# POS3POLY—a MATLAB preprocessor for optimization with positive polynomials

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## Abstract

Positive polynomials, relaxed to sum-of-squares in the multivariate case, are a powerful instrument having applications in signal processing, control and other engineering fields. Hence, appeared the need of a library which can work with positive polynomials as variables in a convex optimization problem. We present here the POS3POLY library, which transforms polynomial positivity into positive semidefinite constraints, thus enabling the user to solve such problems without the need of knowing the parameterization for each type of polynomial. POS3POLY is able to handle three types of polynomials: trigonometric, real and hybrid. The positivity of the polynomials can be global or only on a semialgebraic domain. POS3POLY allows also to define Bounded Real Lemma constraints. The library is written in MATLAB and uses SeDuMi for solving the convex optimization problems. POS3POLY can also work inside CVX. To show the usage of our library we give several examples of 2-D FIR filter design.

## Keywords

Library Convex optimization Positive polynomials Trigonometric/real/hybrid polynomials Filter design## Notes

### Acknowledgements

This work was supported by the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, projects PNII-ID 1033/2007 and PN-II-ID-PCE-2011-3-0400 and the Sectoral Operational Programme Human Resources Development 2007-2013 of the Romanian Ministry of Labour, Family and Social Protection through the Financial Agreement POSDRU/6/1.5/S/16.

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