Related papers: Optimization over the Boolean Hypercube via Sums o…
We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in $n$ variables with roots placed in points $k-1$ and $k$. Functions of this type have played a central role in…
For a non-empty, finite subset $\mathcal{A} \subseteq \mathbb{N}_0^n$, denote by $C_{\text{sonc}}(\mathcal{A}) \in \mathbb{R}[x_1, \ldots, x_n]$ the cone of sums of non-negative circuit polynomials with support $\mathcal{A}$. We derive a…
Certificates of non-negativity such as Putinar's Positivstellensatz have been used to obtain powerful numerical techniques to solve polynomial optimization (PO) problems. Putinar's certificate uses sum-of-squares (sos) polynomials to…
In a first contribution, we revisit two certificates of positivity on (possibly non-compact) basic semialgebraic sets due to Putinar and Vasilescu [Comptes Rendus de l'Acad\'emie des Sciences-Series I-Mathematics, 328(6) (1999) pp.…
We compare four key hierarchies for solving Constrained Polynomial Optimization Problems (CPOP): Sum of Squares (SOS), Sum of Diagonally Dominant Polynomials (SDSOS), Sum of Nonnegative Circuits (SONC), and the Sherali Adams (SA)…
Certifying function nonnegativity is a ubiquitous problem in computational mathematics, with especially notable applications in optimization. We study the question of certifying nonnegativity of signomials based on the recently proposed…
The problem of optimizing over the cone of nonnegative polynomials is a fundamental problem in computational mathematics, with applications to polynomial optimization, control, machine learning, game theory, and combinatorics, among others.…
In (Davis and Papp, 2022), the authors introduced the concept of dual certificates of (weighted) sum-of-squares polynomials, which are vectors from the dual cone of weighted sums of squares (WSOS) polynomials that can be interpreted as…
A SONC polynomial is a sum of finitely many non-negative circuit polynomials, whereas a non-negative circuit polynomial is a non-negative polynomial whose support is a simplicial circuit. We show that there exist non-negative polynomials…
In this paper, we present a computational approach to certify almost sure reachability for discrete-time polynomial stochastic systems by turning drift--variant criteria into sum-of-squares (SOS) programs solved with standard semidefinite…
We provide two hybrid numeric-symbolic optimization algorithms, computing exact sums of nonnegative circuits (SONC) and sums of arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a hybrid numeric-symbolic decision…
We consider numerical certification of approximate solutions to a system of polynomial equations with more equations than unknowns by first certifying solutions to a square subsystem. We give several approaches that certifiably select which…
We consider the problem of minimizing a polynomial $f$ over the binary hypercube. We show that, for a specific set of polynomials, their binary non-negativity can be checked in a polynomial time via minimum cut algorithms, and we construct…
Let $S \subseteq \mathbb{R}^n$ be a compact semialgebraic set and let $f$ be a polynomial nonnegative on $S$. Schm\"udgen's Positivstellensatz then states that for any $\eta > 0$, the nonnegativity of $f + \eta$ on $S$ can be certified by…
We investigate the question whether Subset Sum can be solved by a polynomial-time algorithm with access to a certificate of length poly(k) where k is the maximal number of bits in an input number. In other words, can it be solved using only…
Polynomial optimization problems represent a wide class of optimization problems, with a large number of real-world applications. Current approaches for polynomial optimization, such as the sum of squares (SOS) method, rely on large-scale…
A non-negativity certificate (NNC) is a way to write a polynomial so that its non-negativity on a semialgebraic set becomes evident. Positivstellens\"atze (Ps\"atze) guarantee the existence of NNCs. Both, NNCs and Ps\"atze underlie powerful…
The second-order cone is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a…
We provide a complete and explicit characterization of the real zeros of sums of nonnegative circuit (SONC) polynomials, a recent certificate for nonnegative polynomials independent of sums of squares. As a consequence, we derive an exact…
Sum-of-squares (SOS) optimization provides a computationally tractable framework for certifying polynomial nonnegativity. If the considered problem is convex, the SOS problem can be transcribed into and solved by semi-definite programs.…