Related papers: Univariate representations of solutions to generic…
A new technique for approximating the entire solution set for a nonlinear system of relations (nonlinear equations, inequalities, etc. involving algebraic, smooth, or even continuous functions) is presented. The technique is to first plot…
This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed…
In this work, we study the problem of finding approximate, with minimum support set, solutions to matrix max-plus equations, which we call sparse approximate solutions. We show how one can obtain such solutions efficiently and in polynomial…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
One can reduce the problem of proving that a polynomial is nonnegative, or more generally of proving that a system of polynomial inequalities has no solutions, to finding polynomials that are sums of squares of polynomials and satisfy some…
General methods of solving equations deal with solving N equations in N variables and the solutions are usually a set of discrete values. However, for problems with a softly broken symmetry these methods often first find a point which would…
In this work, we introduce a novel variant of the multivariate quadratic problem, which is at the core of one of the most promising post-quantum alternatives: multivariate cryptography. In this variant, the solution of a given multivariate…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required…
We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement,…
It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a weighted sum of finitely many squares instead of a sum of two squares, then…
We propose a black-box approach to reducing large semidefinite programs to a set of smaller semidefinite programs by projecting to random linear subspaces. We evaluate our method on a set of polynomial optimization problems, demonstrating…
This paper presents a program analysis method that generates program summaries involving polynomial arithmetic. Our approach builds on prior techniques that use solvable polynomial maps for summarizing loops. These techniques are able to…
We combine the known methods for univariate polynomial root-finding and for computations in the Frobenius matrix algebra with our novel techniques to advance numerical solution of a univariate polynomial equation, and in particular…
This paper presents an algorithmic study of a class of covering mixed-integer linear programming problems which encompasses classic cover problems, including multidimensional knapsack, facility location and supplier selection problems. We…
Modelling real world systems frequently requires the solution of systems of nonlinear equations. A number of approaches have been suggested and developed for this computational problem. However, it is also possible to attempt solutions…
We present a generalized formulation for reweighted least squares approximations. The goal of this article is twofold: firstly, to prove that the solution of such problem can be expressed as a convex combination of certain interpolants when…
In this paper, we propose two new methods for solving Set Constraint Problems, as well as a potential polynomial solution for NP-Complete problems using quantum computation. While current methods of solving Set Constraint Problems focus on…
We consider a general statistical linear inverse problem, where the solution is represented via a known (possibly overcomplete) dictionary that allows its sparse representation. We propose two different approaches. A model selection…
We present fully polynomial-time (deterministic or randomised) approximation schemes for Holant problems, defined by a non-negative constraint function satisfying a generalised second order recurrence modulo a couple of exceptional cases.…