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Optimization with orthogonality constraints frequently arises in various fields such as machine learning. Riemannian optimization offers a powerful framework for solving these problems by equipping the constraint set with a Riemannian…
Large-scale optimization problems that seek sparse solutions have become ubiquitous. They are routinely solved with various specialized first-order methods. Although such methods are often fast, they usually struggle with not-so-well…
We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only…
In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and…
We extend Robust Optimization to fractional programming, where both the objective and the constraints contain uncertain parameters. Earlier work did not consider uncertainty in both the objective and the constraints, or did not use Robust…
This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches…
We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Sch{\"o}nhage's efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but…
Finding global optima in high-dimensional optimization problems is extremely challenging since the number of function evaluations required to sufficiently explore the search space increases exponentially with its dimensionality.…
In the context of tree-search stochastic planning algorithms where a generative model is available, we consider on-line planning algorithms building trees in order to recommend an action. We investigate the question of avoiding re-planning…
We study optimization over Riemannian embedded submanifolds, where the objective function is relatively smooth in the ambient Euclidean space. Such problems have broad applications but are still largely unexplored. We introduce two…
Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such…
Many studies have been carried out in order to increase the search efficiency of constraint satisfaction problems; among them, some make use of structural properties of the constraint network; others take into account semantic properties of…
We consider $m \times s$ matrices (with $m\geq s$) in a real affine subspace of dimension $n$. The problem of finding elements of low rank in such spaces finds many applications in information and systems theory, where low rank is…
New algorithms for prime factorization that outperform the existing ones or take advantage of particular properties of the prime factors can have a practical impact on present implementations of cryptographic algorithms that rely on the…
A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified…
In the context of global optimization and mixed-integer non-linear programming, generalizing a technique of D'Ambrosio, Fampa, Lee and Vigerske for handling the square-root function, we develop a virtuous smoothing method, using cubics,…
Integer factorization is a fundamental problem in algorithmic number theory and computer science. It is considered as a one way or trapdoor function in the (RSA) cryptosystem. To date, from elementary trial division to sophisticated methods…
We introduce a new approach to the the asymptotic iteration method (AIM) by means of which we establish the standard AIM connection with the continued fractions technique and we develop a novel termination condition in terms of the…
We examine a wide class of stochastic approximation algorithms for solving (stochastic) nonlinear problems on Riemannian manifolds. Such algorithms arise naturally in the study of Riemannian optimization, game theory and optimal transport,…
Let $F \in \R[X_1,\ldots,X_n]$ and the zero set $V=\zero(\mathcal{P},\R^n)$, where $\mathcal{P}:=\{P_1,\ldots,P_s\} \subset \R[X_1,\ldots,X_n]$ is a finite set of polynomials. We investigate existence of critical points of $F$ on an…