Related papers: Optimized Gr\"obner basis algorithms for maximal d…

We present our latest contributions in terms of mathematical modeling and algorithm development for the global optimization of the ACOPF problem. These contributions allow us to close the optimality gap on a number of open instances in the…

During recent years the field of fine-grained complexity has bloomed to produce a plethora of results, with both applied and theoretical impact on the computer science community. The cornerstone of the framework is the notion of…

For a finite $\mathbb{Z}$-algebra $R$, i.e., for a $\mathbb{Z}$-algebra which is a finitely generated $\mathbb{Z}$-module, we assume that $R$ is explicitly given by a system of $\mathbb{Z}$-module generators $G$, its relation module ${\rm…

In the context of modeling biological systems, it is of interest to generate ideals of points with a unique reduced Groebner basis, and the first main goal of this paper is to identify classes of ideals in polynomial rings which share this…

Low-rank approximation with zeros aims to find a matrix of fixed rank and with a fixed zero pattern that minimizes the Euclidean distance to a given data matrix. We study the critical points of this optimization problem using algebraic…

Let $k$ be an infinite field and $I\subset k [x_1, \ldots ,x_n]$ be an ideal such that dim $V(I)=q$. Denote by $(f_1, \ldots, f_s)$ a set of generators of $I$. One can see that in the set $I\cap k [x_{1},...,x_{q+1}]$ there exist non-zero…

This extended abstract gives a construction for lifting a Gr\"obner basis algorithm for an ideal in a polynomial ring over a commutative ring R under the condition that R also admits a Gr\"obner basis for every ideal in R.

Let f be a polynomial of degree n in ZZ[x_1,..,x_n], typically reducible but squarefree. From the hypersurface {f=0} one may construct a number of other subschemes {Y} by extracting prime components, taking intersections, taking unions, and…

A universal analytic Gr{\"o}bner basis (UAGB) of an ideal of a Tate algebra is a set containing a local Gr{\"o}bner basis for all suitable convergence radii. In a previous article, the authors proved the existence of finite UAGB's for…

In this paper, we extend the characterization of $\mathbb{Z}[x]/\ < f \ >$, where $f \in \mathbb{Z}[x]$ to be a free $\mathbb{Z}$-module to multivariate polynomial rings over any commutative Noetherian ring, $A$. The characterization allows…

Let f be a real or complex polynomial. We give an algorithm to compute the set of generalized critical values. The algorithm uses a finite dimensional space of rational arcs along which we can reach all generalized critical values of f.

Consider the polynomial optimization problem whose objective and constraints are all described by multivariate polynomials. Under some genericity assumptions, %% on these polynomials, we prove that the optimality conditions always hold on…

This paper presents and analyzes the first matrix optimization model which allows general coordinate and spectral constraints. The breadth of problems our model covers is exemplified by a lengthy list of examples from the literature,…

Policy gradient methods can solve complex tasks but often fail when the dimensionality of the action-space or objective multiplicity grow very large. This occurs, in part, because the variance on score-based gradient estimators scales…

The problem of computing the optimum of a function on a finite set is an important problem in mathematics and computer science. Many combinatorial problems such as MAX-SAT and MAXCUT can be recognized as optimization problems on the…

We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…

In this paper, we extend the idea of comprehensive Gr\"{o}bner bases given by Weispfenning (1992) to border bases for zero dimensional parametric polynomial ideals. For this, we introduce a notion of comprehensive border bases and border…

We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An…

Recently, there have been growing interests in solving consensus optimization problems in a multi-agent network. In this paper, we develop a decentralized algorithm for the consensus optimization problem…

We use the lexicographic order to define a hierarchy of primal and dual bounds on the optimum of a bounded integer program. These bounds are constructed using lex maximal and minimal feasible points taken under different permutations. Their…