Related papers: Noncommutative Polynomial Optimization
We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard…
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where…
We discuss the noncommutative generalizations of polynomial algebras which after appropriate completions can be used as coordinate algebras in various noncommutative settings, (noncommutative differential geometry, noncommutative algebraic…
This paper introduces and develops the algebraic framework of moment polynomials, which are polynomial expressions in commuting variables and their formal mixed moments. Their positivity and optimization over probability measures supported…
We demonstrate applications of algebraic techniques that optimize and certify polynomial inequalities to problems of interest in the operations research and transportation engineering communities. Three problems are considered: (i) wireless…
This paper proposes novel algorithm for non-convex multimodal constrained optimisation problems. It is based on sequential solving restrictions of problem to sections of feasible set by random subspaces (in general, manifolds) of low…
This work focuses on minimizing the eigenvalue of a noncommutative polynomial subject to a finite number of noncommutative polynomial inequality constraints. Based on the Helton-McCullough Positivstellensatz, the noncommutative analog of…
We ask whether every polynomial function that is non-negative on a real algebraic curve can be expressed as a sum of squares in the coordinate ring. Scheiderer has classified all irreducible curves for which this is the case. For reducible…
We consider symmetric polynomials, p, in the noncommutative free variables (x_1, x_2, ..., x_g). We define the noncommutative complex hessian of p and we call a noncommutative symmetric polynomial noncommutative plurisubharmonic if it has a…
In this paper we consider multi-objective optimization problems over a box. The problem is very relevant and several computational approaches have been proposed in the literature. They broadly fall into two main classes: evolutionary…
Let X be a finite set of points in R^n. A polynomial p nonnegative on X can be written as a sum of squares of rational functions modulo the vanishing ideal I(X). From the point of view of applications, such as polynomial optimization, we…
We show that every real nonnegative polynomial $f$ can be approximated as closely as desired by a sequence of polynomials $\{f_\epsilon\}$ that are sums of squares. Each $f_\epsilon$ has a simple et explicit form in terms of $f$ and…
We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…
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,…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
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…
We present a general framework to exploit the symmetries present in the Navascu{\'e}s-Pironio-Ac{\'i}n semidefinite relaxations that approximate invariant noncommutative polynomial optimization problems. We put equal emphasis on the moment…
We consider potentially non-convex optimization problems, for which optimal rates of approximation depend on the dimension of the parameter space and the smoothness of the function to be optimized. In this paper, we propose an algorithm…
Semidefinite relaxations are widely used to compute upper bounds on the objective of optimization problems involving noncommutative polynomials. Such optimization problems are prevalent in quantum information. We present an algorithm able…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…