 4 2 2 4 2 2 2 6 semidefinite (psd) if p(x) ≥ 0 for all x 2 R . Clearly, a Mh(x1; x2; x3) := x1x2 + x1x2 − 3x1x2x3 + x3; (3) necessary condition for a polynomial to be psd is for its total which belongs to P3;6 n Σ3;6 as expected. degree to be even. We say that p(x) is a sum of squares (sos), if there exist polynomials q1(x); :::; qm(x) such that B. Sum of squares, semidefinite programming, and duality m Deciding nonnegativity of polynomials is an important X 2 p(x) = qi (x): (1) problem that arises in many areas of systems theory, control, i=1 and optimization [17]. Unfortunately, this problem is known It is clear that p(x) being sos implies that p(x) is psd. In to be NP-hard even when the degree of the polynomial is 1888, David Hilbert [15] proved that the converse is true equal to four [2], [3]. On the other hand, deciding whether for a polynomial in n variables and of degree d only in the a given polynomial admits an sos decomposition turns out following cases: to be a tractable problem. This tractability stems from the underlying convexity of the problem as first pointed out in • n = 1 (univariate polynomials of any degree) [18], [19], [20]. More specifically, it was shown in [20] • d = 2 (quadratic polynomials in any number of vari- that one can reduce the problem of deciding whether a ables) polynomial is sos to feasibility of a semidefinite program • n = 2; d = 4 (bivariate quartics) (SDP). Semidefinite programs are a well-studied subclass of Hilbert showed that in all other cases there exist poly- convex optimization problems that can be efficiently solved nomials that are psd but not sos. Explicit examples of in polynomial time using interior point algorithms. Because such polynomials appeared nearly 80 years later, starting our space is limited, we refrain from further discussing with the celebrated example of Motzkin, followed by more SDPs and refer the interested reader to the excellent review examples by Robinson, Choi and Lam, and Lax-Lax and paper [21].