Semidefinite programming

TL;DR: In this article, a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal, is introduced.

TL;DR: The positive semidefinite rank (psd rank) as discussed by the authors is the smallest integer k for which there exist polyhedra of size k = 1 such that the polyhedron is polyhedrically connected with the rank of k. The psd rank has many appealing geometric interpretations, including semidefinite representations of polyhedras and information-theoretic applications.

TL;DR: A theorem is derived that provides a closed-form analytical optimal solution that appears to be an extension of the Rayleigh quotient minimization for a possibly singular matrix in quadratic form.

TL;DR: A short geometric proof of this result is given, which is used to improve a bound on realizability of weighted graphs as graphs of distances between points in Euclidean space, and its relation to theorems of Bohnenblust, Friedland and Loewy, and Au-Yeung and Poon.

TL;DR: This book describes the first unified theory of polynomial-time interior-point methods, and describes several of the new algorithms described, e.g., the projective method, which have been implemented, tested on "real world" problems, and found to be extremely efficient in practice.

TL;DR: A survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution are given.

TL;DR: In this article, the Fulkerson Prize was won by the Mathematical Programming Society and the American Mathematical Society for proving polynomial time solvability of problems in convexity theory, geometry, and combinatorial optimization.

TL;DR: In this article, a general method is developed to construct higher-dimensional polyhedra (or, in some cases, convex sets) whose projection approximates the convex hull of 0-1 valued solutions of a system of linear inequalities.