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The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by $A \in\mathbb{Z}^{m\times{}n}$ and present an algorithm to solve such problems in polynomial-time provided that both the…
This article presents a validation of a recently proposed strongly polynomial-time algorithm for the general linear programming problem. The proposed algorithm is an implicit reduction procedure that combines primal and dual linear…
We present two new algorithms for the computation of the q-integer linear decomposition of a multivariate polynomial. Such a decomposition is essential for the treatment of q-hypergeometric symbolic summation via creative telescoping and…
General error locator polynomials are polynomials able to decode any correctable syndrome for a given linear code. Such polynomials are known to exist for all cyclic codes and for a large class of linear codes. We provide some decoding…
We consider integer and linear programming problems for which the linear constraints exhibit a (recursive) block-structure: The problem decomposes into independent and efficiently solvable sub-problems if a small number of constraints is…
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
The aim of this study is to find the optimum of a linear fractional function over the efficient set of a multi-objective linear fractional integer program without generating all efficient solutions. By its nature, it is a global…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…
We present new deterministic algorithms for several cases of the maximum rank matrix completion problem (for short matrix completion), i.e. the problem of assigning values to the variables in a given symbolic matrix as to maximize the…
In binary polynomial optimization, the goal is to find a binary point maximizing a given polynomial function. In this paper, we propose a novel way of formulating this general optimization problem, which we call factorized binary polynomial…
Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution.…
This paper considers the multi-parametric linear complementarity problem (pLCP) with sufficient matrices. The main result is an algorithm to find a polyhedral decomposition of the set of feasible parameters and to construct a piecewise…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…
Since the beginning of the development of interior-point methods, there exists a puzzling gap between the results in theory and the observations in numerical experience, i.e., algorithms with good polynomial bound are not computationally…
In this paper, we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities, which depend in a possibly nonlinear way on a vector of uncertain parameters. Motivated by recent results…
We study the problem of minimizing a multivariate polynomial function over the unit hypercube. By representing the polynomial through a hypergraph and exploiting its sparsity structure, we establish a new sufficient condition under which…
In this note, we extend the result of \cite{PoulyG16} about the complexity of solving polynomial differential equations over unbounded domains to work with non-rational input. In order to deal with arbitrary input, we phrase the result in…
Solving algebraic word problems requires executing a series of arithmetic operations---a program---to obtain a final answer. However, since programs can be arbitrarily complicated, inducing them directly from question-answer pairs is a…
Linear programming (LP) is an extremely useful tool and has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the…