Related papers: Dual Bounded Generation: Polynomial, Second-order …
We consider the question of which nonconvex sets can be represented exactly as the feasible sets of mixed-integer convex optimization problems. We state the first complete characterization for the case when the number of possible integer…
Our main result is that every n-dimensional polytope can be described by at most (2n-1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound…
The minimum dominating set problem asks for a dominating set with minimum size. First, we determine some vertices contained in the minimum dominating set of a graph. By applying a particular scheme, we ensure that the resulting graph is…
Polynomial Systems, or at least their algorithms, have the reputation of being doubly-exponential in the number of variables [Mayr and Mayer, 1982], [Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that number of…
Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…
The generalized moment problem (GMP) is an infinite dimensional linear problem over the cone of finite nonnegative Borel measures. When a GMP instance involves finitely many polynomial moment constraints, moment/sum-of-squares hierarchies…
Mutual-visibility sets were motivated by visibility in distributed systems and social networks, and intertwine with several classical mathematical areas. Monotone properties of the variety of mutual-visibility sets, and restrictions of such…
In this paper, we present an approach to integer factorization using distributed representations formed with Vector Symbolic Architectures. The approach formulates integer factorization in a manner such that it can be solved using neural…
Classical results of second order parabolic quasi-linear equations always require that the nonlinear terms are controlled by a power of the unknown functions and their first derivatives. We improve the previous results. More precisely, in…
We consider generalizations of Reed-Muller codes, toric codes, and codes from certain plane curves, such as those defined by norm and trace functions on finite fields. In each case we are interested in codes defined by evaluating arbitrary…
The problem to compute the vertices of a polytope given by affine inequalities is called vertex enumeration. The inverse problem, which is equivalent by polarity, is called the convex hull problem. We introduce `approximate vertex…
This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain…
Two matrices are said to be principal minor equivalent if they have equal corresponding principal minors of all orders. We give a characterization of principal minor equivalence and a deterministic polynomial time algorithm to check if two…
We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box…
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…
The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…
We reformulate the problem of modularity maximization over the set of partitions of a network as a conic optimization problem over the completely positive cone, converting it from a combinatorial optimization problem to a convex continuous…
We present precise bit and degree estimates for the optimal value of the polynomial optimization problem $f^*:=\text{inf}_{x\in \mathscr{X}}~f(x)$, where $\mathscr{X}$ is a semi-algebraic set satisfying some non-degeneracy conditions. Our…
In this paper, we consider the problems of enumerating minimal vertex covers and minimal dominating sets with capacity and/or connectivity constraints. We develop polynomial-delay enumeration algorithms for these problems on bounded-degree…
We study the cone of completely positive (cp) matrices for the first interesting case $n = 5$. This is a semialgebraic set, which means that the polynomial equalities and inequlities that define its boundary can be derived. We characterize…