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Sums of squares, convex optimization, and the Positivstellensatz


P.A. Parrilo

Canada, Third annual colloquiumfest in honour of the birthday of Murray Marshall. Mathematical Sciences Group, University of Saskatchewan.

We present an overview of a recently introduced convex optimization framework for semialgebraic problems. Along the way, we'll learn how to compute sum of squares decompositions for polynomials using semidefinite programming, and the computational shortcuts that are possible whenever additional properties (such as sparsity, symmetries, or an ideal structure) are present. The results are used to develop hierarchies of progressively stronger convex tests, based on the Positivstellensatz, to prove emptiness of semialgebraic sets. The developed techniques unify and generalize many well-known existing results. The ideas and algorithms will be illustrated with examples from a broad range of application domains, such as continuous and combinatorial optimization and systems and control theory.

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