Related papers: Simplification Methods for Sum-of-Squares Programs

Sparse (or toric) elimination exploits the structure of polynomials by measuring their complexity in terms of Newton polytopes instead of total degree. The sparse, or Newton, resultant generalizes the classical homogeneous resultant and its…

The Sum of Squares (\sos{}) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of this hierarchy give…

Suppose f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present an algorithm, with complexity polynomial in log D on average (relative to the stable log-uniform measure), for counting the number of real roots…

We develop the first pure node-differentially-private algorithms for learning stochastic block models and for graphon estimation with polynomial running time for any constant number of blocks. The statistical utility guarantees match those…

In this paper a special piecewise linear system is studied. It is shown that, under a mild assumption, the semi-smooth Newton method applied to this system is well defined and the method generates a sequence that converges linearly to a…

We propose a linear algorithm for determining two function parameters by their linear combination. These functions must satisfy the first order differential equations with polynomial coefficients and our parameters are the coefficients of…

We propose a computationally tractable method for the identification of stable canonical discrete-time rational transfer function models, using frequency domain data. The problem is formulated as a global non-convex optimization problem…

In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation…

Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…

We study a class of combinatorial scheduling problems characterized by a particular type of constraint often associated with electrical power or gas energy. This constraint appears in several practical applications and is expressed as a sum…

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in…

These notes are not intended to substitute for a course in linear algebra on reduction of endomorphisms nor an exhaustive presentation of the Dunford's decomposition. We will limit ourselves to the case where the base is R or C, and the…

We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new…

In this paper we study the problem of maximizing the distance to a given point over an intersection of balls. It was already known that this problem can be solved in polynomial time and space if the given point is not in the convex hull of…

We exhibit a probabilistic symbolic algorithm for solving zero-dimensional sparse systems. Our algorithm combines a symbolic homotopy procedure, based on a flat deformation of a certain morphism of affine varieties, with the polyhedral…

This paper is devoted to the factorization of multivariate polynomials into products of linear forms, a problem which has applications to differential algebra, to the resolution of systems of polynomial equations and to Waring decomposition…

For systems of polynomial equations, we study the problem of computing the Newton polytope of their eliminants. As was shown by Esterov and Khovanskii, such Newton polytopes are mixed fiber polytopes of the Newton polytopes of the input…

In this survey we consider polynomial optimization problems, asking to minimize a polynomial function over a compact semialgebraic set, defined by polynomial inequalities. This models a great variety of (in general, nonlinear nonconvex)…

We revisit two NP-hard geometric partitioning problems - convex decomposition and surface approximation. Building on recent developments in geometric separators, we present quasi-polynomial time algorithms for these problems with improved…

In this chapter we present the sums of Hermitian squares approach to noncommutative polynomial optimization problems. This is an extension of the sums of squares approach for polynomial optimization arising from real algebraic geometry. We…