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 construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
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Received: May 10, 2001 / Accepted May 2002 Published online: April 10, 2003
Key Words. semidefinite programming – convex optimization – sums of squares – polynomial equations – real algebraic geometry
The majority of this research has been carried out while the author was with the Control & Dynamical Systems Department, California Institute of Technology, Pasadena, CA 91125, USA.
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Parrilo, P. Semidefinite programming relaxations for semialgebraic problems. Math. Program., Ser. B 96, 293–320 (2003). https://doi.org/10.1007/s10107-003-0387-5