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In this article we provide an experimental algorithm that in many cases gives us an upper bound of the global infimum of a real polynomial on $\R^{n}$. It is very well known that to find the global infimum of a real polynomial on $\R^{n}$,…
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…
Linear programming (polynomial) techniques are used to obtain lower and upper bounds for the potential energy of spherical designs. This approach gives unified bounds that are valid for a large class of potential functions. Our lower bounds…
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…
We develop a new kind of nonnegativity certificate for univariate polynomials on an interval. In many applications, nonnegative Bernstein coefficients are often used as a simple way of certifying polynomial nonnegativity. Our proposed…
Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on…
We consider a version of geometric programming problem consisting in minimizing a function given by the maximum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption,…
We provide a condition-based analysis of two interior-point methods for unconstrained geometric programs, a class of convex programs that arise naturally in applications including matrix scaling, matrix balancing, and entropy maximization.…
Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…
The problem of characterizing a real polynomial $f$ as a sum of squares of polynomials on a real algebraic variety $V$ dates back to the pioneering work of Hilbert in [Mathematische Annalen 32.3 (1888): 342-350]. In this paper, we…
In polynomial optimization problems, nonnegativity constraints are typically handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and…
We utilize the same technique as in [arXiv:2205.04254 (2022)] to provide some representations of polynomials non-negative on a basic semi-algebraic set, defined by polynomial inequalities, under more general conditions. Based on each…
We consider a generalization of polynomial programs: algebraic programs, which are optimization or feasibility problems with algebraic objectives or constraints. Algebraic functions are defined as zeros of multivariate polynomials. They are…
A Geometric programming (GP) is a type of mathematical problem characterized by objective and constraint functions that have a special form. Many methods have been developed to solve large scale engineering design GP problems. In this paper…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness,…
Assessing non-negativity of multivariate polynomials over the reals, through the computation of {\em certificates of non-negativity}, is a topical issue in polynomial optimization. This is usually tackled through the computation of {\em…
We consider N-fold 4-block decomposable integer programs, which simultaneously generalize N-fold integer programs and two-stage stochastic integer programs with N scenarios. In previous work [R. Hemmecke, M. Koeppe, R. Weismantel, A…
The realization space of geometric constraint systems is given by the vanishing locus of polynomials corresponding to natural geometric constraints. Such geometric constraint systems arise in many real-world scenarios such as structural…
Global polynomial optimization methods typically rely on compactness of the feasible region in order to find solutions. These methods can incur considerable computational expense and most commercially available solvers do not verify the…