Related papers: A Gr\"obner bases methodology for solving multiobj…
In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the…
We study the complexity of Gr\"obner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. We give a bound on the number of polynomials of degree $d$…
Multiobjective optimization plays an increasingly important role in modern applications, where several objectives are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to…
We study how to solve semidefinite programming relaxations for large scale polynomial optimization. When interior-point methods are used, typically only small or moderately large problems could be solved. This paper studies regularization…
Complex-variable matrix optimization problems (CMOPs) in Frobenius norm emerge in many areas of applied mathematics and engineering applications. In this letter, we focus on solving CMOPs by iterative methods. For unconstrained CMOPs, we…
Computations over the rational numbers often encounter the problem of intermediate coefficient growth. A solution to this is provided by modular methods, which apply the algorithm under consideration modulo a number of primes and then lift…
We consider the problem of computing critical points of the restriction of a polynomial map to an algebraic variety. This is of first importance since the global minimum of such a map is reached at a critical point. Thus, these points…
Let $\K$ be a field and $(f_1, \ldots, f_n)\subset \K[X_1, \ldots, X_n]$ be a sequence of quasi-homogeneous polynomials of respective weighted degrees $(d_1, \ldots, d_n)$ w.r.t a system of weights $(w_{1},\dots,w_{n})$. Such systems are…
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
We describe how Groebner bases can be used to solve the reduction problem for Feynman integrals, i.e. to construct an algorithm that provides the possibility to express a Feynman integral of a given family as a linear combination of some…
We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…
In the present work, we study how to develop an efficient solver for the fast resolution of large and sparse linear systems that occur while discretizing elliptic partial differential equations using isogeometric analysis. Our new approach…
The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…
Linear programming is the seminal optimization problem that has spawned and grown into today's rich and diverse optimization modeling and algorithmic landscape. This article provides an overview of the recent development of first-order…
We propose an exact algorithm for solving biobjective integer programming problems, which arise in various applications of operations research. The algorithm is based on solving Pascoletti-Serafini scalarizations to search specified regions…
Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced…
We consider the problem of mutually unbiased bases as a polynomial optimization problem over the reals. We heavily reduce it using known symmetries before exploring it using two methods, combining a number of optimization techniques. The…
A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally…
In this paper, we tackle the parametric complete multiplicity problem for a univariate polynomial. Our approach to the parametric complete multiplicity problem has a significant difference from the classical method, which relies on repeated…
In this paper we introduce a multistep generalization of the guess-and-determine or hybrid strategy for solving a system of multivariate polynomial equations over a finite field. In particular, we propose performing the exhaustive…