This chapter surveys how semidefinite programming can be used for finding good
approximative solutions to hard combinatorial optimization problems. The chapter begins …

The present volume is a translation, revision and updating of our book (pub lished in French)
with the title" Geometrie Algebrique Reelle". Since its pub lication in 1987 the theory has …

Since a real univariate polynomial does not always have real roots, a very natural
algorithmic problem, is to design a method to count the number of real roots of a given …

A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions
reducible to a finite number of polynomial equalities and inequalities, it is shown how to …

The Nonnegative Matrix Factorization (NMF) problem has a rich history spanning quantum
mechanics, probability theory, data analysis, polyhedral combinatorics, communication …

Nonnegative matrix factorization (NMF) has become a widely used tool for the analysis of
high-dimensional data as it automatically extracts sparse and meaningful features from a set …

The purpose of this paper is twofold:(a) to provide a tutorial introduction to some key
concepts from the theory of computational complexity, highlighting their relevance to …

We show that some basic linear control design problems are NP-hard, implying that, unless
P= NP, they cannot be solved by polynomial time algorithms. The problems that we consider …

Identifying the underlying structure of a data set and extracting meaningful information is a
key problem in data analysis. Simple and powerful methods to achieve this goal are linear …

We study the ℓ1-low rank approximation problem, where for a given nxd matrix A and
approximation factor α≤ 1, the goal is to output a rank-k matrix  for which‖ A-Â‖ 1≤ α …