Related papers: Which constraints of a numerical problem cause ill…
This paper provides a framework for deriving a new set of necessary conditions for adverse control problems among two players. The distinguish feature of such problems is that the first player has a priori knowledge on the second player…
An important measure of conditioning of a conic linear system is the size of the smallest structured perturbation making the system ill-posed. We show that this measure is unchanged if we restrict to perturbations of low rank. We thereby…
We are interested in the relative conditioning of the problem $y_0\mapsto \mathrm{e}^{tA}y_0$, i.e., the relative conditioning of the action of the matrix exponential $\mathrm{e}% ^{tA}$ on a vector with respect to perturbations of this…
When computing the expected value of time till extinction of a Birth and Death process, the usual textbook approach results in an extreme case of numerical ill-conditioning, which prevents us from getting accurate answers beyond the first…
We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell_1$-analysis setting. Using results from conic integral geometry, we show…
The tensor rank decomposition, or canonical polyadic decomposition, is the decomposition of a tensor into a sum of rank-1 tensors. The condition number of the tensor rank decomposition measures the sensitivity of the rank-1 summands with…
In this work, we investigate the condition number for a system of coupled non-local reaction-diffusion-advection equations developed in the context of modelling normal and abnormal wound healing. Following a finite element discretisation of…
This paper is devoted to a structured perturbation analysis of the symmetric algebraic Riccati equations by exploiting the symmetry structure. Based on the analysis, the upper bounds for the structured normwise, mixed and componentwise…
Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a…
Let $F$ be a quadratic form in four variables, let $m\in\mathbb{N}$ and let $\mathbf{k}\in \mathbb{Z}^4$. We count integer solutions to $F(\mathbf{x})=0$ with $\mathbf{x}\equiv \mathbf{k}\:\mathrm{mod}(m)$. One can compare this to the…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
The well-posedness of a class of optimal control problems is analysed, where the state equation couples a nonlinear degenerate Fokker-Planck equation with a system of Ordinary Differential Equations (ODEs). Such problems naturally arise as…
In this paper, we study the general problem of optimizing a convex function $F(L)$ over the set of $p \times p$ matrices, subject to rank constraints on $L$. However, existing first-order methods for solving such problems either are too…
A numerical method for approximating weak solutions of an aggregation equation with degenerate diffusion is introduced. The numerical method consists of a stabilized finite element method together with a mass lumping technique and an extra…
This paper considers the problem of testing whether there exists a solution satisfying certain non-negativity constraints to a linear system of equations. Importantly and in contrast to some prior work, we allow all parameters in the system…
Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible…
This paper deals with optimal control problems of integral equations, with initial-final and running state constraints. The order of a running state constraint is defined in the setting of integral dynamics, and we work here with…
Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as…
This paper provides a theoretical and numerical investigation of a penalty decomposition scheme for the solution of optimization problems with geometric constraints. In particular, we consider some situations where parts of the constraints…
We introduce statistical constraints, a declarative modelling tool that links statistics and constraint programming. We discuss two statistical constraints and some associated filtering algorithms. Finally, we illustrate applications to…