Related papers: The algebraic degree of sparse polynomial optimiza…

We minimize elastic energies on framed curves which penalize both curvature and torsion. We also discuss critical points using the infinite dimensional version of the Lagrange multipliers' method. Finally, some examples arising from the…

Finding the product of two polynomials is an essential and basic problem in computer algebra. While most previous results have focused on the worst-case complexity, we instead employ the technique of adaptive analysis to give an improvement…

This paper investigates a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both…

The works presented in this habilitation concern the algorithmics of polynomials. This is a central topic in computer algebra, with numerous applications both within and outside the field - cryptography, error-correcting codes, etc. For…

We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector…

Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this…

We study algebraic points of bounded degree on polarized projective varieties. To do so, we refine further the filtration construction and Subspace Theorem approach, for the study of integral points, which has origins in the work of…

Minimizing a convex function of a measure with a sparsity-inducing penalty is a typical problem arising, e.g., in sparse spikes deconvolution or two-layer neural networks training. We show that this problem can be solved by discretizing the…

Constrained optimization problems exist in many domains of science, such as thermodynamics, mechanics, economics, etc. These problems are classically solved with the help of the Lagrange multipliers and the Lagrangian function. However, the…

Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume…

Standard quadratic optimization problems (StQPs) provide a versatile modelling tool in various applications. In this paper, we consider StQPs with a hard sparsity constraint, referred to as sparse StQPs. We focus on various tractable convex…

We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are…

We present an efficient framework for solving algebraically-constrained global non-convex polynomial optimization problems over subsets of the hypercube. We prove the existence of an equivalent nonlinear reformulation of such problems that…

Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This…

We consider the problem of sparse coding, where each sample consists of a sparse linear combination of a set of dictionary atoms, and the task is to learn both the dictionary elements and the mixing coefficients. Alternating minimization is…

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…

We consider the maximum likelihood estimation of sparse inverse covariance matrices. We demonstrate that current heuristic approaches primarily encourage robustness, instead of the desired sparsity. We give a novel approach that solves the…

We study polynomial systems with prescribed monomial supports in the Cox rings of toric varieties built from complete polyhedral fans. We present combinatorial formulas for the dimensions of their associated subvarieties under genericity…

Grassmann and flag varieties lead many lives in pure and applied mathematics. Here we focus on the algebraic complexity of solving various problems in linear algebra and statistics as optimization problems over these varieties. The measure…

The real solutions to a system of sparse polynomial equations may be realized as a fiber of a projection map from a toric variety. When the toric variety is orientable, the degree of this map is a lower bound for the number of real…