## Random projections for quadratic programs

Random projections map a set of points in a high dimensional space to a lower dimen- sional one while approximately preserving all pairwise Euclidean distances. While random projections are usually applied to numerical data, we show they can be successfully applied to quadratic programming formulations over a set of linear inequality constraints. Instead of solving … Read more

## Random projections for trust region subproblems

The trust region method is an algorithm traditionally used in the field of derivative free optimization. The method works by iteratively constructing surrogate models (often linear or quadratic functions) to approximate the true objective function inside some neighborhood of a current iterate. The neighborhood is called “trust region” in the sense that the model is … Read more

## Random projections for linear programming

Random projections are random linear maps, sampled from appropriate distributions, that approximately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well-known Johnson-Lindenstrauss lemma states that there are \LL{random matrices with surprisingly few rows} that approximately preserve pairwise Euclidean distances among a set of points. This is … Read more

## Fast approximate solution of large dense linear programs

We show how random projections can be used to solve large-scale dense linear programs approximately. This is a new application of techniques which are now fairly well known in probabilistic algorithms, but have never yet been systematically applied to the fundamental class of Linear Programming. We develop the necessary theoretical framework, and show that this … Read more

## New error measures and methods for realizing protein graphs from distance data

The interval Distance Geometry Problem (iDGP) consists in finding a realization in R^K of a simple undirected graph G=(V,E) with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. Our aim is to determine … Read more

## Using the Johnson-Lindenstrauss lemma in linear and integer programming

The Johnson-Lindenstrauss lemma allows dimension reduction on real vectors with low distortion on their pairwise Euclidean distances. This result is often used in algorithms such as \$k\$-means or \$k\$ nearest neighbours since they only use Euclidean distances, and has sometimes been used in optimization algorithms involving the minimization of Euclidean distances. In this paper we … Read more