Abstract
We study the representation of nonnegative polynomials in two variables on a certain class of unbounded closed basic semi-algebraic sets (which are called generalized strips). This class includes the strip \([a,b] \times {\mathbb {R}}\) which was studied by Marshall in (Proc Am Math Soc 138(5):1559–1567, 2010). A denominator-free Nichtnegativstellensätz holds true on a generalized strip when the width of the generalized strip is constant and fails otherwise. As a consequence, we confirm that the standard hierarchy of semidefinite programming relaxations defined for the compact case can indeed be adapted to the generalized strip with constant width. For polynomial optimization problems on the generalized strip with non-constant width, we follow Ha-Pham’s work: Solving polynomial optimization problems via the truncated tangency variety and sums of squares.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. Emerging Applications of Algebraic Geometry, vol. 149 of IMA Volumes in Mathematics and its Applications, M. Putinar and S. Sullivant (eds.), Springer, 157–270 (2009)
Scheiderer, C.: Positivity and sums of squares: a guide to recent results. Emerging applications of algebraic geometry, IMA Vol. Math. Appl., Springer, New York, 149, 271–324 (2009)
Lasserre, J.B.: Moments. Positive Polynomials and their Applications. Imperial College Press, London (2009)
Hà, H.V., Phạm, T.S.: Genericity in polynomial optimization, vol. 3 of Series on Optimization and Its Applications, World Scientific (2017)
Blekherman, G., Smith, G.G., Velasco, M.: Sums of squares and varieties of minimal degree. J. Am. Math. Soc. 29, 893–913 (2016)
Schmüdgen, K.: The K-moment problem for compact semi-algebraic sets. Math. Ann. 289, 203–206 (1991)
Hà, H.V., Ho, T.M.: Positive polynomials on nondegenerate basic semi-algebraic sets. Adv. Geom. 16(4), 497–510 (2016)
Marshall, M.: Polynomials nonnegative on a strip. Proc. Am. Math. Soc. 138(5), 1559–1567 (2010)
Nguyen, H., Powers, V.: Polynomials nonnegative on strips and half-strips. J. Pure Appl. Algebra 216(10), 2225–2232 (2012)
Powers, V.: Positive polynomials and the moment problem for cylinders with compact cross-section. J. Pure Appl. Algebra 188(1–3), 217–226 (2004)
Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42(3), 969–984 (1993)
Scheiderer, C.: Sums of squares on real algebraic surfaces. Manuscr. Math. 119(4), 395–410 (2006)
Scheiderer, C.: Sums of squares of regular functions on real algebraic varieties. Trans. Am. Math. Soc. 352(3), 1039–1069 (2003)
Scheiderer, C.: Sums of squares on real algebraic curves. Math. Z. 245, 725–760 (2003)
Scheiderer, C., Wenzel, S.: Polynomials nonnegative on the cylinder. In: Ordered Algebraic Structures and Related Topics, F. Broglia et al (eds.), Contemp. Math. 697, AMS, Providence, RI, 291–300 (2017)
Michalska, M.: Curves testing boundedness of polynomials on subsets of the real plane. J. Symb. Comput. 56, 107–124 (2013)
Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Springer-Verlag, Berlin (1998)
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Algorithms Comput. Math., vol. 10. Springer-Verlag, Berlin (2003)
Hà, H.V., Phạm, T.S.: Solving polynomial optimization problems via the truncated tangency variety and sums of squares. J. Pure Appl. Algebra 213, 2167–2176 (2009)
Hà, H.V., Phạm, T.S.: Global optimization of polynomials using the truncated tangency variety and sums of squares. SIAM J. Optim. 19, 941–951 (2008)
Michalska, M.: Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated. Bull. Sci. Math. 137, 705–715 (2013)
Marshall, M.: Positive polynomials and sum of squares. Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 146 (2008)
Plaumann, D.: Sums of squares on reducible real curves. Math. Z. 265(4), 777–797 (2010)
Schweighofer, M.: Global optimization of polynomials using gradient tentacles and sums of squares. SIAM J. Optim. 17(3), 920–942 (2006)
Trang, D.T.T., Ngoc, N.A.: Algebra of polynomials bounded on some strips. Transp. Commun. Sci. J. 73(1), 90–99 (2022)
Papachristodoulou, A., Anderson, J., Valmorbida, G., Prajna, S., Seiler, P., Parrilo, P.A.: SOSTOOLS, Sum of Squares Optimization Toolbox for MATLAB, User’s guide., version 3.01, http://www.mit.edu/parrilo/sostools/
Acknowledgements
We are grateful to the referees for careful reading and corrections of the manuscript. This work of the first and the second authors is supported by Grant number ICRTM.02_2021.01 of International Center for Research and Postgraduate Training in Mathematics, HIM, VAST.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Du, T.T.T., Ho, T.M. & Hoang, PD. Representation of positive polynomials on a generalized strip and its application to polynomial optimization. Optim Lett (2024). https://doi.org/10.1007/s11590-023-02087-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11590-023-02087-5