相关论文: The Duality of SONC: Advances in Circuit-based Cer…
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…
We study the problem of computing weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. Building on the theory of interior-point methods for convex optimization, we introduce the concept of…
Finding the minimum of a multivariate real polynomial is a well-known hard problem with various applications. We present a polynomial time algorithm to approximate such lower bounds via sums of nonnegative circuit polynomials (SONC). As a…
A polynomial that is a sum of squares (SOS) of other polynomials is evidently positive. The converse is not true, there are positive polynomials which are not SOS. This note focuses on the problem of certifying, in exact arithmetic, that a…
In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semi-algebraic sets…
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…
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 investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a…
We analyze self-dual polyhedral cones and prove several properties about their slack matrices. In particular, we show that self-duality is equivalent to the existence of a positive semidefinite (PSD) slack. Beyond that, we show that if the…
In this article, we are interested in developing polynomial decomposition techniques based on sums-of-squares (SOS), namely the difference-of-sums-of-squares (D-SOS) and the difference-of-convex-sums-of-squares (DC-SOS). In particular, the…
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…
We study cyclic sieving phenomena (CSP) on combinatorial objects from an abstract point of view by considering a rational polyhedral cone determined by the linear equations that define such phenomena. Each lattice point in the cone…
The authors in a previous paper devised certain subcones of the semidefinite plus nonnegative cone and showed that satisfaction of the requirements for membership of those subcones can be detected by solving linear optimization problems…
We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases…
One-counter nets (OCN) are finite automata equipped with a counter that can store non-negative integer values, and that cannot be tested for zero. Equivalently, these are exactly 1-dimensional vector addition systems with states. We show…
We investigate structural properties of the completely positive semidefinite cone $\mathcal{CS}_+^n$, consisting of all the $n \times n$ symmetric matrices that admit a Gram representation by positive semidefinite matrices of any size. This…
In recent years, optimization theory has been greatly impacted by the advent of sum of squares (SOS) optimization. The reliance of this technique on large-scale semidefinite programs however, has limited the scale of problems to which it…
We introduce and study a generalized concept of boundedness of a subset of a normed vector space with respect to a cone, which is defined as lower boundedness of the images of the underlying set through all the positive functionals of the…
Arithmetic circuits (AC) are circuits over the real numbers with 0/1-valued input variables whose gates compute the sum or the product of their inputs. Positive AC -- that is, AC representing non-negative functions -- subsume many…
The second-order tangent set is an important concept in describing the curvature of the set involved. Due to the existence of the complementarity condition, the second-order cone (SOC) complementarity set is a nonconvex set. Moreover,…