Convexity in SemiAlgebraic Geometry and Polynomial Optimization

Convexity in SemiAlgebraic Geometry and Polynomial Optimization

We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence, a highly desirable feature as convex problems are in principle easier to solve. In addition, if a ba...

Entropy, homology and semialgebraic geometry

© Association des collaborateurs de Nicolas Bourbaki, 1985-1986, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier ...

A Polynomial Regularity Lemma for Semialgebraic Hypergraphs and Its Applications in Geometry and Property Testing

Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for semi-algebraic kuniform hypergraphs of bounded complexity, showing that for each ǫ > 0 the vertex set can be equitably partitioned into a bounded number of parts (in terms of ǫ and the complexity) so that all but an ǫ-fraction of the k-tuples of parts are homogeneous. We prove that the number of parts can be taken to be polyno...

An Introduction to Semialgebraic Geometry

Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization

In the first part of this thesis, we introduce a specific class of Linear Matrix Inequalities (LMI) whose optimal solution can be characterized exactly. This family corresponds to the case where the associated linear operator maps the cone of positive semidefinite matrices onto itself. In this case, the optimal value equals the spectral radius of the operator. It is shown that some rank minimiz...