# POS3POLY—a MATLAB preprocessor for optimization with positive polynomials

## 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.

## References

- Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, Cambridge MATHGoogle Scholar
- Choi MD, Lam TY, Reznick B (1995) Sums of squares of real polynomials. Proc Symp Pure Math 58(2):103–126 MathSciNetGoogle Scholar
- Davidson TN, Luo ZQ, Sturm JF (2002) Linear matrix inequality formulation of spectral mask constraints with applications to fir filter design. IEEE Trans Signal Process 50(11):2702–2715 MathSciNetCrossRefGoogle Scholar
- Du C, Xie L, Soh YC (2000)
*H*_{∞}filtering of 2-D discrete systems. IEEE Trans Signal Process 48(6):1760–1768 MATHCrossRefGoogle Scholar - Dumitrescu B (2006a) Multidimensional stability test using sum-of-squares decomposition. IEEE Trans Autom Circuits Syst I 53(4):928–936 MathSciNetCrossRefGoogle Scholar
- Dumitrescu B (2006b) Trigonometric polynomials positive on frequency domains and applications to 2-D FIR filter design. IEEE Trans Signal Process 54(11):4282–4292 MathSciNetCrossRefGoogle Scholar
- Dumitrescu B (2007b) Positive trigonometric polynomials and signal processing applications. Springer, Berlin MATHGoogle Scholar
- Dumitrescu B (2009) Bouded real lemma for multivariate trigonometric matrix polynomials and FIR filter design applications. In: Proc European sign proc conf (EUSIPCO), Glasgow, Scotland, pp 676–680 Google Scholar
- Dumitrescu B, Şicleru BC, Ştefan R (2010) Positive hybrid real-trigonometric polynomials and applications to adjustable filter design and absolute stability analysis. Circuits Syst Signal Process 29(5):881–899 MATHCrossRefGoogle Scholar
- Grant M, Boyd S (2010) CVX: MATLAB software for disciplined convex programming, version 1.21. http://cvxr.com/cvx, May
- Henrion D, Lasserre JB (2003) GloptiPoly: Global optimization over polynomials with MATLAB and SeDuMi. ACM Trans Math Softw 29(2):165–194 MathSciNetMATHCrossRefGoogle Scholar
- Ichihara H (2009) Optimal control for polynomial systems using matrix sum of squares relaxations. IEEE Trans Autom Control 54(5):1048–1053 MathSciNetCrossRefGoogle Scholar
- Jacobi T (2001) A representation theorem for certain partially ordered commutative rings. Math Z 237:259–273 MathSciNetMATHCrossRefGoogle Scholar
- Lasserre JB (2001) Global optimization with polynomials and the problem of moments. SIAM J Optim 11(3):796–814 MathSciNetMATHCrossRefGoogle Scholar
- Li SJ, Teo KL, Yang XQ, Wu SY (2007) Robust envelope-constrained filter with orthonormal bases and semi-definite and semi-infinite programming. Optim Eng 8(3):299–319 MathSciNetMATHCrossRefGoogle Scholar
- Löfberg J (2004) YALMIP: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD conference, Taipei, Taiwan Google Scholar
- Magnani A, Boyd SP (2009) Convex piecewise-linear fitting. Optim Eng 10(1):1–17 MathSciNetCrossRefGoogle Scholar
- McLean JW, Woerdeman HJ (2001) Spectral factorizations and sum of squares representations via semidefinite programming. SIAM J Matrix Anal Appl 23(3):646–655 MathSciNetMATHCrossRefGoogle Scholar
- Parrilo PA (2003) Semidefinite programming relaxations for semialgebraic problems. Math Program, Ser B 96:293–320 MathSciNetMATHCrossRefGoogle Scholar
- Popeea C, Dumitrescu B (2001) Optimal compaction gain by eigenvalue minimization. Signal Process 81(5):1113–1116 MATHCrossRefGoogle Scholar
- Prajna S, Papachristodoulou A (2004) SOSTOOLS: Sum of squares optimization toolbox for Matlab Google Scholar
- Putinar M (1993) Positive polynomials on compact semialgebraic sets. Indiana Univ Math J 42(3):969–984 MathSciNetMATHCrossRefGoogle Scholar
- Şicleru BC, Dumitrescu B (2011) FIR filter optimization with POS3POLY in CVX. In: Proc European sign proc conf (EUSIPCO), Barcelona, Spain, August–September, pp 1405–1409 Google Scholar
- Sturm JF (1999) Using SeDuMi, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw 11–12:625–653 MathSciNetCrossRefGoogle Scholar
- Tan W, Packard A (2008) Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming. IEEE Trans Autom Control 53(2):565–571 MathSciNetCrossRefGoogle Scholar
- Toh KC, Todd MJ, Tütüncü RH (1999) SDPT3—a MATLAB software package for semidefinite programming. Optim Methods Softw 11(1):545–581 MathSciNetCrossRefGoogle Scholar
- Xie L, de Souza CE (1990) Robust
*H*_{∞}control for linear time-invariant systems with norm-bounded uncertainty in the input matrix. Syst Control Lett 14(5):389–396 MATHCrossRefGoogle Scholar