Study of the cone of sums of squares plus sums of nonnegative circuit forms
-
Mareike Dressler
Abstract
In this article, we combine sums of squares (SOS) and sums of nonnegative circuit (SONC) forms, two independent nonnegativity certificates for real homogeneous polynomials. We consider the convex cone SOS+SONC of forms that decompose into a sum of an SOS and a SONC form and study it from a geometric point of view. We show that the SOS+SONC cone is proper and neither closed under multiplication nor under linear transformation of variables. Moreover, we present an alternative proof of an analog of Hilbert’s 1888 Theorem for the SOS+SONC cone and prove that in the non-Hilbert cases it provides a proper superset of the union of the SOS and SONC cones. This follows by exploiting a new necessary condition for membership in the SONC cone.
Acknowledgements
We would like to thank the Mathematisches Forschungsinstitut Oberwolfach for its hospitality. Mareike Dressler was supported by a Simon's Visiting Professorship 2023. Her research stay was partially supported by the Simons Foundation and by the Mathematisches Forschungsinstitut Oberwolfach. Since 2024, Mareike is supported by the Australian Research Council Discovery Early Career Award DE240100674. Salma Kuhlmann received support from the Ausschuss för Forschungsfragen, University of Konstanz. Moritz Schick acknowledges support by the scholarship program of the University of Konstanz under the Landesgraduiertenfördergesetz, the Studienstiftung des deutschen Volkes, as well as the Oberwolfach Leibniz Graduate Students Project.
We would also like to thank Janin Heuer and Khazhgali Kozhasov for various fruitful comments on different occasions. We thank the anonymous referee for valuable comments.
References
[1] A. A. Ahmadi, G. Hall, A. Papachristodoulou, J. Saunderson, Y. Zheng, Improving efficiency and scalability of sum of squares optimization: Recent advances and limitations. IEEE 56th Annual Conference on Decision and Control 2017, 453–462.10.1109/CDC.2017.8263706Search in Google Scholar
[2] G. Averkov, Optimal size of linear matrix inequalities in semidefinite approaches to polynomial optimization. SIAM J. Appl. Algebra Geom. 3 (2019), 128–151. MR3922906 Zbl 1420.9004310.1137/18M1201342Search in Google Scholar
[3] M. Bellare, P. Rogaway, The complexity of approximating a nonlinear program. Math. Programming 69 (1995), 429–441. MR1355698 Zbl 0839.9010410.1007/BF01585569Search in Google Scholar
[4] C. Berg, J. P. R. Christensen, C. U. Jensen, A remark on the multidimensional moment problem. Math. Ann. 243 (1979), 163–169. MR543726 Zbl 0416.4600310.1007/BF01420423Search in Google Scholar
[5] G. Blekherman, There are significantly more nonnegative polynomials than sums of squares. Israel J. Math. 153 (2006), 355–380. MR2254649 Zbl 1139.1404410.1007/BF02771790Search in Google Scholar
[6] G. Blekherman, P. A. Parrilo, R. R. Thomas (editors), Semidefinite optimization and convex algebraic geometry, volume 13 of MOS-SIAM Series on Optimization. Society for Industrial and Applied Mathematics, Mathematical Optimization Society, Philadelphia, PA 2013. MR3075433 Zbl 1260.9000610.1137/1.9781611972290Search in Google Scholar
[7] S. Boyd, L. Vandenberghe, Convex optimization. Cambridge Univ. Press 2004. MR2061575 Zbl 1058.9004910.1017/CBO9780511804441Search in Google Scholar
[8] V. Chandrasekaran, P. Shah, Relative entropy relaxations for signomial optimization. SIAM J. Optim. 26 (2016), 1147–1173. MR3499559 Zbl 1345.9006610.1137/140988978Search in Google Scholar
[9] M. D. Choi, T. Y. Lam, Extremal positive semidefinite forms. Math. Ann. 231 (1977/78), 1–18. MR498384 Zbl 0347.1500910.1007/BF01360024Search in Google Scholar
[10] M. Dressler, Sums of nonnegative circuit polynomials: Geometry and Optimization. PhD thesis, Goethe-Universität Frankfurt am Main, 2018.Search in Google Scholar
[11] M. Dressler, Real zeros of SONC polynomials. J. Pure Appl. Algebra 225 (2021), Paper No. 106602, 23 pages. MR4217923 Zbl 1457.1200210.1016/j.jpaa.2020.106602Search in Google Scholar
[12] M. Dressler, S. Iliman, T. de Wolff, A Positivstellensatz for sums of nonnegative circuit polynomials. SIAM J. Appl. Algebra Geom. 1 (2017), 536–555. MR3691721 Zbl 1372.1405110.1137/16M1086303Search in Google Scholar
[13] M. Dressler, A. Kurpisz, T. de Wolff, Optimization over the boolean hypercube via sums of nonnegative circuit polynomials. Found. Comput. Math. 22 (2022), 365–387. MR4407746 Zbl 1535.1413610.1007/s10208-021-09496-xSearch in Google Scholar
[14] M. Dressler, R. Murray, Algebraic perspectives on signomial optimization. SIAM J. Appl. Algebra Geom. 6 (2022), 650–684. MR4521654 Zbl 1524.1412510.1137/21M1462568Search in Google Scholar
[15] C. Fidalgo, A. Kovacec, Positive semidefinite diagonal minus tail forms are sums of squares. Math. Z. 269 (2011), 629–645. MR2860255 Zbl 1271.1104510.1007/s00209-010-0753-ySearch in Google Scholar
[16] J. Forsgård, T. de Wolff, The algebraic boundary of the sonc-cone. SIAM J. Appl. Algebra Geom. 6 (2022), 468–502. MR4477805 Zbl 1506.1410710.1137/20M1325484Search in Google Scholar
[17] G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities. Cambridge Univ. Press 1952. MR46395 Zbl 0047.05302Search in Google Scholar
[18] J. Hartzer, O. Röhrig, T. de Wolff, O. Yürük, Initial steps in the classification of maximal mediated sets. J. Symbolic Comput. 109 (2022), 404–425. MR4316048 Zbl 1475.5202310.1016/j.jsc.2020.07.013Search in Google Scholar
[19] D. Hilbert, Über die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann. 32 (1888), 342–350 JFM 20.0198.0210.1007/BF01443605Search in Google Scholar
[20] S. Iliman, T. de Wolff, Amoebas, nonnegative polynomials and sums of squares supported on circuits. Res. Math. Sci. 3 (2016), Paper No. 9, 35 pages. MR3481195 Zbl 1415.1107110.1186/s40687-016-0052-2Search in Google Scholar
[21] S. Iliman, T. de Wolff, Lower bounds for polynomials with simplex Newton polytopes based on geometric programming. SIAM J. Optim. 26 (2016), 1128–1146. MR3498517 Zbl 1380.1200110.1137/140962425Search in Google Scholar
[22] O. Karaca, G. Darivianakis, P. Beuchat, A. Georghiou, J. Lygeros, The REPOP toolbox: tackling polynomial optimization using relative entropy relaxations. IFAC-Pap. no. 50 (2017), 11652–11657.10.1016/j.ifacol.2017.08.1669Search in Google Scholar
[23] J. B. Lasserre, Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11 (2000/01), 796–817. MR1814045 Zbl 1010.9006110.1137/S1052623400366802Search in Google Scholar
[24] J. B. Lasserre, Moments, positive polynomials and their applications, volume 1 of Imperial College Press Optimization Series. Imperial College Press, London 2010. MR2589247 Zbl 1211.90007Search in Google Scholar
[25] M. Laurent, Sums of squares, moment matrices and optimization over polynomials. In: Emerging applications of algebraic geometry, volume 149 of IMA Vol. Math. Appl., 157–270, Springer 2009. MR2500468 Zbl 1163.1302110.1007/978-0-387-09686-5_7Search in Google Scholar
[26] V. Magron, J. Wang, SONC optimization and exact nonnegativity certificates via second-order cone programming. J. Symbolic Comput. 115 (2023), 346–370. MR4473041 Zbl 1500.9004510.1016/j.jsc.2022.08.002Search in Google Scholar
[27] M. Marshall, Positive polynomials and sums of squares, volume 146 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2008. MR2383959 Zbl 1169.1300110.1090/surv/146Search in Google Scholar
[28] T. Motzkin, The arithmetic-geometric inequalities. In: Inequalities, Proc. Sympos. Wright-Patterson Air Force Base, Ohio, 1965, 205–224. Academic Press 1967. MR0223521 Zbl 0178.00102Search in Google Scholar
[29] R. Murray, V. Chandrasekaran, A. Wierman, Newton polytopes and relative entropy optimization. Found. Comput. Math. 21 (2021), 1703–1737. MR4343021 Zbl 1483.9012110.1007/s10208-021-09497-wSearch in Google Scholar
[30] R. Murray, V. Chandrasekaran, A. Wierman, Signomial and polynomial optimization via relative entropy and partial dualization. Math. Program. Comput. 13 (2021), 257–295. MR4266925 Zbl 1530.9007110.1007/s12532-020-00193-4Search in Google Scholar
[31] K. G. Murty, S. N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming. Math. Programming 39 (1987), 117–129. MR916001 Zbl 0637.9007810.1007/BF02592948Search in Google Scholar
[32] C. Pantea, H. Koeppl, G. Craciun, Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks. Discrete Contin. Dyn. Syst. Ser. B 17 (2012), 2153–2170. MR2924455 Zbl 1253.8002310.3934/dcdsb.2012.17.2153Search in Google Scholar
[33] D. Papp, Duality of sum of nonnegative circuit polynomials and optimal SONC bounds. J. Symbolic Comput. 114 (2023), 246–266. MR4421048 Zbl 1491.9011610.1016/j.jsc.2022.04.015Search in Google Scholar
[34] A. R. Rajwade, Squares. Cambridge Univ. Press 1993. MR1253071 Zbl 0785.1102210.1017/CBO9780511566028Search in Google Scholar
[35] B. Reznick, Extremal PSD forms with few terms. Duke Math. J. 45 (1978), 363–374. MR480338 Zbl 0395.1003710.1215/S0012-7094-78-04519-2Search in Google Scholar
[36] B. Reznick, Forms derived from the arithmetic-geometric inequality. Math. Ann. 283 (1989), 431–464. MR985241 Zbl 0637.1001510.1007/BF01442738Search in Google Scholar
[37] B. Reznick, Sums of even powers of real linear forms. Mem. Amer. Math. Soc. 96 (1992), viii+155 pages. MR1096187 Zbl 0762.1101910.1090/memo/0463Search in Google Scholar
[38] B. Reznick, On Hilbert’s construction of positive polynomials. Preprint 2017, arXiv:0707.2156Search in Google Scholar
[39] B. Reznick, The odd powers of the Motzkin polynomial. In: Real Algebraic Geometry with a View toward Koopman Operator Methods, 38–40, Mathematisches Forschungsinstitut Oberwolfach 2023.Search in Google Scholar
[40] R. M. Robinson, Some definite polynomials which are not sums of squares of real polynomials. In: Selected questions of algebra and logic (a collection dedicated to the memory of A. I. Mal’cev) (Russian), 264–282, Izdat. “Nauka” Sibirsk. Otdel., Novosibirsk 1973. MR337878 Zbl 0277.10019Search in Google Scholar
[41] R. T. Rockafellar, Convex analysis. Princeton Univ. Press 1970. MR274683 Zbl 0193.1840110.1515/9781400873173Search in Google Scholar
[42] M. Schick, The Convex Cone of Sums of Squares plus Sums of Nonnegative Circuit Polynomials. PhD thesis, Universität Konstanz, 2024 (work in progress).Search in Google Scholar
[43] K. Schmüdgen, An example of a positive polynomial which is not a sum of squares of polynomials. A positive, but not strongly positive functional. Math. Nachr. 88 (1979), 385–390. MR543417 Zbl 0424.4604110.1002/mana.19790880130Search in Google Scholar
[44] J. Wang, Nonnegative polynomials and circuit polynomials. SIAM J. Appl. Algebra Geom. 6 (2022), 111–133. MR4405183 Zbl 1498.1414510.1137/20M1313969Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
Articles in the same Issue
- Frontmatter
- New rigidity results for critical metrics of some quadratic curvature functionals
- Extremizers of the Alexandrov–Fenchel inequality within a new class of convex bodies
- Perturbation theory of asymptotic operators of contact instantons and pseudoholomorphic curves on symplectization
- Cylinders in smooth del Pezzo surfaces of degree 2
- The total absolute curvature of closed curves with singularities
- On some relations between the perimeter, the area and the visual angle of a convex set
- Fundamental polyhedra of projective elementary groups
- Study of the cone of sums of squares plus sums of nonnegative circuit forms
