Non-SOS Positivstellensätze for Semialgebraic Sets Defined by Polynomial Matrix Inequalities

Published Online:https://doi.org/10.1287/moor.2025.1239

References

  • [1] Ahmadi AA, Majumdar A (2019) DSOS and SDSOS optimization: More tractable alternatives to sum of squares and semidefinite optimization. SIAM J. Appl. Algebra Geometry 3(2):193–230.CrossrefGoogle Scholar
  • [2] Andersen ED, Andersen KD (2000) The Mosek interior point optimizer for linear programming: An implementation of the homogeneous algorithm. Frenk H, Roos K, Terlaky T, Zhang S, eds. High Performance Optimization, Applied Optimization, vol. 33 (Springer US, Boston), 197–232.CrossrefGoogle Scholar
  • [3] Barvinok A (2002) A Course in Convexity (American Mathematical Society, Providence, RI).CrossrefGoogle Scholar
  • [4] Basu S, Pollack R, Roy MF (2006) Algorithms in Real Algebraic Geometry (Springer, Berlin).CrossrefGoogle Scholar
  • [5] Blekherman G, Parrilo PA, Thomas RR (2012) Semidefinite Optimization and Convex Algebraic Geometry (Society for Industrial and Applied Mathematics, Philadelphia).CrossrefGoogle Scholar
  • [6] Bochnak J, Coste M, Roy MF (1998) Real Algebraic Geometry (Springer, Berlin).CrossrefGoogle Scholar
  • [7] Chandrasekaran V, Shah P (2016) Relative entropy relaxations for signomial optimization. SIAM J. Optim. 26(2):1147–1173.CrossrefGoogle Scholar
  • [8] Dias da Silva JA, Machado A (2013) Multilinear algebra. Hogben L, ed. Handbook of Linear Algebra, 2nd ed. (Chapman and Hall/CRC, New York), 14-1.Google Scholar
  • [9] Dickinson PJC, Povh J (2015) On an extension of Pólya’s Positivstellensatz. J. Global Optim. 61(4):615–625.CrossrefGoogle Scholar
  • [10] Dinh TH, Ho MT, Le CT (2021) Positivstellensätze for polynomial matrices. Positivity 25(4):1295–1312.CrossrefGoogle Scholar
  • [11] Dressler M, Iliman S, de Wolff T (2017) A Positivstellensatz for sums of nonnegative circuit polynomials. SIAM J. Appl. Algebra Geometry 1(1):536–555.CrossrefGoogle Scholar
  • [12] Handelman D (1988) Representing polynomials by positive linear functions on compact convex polyhedra. Pacific J. Math. 132(1):35–62.CrossrefGoogle Scholar
  • [13] Henrion D, Lasserre JB (2006) Convergent relaxations of polynomial matrix inequalities and static output feedback. IEEE Trans. Automatic Control 51(2):192–202.CrossrefGoogle Scholar
  • [14] Henrion D, Lasserre JB (2012) Inner approximations for polynomial matrix inequalities and robust stability regions. IEEE Trans. Automatic Control 57(6):1456–1467.CrossrefGoogle Scholar
  • [15] Henrion D, Korda M, Lasserre JB (2020) The Moment-SOS Hierarchy (World Scientific (Europe), Singapore).CrossrefGoogle Scholar
  • [16] Huang L (2025) On the complexity of matrix Putinar’s Positivstellensätz. SIAM J. Optim. 35(1):567–591.CrossrefGoogle Scholar
  • [17] Huang L, Nie J (2025) Finite convergence of the moment-SOS hierarchy for polynomial matrix optimization. Math. Programming 214:685–722.CrossrefGoogle Scholar
  • [18] Huang L, Nie J, Yuan Y (2023) Homogenization for polynomial optimization with unbounded sets. Math. Programming 200(1):105–145.CrossrefGoogle Scholar
  • [19] Ichihara H (2009) Optimal control for polynomial systems using matrix sum of squares relaxations. IEEE Trans. Automatic Control 54(5):1048–1053.CrossrefGoogle Scholar
  • [20] Klep I, Nie J (2020) A matrix Positivstellensatz with lifting polynomials. SIAM J. Optim. 30(1):240–261.CrossrefGoogle Scholar
  • [21] Kojima M (2003) Sums of squares relaxations of polynomial semidefinite programs. Research Report No. B-397, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo.Google Scholar
  • [22] Kojima M, Muramatsu M (2009) A note on sparse SOS and SDP relaxations for polynomial optimization problems over symmetric cones. Comput. Optim. Appl. 42(1):31–41.CrossrefGoogle Scholar
  • [23] Krivine JL (1964) Anneaux préordonnés. J. d’Analyse Mathématique 12(1):307–326.CrossrefGoogle Scholar
  • [24] Kuryatnikova O, Vera JC, Zuluaga LF (2024) Reducing nonnegativity over general semialgebraic sets to nonnegativity over simple sets. SIAM J. Optim. 34(2):1970–2006.CrossrefGoogle Scholar
  • [25] Lasserre JB (2001) Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3):796–817.CrossrefGoogle Scholar
  • [26] Lasserre JB (2002) Semidefinite programming vs. LP relaxations for polynomial programming. Math. Oper. Res. 27(2):347–360.LinkGoogle Scholar
  • [27] Lasserre JB (2005) Polynomial programming: LP-relaxations also converge. SIAM J. Optim. 15(2):383–393.CrossrefGoogle Scholar
  • [28] Lasserre JB (2005) A unified criterion for positive definiteness and semidefiniteness. Research Report No. 05-283, LAAS-CNRS, Toulouse, France.Google Scholar
  • [29] Lasserre JB (2009) Moments, Positive Polynomials and Their Applications (Imperial College Press, London).CrossrefGoogle Scholar
  • [30] Lasserre JB (2015) An Introduction to Polynomial and Semi-Algebraic Optimization, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, UK).CrossrefGoogle Scholar
  • [31] Lasserre JB, Toh KC, Yang S (2017) A bounded degree SOS hierarchy for polynomial optimization. EURO J. Comput. Optim. 5(1):87–117.CrossrefGoogle Scholar
  • [32] Laurent M (2009) Sums of squares, moment matrices and optimization over polynomials. Putinar M, Sullivant S, eds. Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and Its Applications, vol. 149 (Springer, New York), 157–270.CrossrefGoogle Scholar
  • [33] Lê CT, Dư THB (2018) Handelman’s Positivstellensatz for polynomial matrices positive definite on polyhedra. Positivity 22(2):449–460.CrossrefGoogle Scholar
  • [34] Löfberg J (2004) YALMIP: A toolbox for modeling and optimization in MATLAB. 2004 IEEE Internat. Sympos. Comput. Aided Control System Design (IEEE, Taipei, Taiwan), 284–289.Google Scholar
  • [35] Mai NHA, Lasserre JB, Magron V (2022) Positivity certificates and polynomial optimization on non-compact semialgebraic sets. Math. Programming 194(1):443–485.CrossrefGoogle Scholar
  • [36] Marshall M (2008) Positive Polynomials and Sums of Squares, Mathematical Surveys and Monographs, vol. 146 (American Mathematical Society, Providence, RI).CrossrefGoogle Scholar
  • [37] Nie J (2023) Moment and Polynomial Optimization (Society for Industrial and Applied Mathematics, Philadelphia).CrossrefGoogle Scholar
  • [38] Pólya G (1928) Über positive darstellung von polynomen. Vierteljahrsschrift Naturforschenden Gesellschaft Zürich 73:141–145.Google Scholar
  • [39] Pozdyayev VV (2014) Atomic optimization. II. Multidimensional problems and polynomial matrix inequalities. Automation Remote Control 75(6):1155–1171.CrossrefGoogle Scholar
  • [40] Putinar M (1993) Positive polynomials on compact semialgebraic sets. Indiana Univ. Math. J. 42(3):969–984.CrossrefGoogle Scholar
  • [41] Putinar M, Vasilescu FH (1999) Solving moment problems by dimensional extension. Comptes Rendus l’Académie Sci. Ser. I Math. 328(6):495–499.Google Scholar
  • [42] Ramana M, Goldman AJ (1995) Some geometric results in semidefinite programming. J. Global Optim. 7(1):33–50.CrossrefGoogle Scholar
  • [43] Reznick B (1995) Uniform denominators in Hilbert’s seventeenth problem. Mathematische Zeitschrift 220(1):75–97.CrossrefGoogle Scholar
  • [44] Roebers LM, Vera JC, Zuluaga LF (2021) Sparse non-SOS Putinar-type Positivstellensätze. Preprint October 12, https://arxiv.org/abs/2110.10079.Google Scholar
  • [45] Scheiderer C (2009) Positivity and sums of squares: A guide to recent results. Putinar M, Sullivant S, eds. Emerging Applications of Algebraic Geometry, IMA Volumes in Mathematics and Its Applications, vol. 149 (Springer, New York), 1–54.CrossrefGoogle Scholar
  • [46] Scherer CW, Hol CWJ (2006) Matrix sum-of-squares relaxations for robust semi-definite programs. Math. Programming 107(1):189–211.CrossrefGoogle Scholar
  • [47] Schmüdgen K (1991) The K-moment problem for compact semi-algebraic sets. Mathematische Annalen 289(1):203–206.CrossrefGoogle Scholar
  • [48] Stengle G (1974) A nullstellensatz and a positivstellensatz in semialgebraic geometry. Mathematische Annalen 207(2):87–97.CrossrefGoogle Scholar
  • [49] VanAntwerp JG, Braatz RD (2000) A tutorial on linear and bilinear matrix inequalities. J. Process Control 10(4):363–385.CrossrefGoogle Scholar
  • [50] Vasilescu FH (2003) Spectral measures and moment problems. Gheondea A, Şabac M, eds. Spectral Analysis and Its Applications (Theta Foundation, Bucharest, Romania), 173–215.Google Scholar
  • [51] Weisser T, Lasserre JB, Toh KC (2018) Sparse-BSOS: A bounded degree SOS hierarchy for large scale polynomial optimization with sparsity. Math. Programming Comput. 10(1):1–32.CrossrefGoogle Scholar
  • [52] Zheng Y (2019) Chordal sparsity in control and optimization of large-scale systems. PhD thesis, University of Oxford, Oxford, UK.Google Scholar
  • [53] Zheng Y, Fantuzzi G (2023) Sum-of-squares chordal decomposition of polynomial matrix inequalities. Math. Programming 197(1):71–108.CrossrefGoogle Scholar
INFORMS site uses cookies to store information on your computer. Some are essential to make our site work; Others help us improve the user experience. By using this site, you consent to the placement of these cookies. Please read our Privacy Statement to learn more.