On the sum-of-squares degree of symmetric quadratic functions

We study how well functions over the boolean hypercube of the form $f_k(x)=(|x|-k)(|x|-k-1)$ can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in $\ell_{\infty}$-norm as well as in $\ell_1$-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on $\ell_1$-approximation of $f_k$; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from his work; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions.Comment: 33 pages. Second version fixes some typos and adds reference

On the sum-of-squares degree of symmetric quadratic functions

We study how well functions over the boolean hypercube of the form fk(x) = (|x|-k)(|x|-k-1) can be approximated by sums of squares of low-degree polynomials, obtaining good bounds for the case of approximation in '1-norm as well as in '1-norm. We describe three complexity-theoretic applications: (1) a proof that the recent breakthrough lower bound of Lee, Raghavendra, and Steurer [19] on the positive semidefinite extension complexity of the correlation and TSP polytopes cannot be improved further by showing better sum-of-squares degree lower bounds on '1-approximation of fk; (2) a proof that Grigoriev's lower bound on the degree of Positivstellensatz refutations for the knapsack problem is optimal, answering an open question from [12]; (3) bounds on the query complexity of quantum algorithms whose expected output approximates such functions

A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure

Yuan's theorem of the alternative is an important theoretical tool in optimization, which provides a checkable certificate for the infeasibility of a strict inequality system involving two homogeneous quadratic functions. In this paper, we provide a tractable extension of Yuan's theorem of the alternative to the symmetric tensor setting. As an application, we establish that the optimal value of a class of nonconvex polynomial optimization problems with suitable sign structure (or more explicitly, with essentially non-positive coefficients) can be computed by a related convex conic programming problem, and the optimal solution of these nonconvex polynomial optimization problems can be recovered from the corresponding solution of the convex conic programming problem. Moreover, we obtain that this class of nonconvex polynomial optimization problems enjoy exact sum-of-squares relaxation, and so, can be solved via a single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW preprint, 22 page

Sparse sum-of-squares certificates on finite abelian groups

Let G be a finite abelian group. This paper is concerned with nonnegative functions on G that are sparse with respect to the Fourier basis. We establish combinatorial conditions on subsets S and T of Fourier basis elements under which nonnegative functions with Fourier support S are sums of squares of functions with Fourier support T. Our combinatorial condition involves constructing a chordal cover of a graph related to G and S (the Cayley graph Cay($\hat{G}$,S)) with maximal cliques related to T. Our result relies on two main ingredients: the decomposition of sparse positive semidefinite matrices with a chordal sparsity pattern, as well as a simple but key observation exploiting the structure of the Fourier basis elements of G. We apply our general result to two examples. First, in the case where $G = \mathbb{Z}_2^n$, by constructing a particular chordal cover of the half-cube graph, we prove that any nonnegative quadratic form in n binary variables is a sum of squares of functions of degree at most $\lceil n/2 \rceil$, establishing a conjecture of Laurent. Second, we consider nonnegative functions of degree d on $\mathbb{Z}_N$ (when d divides N). By constructing a particular chordal cover of the d'th power of the N-cycle, we prove that any such function is a sum of squares of functions with at most $3d\log(N/d)$ nonzero Fourier coefficients. Dually this shows that a certain cyclic polytope in $\mathbb{R}^{2d}$ with N vertices can be expressed as a projection of a section of the cone of psd matrices of size $3d\log(N/d)$. Putting $N=d^2$ gives a family of polytopes $P_d \subset \mathbb{R}^{2d}$ with LP extension complexity $\text{xc}_{LP}(P_d) = \Omega(d^2)$ and SDP extension complexity $\text{xc}_{PSD}(P_d) = O(d\log(d))$. To the best of our knowledge, this is the first explicit family of polytopes in increasing dimensions where $\text{xc}_{PSD}(P_d) = o(\text{xc}_{LP}(P_d))$.Comment: 34 page

Symmetric nonnegative forms and sums of squares

We study symmetric nonnegative forms and their relationship with symmetric sums of squares. For a fixed number of variables $n$ and degree $2d$, symmetric nonnegative forms and symmetric sums of squares form closed, convex cones in the vector space of $n$-variate symmetric forms of degree $2d$. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree $2d$ is fixed and the number of variables $n$ grows. Here, we show that, in sharp contrast to the general case, the difference between symmetric nonnegative forms and sums of squares does not grow arbitrarily large for any fixed degree $2d$. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree $4$ the cones of nonnegative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree $2d$.Comment: (v4) minor revision and small reorganizatio

Positive Forms and Stability of Linear Time-Delay Systems

We consider the problem of constructing Lyapunov functions for linear differential equations with delays. For such systems it is known that exponential stability implies the existence of a positive Lyapunov function which is quadratic on the space of continuous functions. We give an explicit parametrization of a sequence of finite-dimensional subsets of the cone of positive Lyapunov functions using positive semidefinite matrices. This allows stability analysis of linear time-delay systems to be formulated as a semidefinite program.Comment: journal version, 14 page

Isoparametric polynomials and sums of squares

The Hilbert's 17th problem asks that whether every nonnegative polynomial can be a sum of squares of rational functions. It has been answered affirmatively by Artin. However, as to the question whether a given nonnegative polynomial is a sum of squares of polynomials is still a central question in real algebraic geometry. In this paper, we solve this question completely for the nonnegative polynomials associated with isoparametric polynomials (initiated by E. Cartan) which define the focal submanifolds of the corresponding isoparametric hypersurfaces.Comment: 38 page