Related papers: Improved Polynomial Bounds and Acceleration of GMR…
We consider a convex minimization problem for which the objective is the sum of a homogeneous polynomial of degree four and a linear term. Such task arises as a subproblem in algorithms for quadratic inverse problems with a…
We consider min-max optimization problems for polynomial functions, where a multivariate polynomial is maximized with respect to a subset of variables, and the resulting maximal value is minimized with respect to the remaining variables.…
Extension problems for polynomial valuations on different cones of convex functions are investigated. It is shown that for the classes of functions under consideration, the extension problem reduces to a simple geometric obstruction on the…
In this paper, two efficient iterative algorithms based on the simpler GMRES method are proposed for solving shifted linear systems. To make full use of the shifted structure, the proposed algorithms utilizing the deflated restarting…
The motivation of this work stems from the numerical approximation of bounded functions by polynomials satisfying the same bounds. The present contribution makes use of the recent algebraic characterization found in [B. Despr\'es, Numer.…
In this article, we propose a geometric programming method in order to compute lower bounds for real polynomials. We provide new sufficient conditions for polynomials to be nonnegative as well as to have a sum of binomial squares…
In order to verify programs or hybrid systems, one often needs to prove that certain formulas are unsatisfiable. In this paper, we consider conjunctions of polynomial inequalities over the reals. Classical algorithms for deciding these not…
Work on generalizing the deflated, restarted GMRES algorithm, useful in lattice studies using stochastic noise methods, is reported. We first show how the multi-mass extension of deflated GMRES can be implemented. We then give a deflated…
Adaptive regularization methods pre-multiply a descent direction by a preconditioning matrix. Due to the large number of parameters of machine learning problems, full-matrix preconditioning methods are prohibitively expensive. We show how…
Many important combinatorial problems can be modeled as constraint satisfaction problems. Hence identifying polynomial-time solvable classes of constraint satisfaction problems has received a lot of attention. In this paper, we are…
This paper proposes a two-level restricted additive Schwarz (RAS) method for multiscale PDEs, built on top of a multiscale spectral generalized finite element method (MS-GFEM). The method uses coarse spaces constructed from optimal local…
We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable…
In a multidimensional infinite layer bounded by two hyperplanes, the inhomogeneous Helmholtz equation with a polynomial right-hand side is considered. It is shown that the Dirichlet and Dirichlet-Neumann boundary-value problems with…
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…
Conic optimization has recently emerged as a powerful tool for designing tractable and guaranteed algorithms for non-convex polynomial optimization problems. On the one hand, tractability is crucial for efficiently solving large-scale…
This paper is about GMRES algorithms for the solution of nonsingular linear systems. We first consider basic algorithms and study their convergence. We then focus on acceleration strategies and parallel algorithms that are useful for…
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our…
We investigate the problem of certifying optimality for sparse generalized linear models (GLMs), where sparsity is enforced through a cardinality constraint. While Branch-and-Bound (BnB) frameworks can certify optimality using perspective…
When a linear system Ax = y is solved by means of iterative methods (mainly CG and GMRES) and the convergence rate is slow, one may consider a preconditioner P. The use of such preconditioner changes the spectrum of the matrix defining the…
With the recent emergence of mixed precision hardware, there has been a renewed interest in its use for solving numerical linear algebra problems fast and accurately. The solution of least squares (LS) problems $\min_x\|b-Ax\|_2$, where $A…