The notion of sos-convexity has recently been proposed as a tractable sufficient condition for convexity of polynomials based on sum of squares decomposition. A multivariate polynomial p(x) = p(x1, . . . , xn) is said to be sos-convex if its Hessian H(x) can be factored as H(x) = M (x) M (x) with a possibly nonsquare polynomial matrix M(x). It turns out that one can reduce the problem of decidi...

Digital signal processing and multimedia applications often profit from the use of a residue number system. Among the most commonly used moduli, in such systems, are those of 221 and 2þ 1 forms and among the most commonly used operations are multiplication and sum-of-squares. These operations are currently performed using distinct design units and/or consecutive machine cycles. Novel architectu...

Checking non-negativity of polynomials using sum-of-squares has recently been popularized and found many applications in control. Although the method is based on convex programming, the optimization problems rapidly grow and result in huge semidefinite programs. The paper [4] describes how symmetry is exploited in sum-of-squares problems in the MATLAB toolbox YALMIP, but concentrates on the sca...

In previous chapters we analyzed data from so-called one-way experimental designs, in which subjects were randomly assigned to groups that di↵ered on a single treatment or grouping variable. In this chapter we will analyze data from factorial experiments. Factorial experiments contain two or more experimental variables. In a completely crossed factorial experiment, each level of every variable ...

A sum-of-squares is a polynomial that can be expressed as a sum of squares of other polynomials. Determining if a sum-of-squares decomposition exists for a given polynomial is equivalent to a linear matrix inequality feasibility problem. The computation required to solve the feasibility problem depends on the number of monomials used in the decomposition. The Newton polytope is a method to prun...

1 Abstract This vignette shows the enhancements made for GlobalAncova. Basically, there are four ideas implemented: ˆ decomposition of the sum of squares of a linear model ([2]) ˆ a plotting function for the sequential decomposition ˆ pairwise comparison for factor levels ˆ adjustment for global covariates The decomposition of the model sum of squares results in an ANOVA table, which shows the ...

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a combining algorithm—an algorithm that maps a distribution over solutions into a (possibly weaker) solution—into a rounding algorithm th...

Sum of squares optimization is an active area of research at the interface of algorithmic algebra and convex optimization. Over the last decade, it has made significant impact on both discrete and continuous optimization, as well as several other disciplines, notably control theory. A particularly exciting aspect of this research area is that it leverages classical results from real algebraic g...

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with r incoherent, orthogonal components in ’ from r · Õ(n1.5) randomly observed entries of the tensor. This bound improves over the previous best one of r · Õ(n2) by reduction to exact matrix completion. Our...

How can one find real solutions (x1, x2)? How to prove that they do not exist? And if the solution set is nonempty, how to optimize a polynomial function over this set? Until a few years ago, the default answer to these and similar questions would have been that the possi­ ble nonconvexity of the feasible set and/or objective function precludes any kind of analytic global results. Even today, t...