Related papers: Positively not SOS: pseudo-moments and extreme ray…
We provide explicit conditions for a real polynomial $f$ of degree 2d to be a sum of squares (s.o.s.), stated only in terms of the coefficients of $f$, i.e. with no lifting. All conditions are simple and provide an explicit description of a…
The truncated moment problem consists of determining whether a given finitedimensional vector of real numbers y is obtained by integrating a basis of the vector space of polynomials of bounded degree with respect to a non-negative measure…
New results on computing certificates of strictly positive polynomials in Archimedean quadratic modules are presented. The results build upon (i) Averkov's method for generating a strictly positive polynomial for which a membership…
We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials…
One considers polynomial optimization problems with compact feasible set $\mathbf{\Omega}$ defined by SOS-concave polynomials $g_j$, and with a globally non-convex polynomial objective $f$. We show that if $f$ is strongly convex on…
We completely characterize sections of the cones of nonnegative polynomials, convex polynomials and sums of squares with polynomials supported on circuits, a genuine class of sparse polynomials. In particular, nonnegativity is characterized…
Recently, non-SOS Positivstellens\"atze for polynomials on compact semialgebraic sets, following the general form of Schm\"{u}dgen's Positivstellensatz, have been derived by appropriately replacing the SOS polynomials with other classes of…
Real algebraic geometry provides certificates for the positivity of polynomials on semi-algebraic sets by expressing them as a suitable combination of sums of squares and the defining inequalitites. We show how Putinar's theorem for…
We present an example of a strictly positive polynomial with rational coefficients that can be decomposed as a sum of squares of polynomials over $\R$ but not over $\Q$. This answers an open question by C. Scheiderer posed as the second…
Let $\mathbb{Q}$ (resp. $\mathbb{R}$) be the field of rational (resp. real) numbers and $X = (X_1, \ldots, X_n)$ be variables. Deciding the non-negativity of polynomials in $\mathbb{Q}[X]$ over $\mathbb{R}^n$ or over semi-algebraic domains…
The standard moment-sum-of-squares (SOS) hierarchy is a powerful method for solving global polynomial optimization problems. However, its convergence relies on Putinar's Positivstellensatz, which requires the feasible set to satisfy the…
Recently, the second and the third author developed sums of nonnegative circuit polynomials (SONC) as a new certificate of nonnegativity for real polynomials, which is independent of sums of squares. In this article we show that the SONC…
Optimization over non-negative polynomials is fundamental for nonlinear systems analysis and control. We investigate the relation between three tractable relaxations for optimizing over sparse non-negative polynomials: sparse sum-of-squares…
In the smallest cases where there exist nonnegative polynomials that are not sums of squares we present a complete explanation of this distinction. The fundamental reason that the cone of sums of squares is strictly contained in the cone of…
We revisit Stengle's classical univariate polynomial optimization example $min 1 - x^2 s.t. (1 - x^2)^3 \geq 0$ whose constraint description is degenerate at the minimizers. We prove that the moment-SOS hierarchy of relaxation order $r \geq…
It is well-known that any sum of squares (SOS) program can be cast as a semidefinite program (SDP) of a particular structure and that therein lies the computational bottleneck for SOS programs, as the SDPs generated by this procedure are…
The dynamics of many systems from physics, economics, chemistry, and biology can be modelled through polynomial functions. In this paper, we provide a computational means to find positively invariant sets of polynomial dynamical systems by…
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.…
Let $g_1,\dots, g_s \in \mathbb{R}[X_1,\dots, X_n,Y]$ and $S = \{(\bar{x},y)\in \mathbb{R}^{n+1} \mid g_1(\bar{x},y) \ge 0, \dots, g_s(\bar{x}, y) \ge 0\}$ be a non-empty, possibly unbounded, subset of a cylinder in $\mathbb{R}^{n+1}$. Let…
One of the desirable objectives in feedback control design is to formulate and solve the design problem as an optimisation problem that is convex, so that an optimal solution can be found efficiently. Unfortunately many control design…