Journal of Systems Science and Complexity

, Volume 32, Issue 1, pp 158–184 | Cite as

Global Optimization of Polynomials over Real Algebraic Sets

  • Chu WangEmail author
  • Zhi-Hong Yang
  • Lihong Zhi


Let f, g1,..., gs be polynomials in R[X1,..., Xn]. Based on topological properties of generalized critical values, the authors propose a method to compute the global infimum f* of f over an arbitrary given real algebraic set V = {x ∈ Rn | g1(x) = 0,..., gs(x) = 0}, where V is not required to be compact or smooth. The authors also generalize this method to solve the problem of optimizing f over a basic closed semi-algebraic set S = {x ∈ Rn | g1(x) ≥ 0,..., gs(x) ≥ 0}.


Polynomial optimization real algebraic set generalized critical value 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We would like to thank Mohab Safey El Din for valuable comments and suggestions on the topic of the paper.


  1. [1]
    Kamyar R and Peet M, Polynomial optimization with applications to stability analysis and controlalternatives to sum of squares, Discrete and Continuous Dynamical Systems-Series B, 2015, 20(8): 2383–2417.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Ahmadi A A and Majumdar A, Some applications of polynomial optimization in operations research and real-time decision making, Optimization Letters, 2016, 10(4): 709–729.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Qi L and Teo K L, Multivariate polynomial minimization and its application in signal processing, Journal of Global Optimization, 2003, 26(4): 419–433.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Kahl F and Henrion D, Globally optimal estimates for geometric reconstruction problems, International Journal of Computer Vision, 2007, 74(1): 3–15.Google Scholar
  5. [5]
    Nesterov Y, Squared functional systems and optimization problems, High Performance Optimization, Springer, 2000, 405–440.zbMATHGoogle Scholar
  6. [6]
    Wu W T, On problems involving inequalities, Mathematics-Mechanization Research Preprints, 1992, 7: 103–138.Google Scholar
  7. [7]
    Wu W T, On zeros of algebraic equations — An application of Ritt principle, Chinese Science Bulletin, 1986, 31(1): 1–5.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Wu W T, A zero structure theorem for polynomial-equations-solving and its applications, Mathematics-Mechanization Research Preprints, 1987, 1: 2–12.Google Scholar
  9. [9]
    Wu W T, A zero structure theorem for polynomial-equations-solving and its applications, European Conference on Computer Algebra, 1987.Google Scholar
  10. [10]
    Wu W T, On a finiteness theorem about optimization problems, Mathematics-Mechanization Research Preprints, 1992, 81–18.Google Scholar
  11. [11]
    Wu W T, On a finite kernel theorem for polynomial-type optimization problems and some of its applications, Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation, ISSAC’05, (New York, USA), ACM, 2005, 4–4.Google Scholar
  12. [12]
    Morse A P, The behavior of a function on its critical set, Annals of Mathematics, 1939, 40(1): 62–70.MathSciNetzbMATHGoogle Scholar
  13. [13]
    Sard A, The measure of the critical values of differentiable maps, Bulletin of the American Mathematical Society, 1942, 48(1942): 883–890.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Sard A, Hausdorff measure of critical images on banach manifolds, American Journal of Mathematics, 1965, 87(1): 158–174.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Wu T, Some test problems on applications of Wu’s method in nonlinear programming problems, Chinese Quarterly Journal of Mathematics, 1994, 9(2): 8–17.zbMATHGoogle Scholar
  16. [16]
    Wu T, On a collision problem, Acta Mathematica Scientia, 1995, 15(Supp.): 32–38.MathSciNetGoogle Scholar
  17. [17]
    Yang L, Recent advances in automated theorem proving on inequalities, Journal of Computer Science & Technology, 1999, 14(5): 434–446.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Fang W, Wu T, and Chen J, An algorithm of global optimization for rational functions with rational constraints, Journal of Global Optimization, 2000, 18(3): 211–218.MathSciNetzbMATHGoogle Scholar
  19. [19]
    Yang L and Zhang J, A Practical Program of Automated Proving for a Class of Geometric Inequalities, Springer Berlin Heidelberg, 2001, 41–57.zbMATHGoogle Scholar
  20. [20]
    Xiao S J and Zeng G X, Algorithms for computing the global infimum and minimum of a polynomial function, Science China Mathematics, 2012, 55(4): 881–891.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Zeng G and Xiao S, Global minimization of multivariate polynomials using nonstandard methods, Journal of Global Optimization, 2012, 53(3): 391–415.MathSciNetzbMATHGoogle Scholar
  22. [22]
    Collins G E, Quantifier elimination for real closed fields by cylindrical algebraic decomposition, Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, May 20–23, 1975, Springer, 1975, 134–183.Google Scholar
  23. [23]
    Hong H, An improvement of the projection operator in cylindrical algebraic decomposition, Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC’90, (New York, USA), ACM, 1990, 261–264.Google Scholar
  24. [24]
    Collins G E and Hong H, Partial cylindrical algebraic decomposition for quantifier elimination, Journal of Symbolic Computation, 1991, 12(3): 299–328.MathSciNetzbMATHGoogle Scholar
  25. [25]
    Hong H, Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination, Proceedings of the International Symposium on Symbolic and Algebraic Computation, ISSAC’92, (New York, USA), ACM, 1992, 177–188.Google Scholar
  26. [26]
    McCallum S, An improved projection operation for cylindrical algebraic decomposition, Quantifier Elimination and Cylindrical Algebraic Decomposition, Eds. by Caviness B F and Johnson J R, Springer Vienna, 1998, 242–268.zbMATHGoogle Scholar
  27. [27]
    McCallum S, On projection in CAD-based quantifier elimination with equational constraint, Proceedings of the 1999 International Symposium on Symbolic and Algebraic Computation, ISSAC’99, (New York, USA), ACM, 1999, 145–149.Google Scholar
  28. [28]
    Brown C W, Improved projection for cylindrical algebraic decomposition, Journal of Symbolic Computation, 2001, 32(5): 447–465.MathSciNetzbMATHGoogle Scholar
  29. [29]
    Brown C W, QEPCAD B: A program for computing with semi-algebraic sets using CADs, SIGSAM Bull., 2003, 37(4): 97–108.zbMATHGoogle Scholar
  30. [30]
    Han J, Dai L, and Xia B, Constructing fewer open cells by GCD computation in CAD projection, Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation, ISSAC’14, (New York, USA), ACM, 2014, 240–247.Google Scholar
  31. [31]
    Han J, Jin Z, and Xia B, Proving inequalities and solving global optimization problems via simplified CAD projection, Journal of Symbolic Computation, 2016, 72: 206–230.MathSciNetzbMATHGoogle Scholar
  32. [32]
    Han J, Dai L, Hong H, et al., Open weak CAD and its applications, Journal of Symbolic Computation, 2017, 80: 785–816.MathSciNetzbMATHGoogle Scholar
  33. [33]
    Basu S, Pollack R, and Roy M F, Algorithms in Real Algebraic Geometry, Springer Berlin Heidelberg, Berlin, 2006.zbMATHGoogle Scholar
  34. [34]
    Hong H and Safey El Din M, Variant real quantifier elimination: Algorithm and application, Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, ISSAC’09, (New York, USA), ACM, 2009, 183–190.Google Scholar
  35. [35]
    Hong H and Safey El Din M, Variant quantifier elimination, Journal of Symbolic Computation, 2012, 47(7): 883–901, International Symposium on Symbolic and Algebraic Computation (ISSAC 2009).MathSciNetzbMATHGoogle Scholar
  36. [36]
    Safey El Din M, Computing the global optimum of a multivariate polynomial over the reals, Proceedings of the Twenty-First International Symposium on Symbolic and Algebraic Computation, ISSAC’08, (New York, NY, USA), ACM, 2008, 71–78.Google Scholar
  37. [37]
    Greuet A and Safey El Din M, Probabilistic algorithm for polynomial optimization over a real algebraic set, SIAM Journal on Optimization, 2014, 24(3): 1313–1343.MathSciNetzbMATHGoogle Scholar
  38. [38]
    Lasserre J B, Global optimization with polynomials and the problem of moments, SIAM Journal on Optimization, 2001, 11(3): 796–817.MathSciNetzbMATHGoogle Scholar
  39. [39]
    Nie J, Demmel J, and Sturmfels B, Minimizing polynomials via sum of squares over the gradient ideal, Mathematical Programming, 2006, 106(3): 587–606.MathSciNetzbMATHGoogle Scholar
  40. [40]
    Demmel J, Nie J, and Powers V, Representations of positive polynomials on noncompact semialgebraic sets via KKT ideals, Journal of Pure and Applied Algebra, 2007, 209(1): 189–200.MathSciNetzbMATHGoogle Scholar
  41. [41]
    Nie J, An exact Jacobian SDP relaxation for polynomial optimization, Mathematical Programming, 2013, 137(1–2): 225–255.MathSciNetzbMATHGoogle Scholar
  42. [42]
    Schweighofer M, Global optimization of polynomials using gradient tentacles and sums of squares, SIAM Journal on Optimization, 2006, 17(3): 920–942.MathSciNetzbMATHGoogle Scholar
  43. [43]
    Hà H V and Phạm T S, Solving polynomial optimization problems via the truncated tangency variety and sums of squares, J. Pure Appl. Algebra, 2009, 213(11): 2167–2176.MathSciNetzbMATHGoogle Scholar
  44. [44]
    Hà H V and Phạm T S, Representations of positive polynomials and optimization on noncompact semialgebraic sets, SIAM Journal on Optimization, 2010, 20(6): 3082–3103.MathSciNetzbMATHGoogle Scholar
  45. [45]
    Guo F, Safey El Din M, and Zhi L, Global optimization of polynomials using generalized critical values and sums of squares, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ACM, 2010, 107–114.zbMATHGoogle Scholar
  46. [46]
    Greuet A, Guo F, Safey El Din M, et al., Global optimization of polynomials restricted to a smooth variety using sums of squares, Journal of Symbolic Computation, 2012, 47(5): 503–518.MathSciNetzbMATHGoogle Scholar
  47. [47]
    Rabier P J, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Annals of Mathematics, 1997, 647–691.zbMATHGoogle Scholar
  48. [48]
    Jelonek Z and Kurdyka K, Quantitative generalized Bertini-Sard theorem for smooth affine varieties, Discrete and Computational Geometry, 2005, 34(4): 659–678.MathSciNetzbMATHGoogle Scholar
  49. [49]
    Bochnak J, Coste M, and Roy M F, Real Algebraic Geometry, 36, Springer Science and Business Media, 1998.zbMATHGoogle Scholar
  50. [50]
    Greuel G M and Pfister G, A Singular Introduction to Commutative Algebra, Springer Science and Business Media, 2012.zbMATHGoogle Scholar
  51. [51]
    Eisenbud D, Commutative Algebra: With a View Toward Algebraic Geometry, 150, Springer New York, 1995.zbMATHGoogle Scholar
  52. [52]
    Yang Z H, Computation of real radicals and global optimization of polynomials, PhD thesis, University of Chinese Academy of Sciences, 2018.Google Scholar
  53. [53]
    Valette A and Valette G, A generalized Sard theorem on real closed fields, Mathematische Nachrichten, 2016, 289(5–6): 748–755.MathSciNetzbMATHGoogle Scholar
  54. [54]
    Kurdyka K, Orro P, and Simon S, Semialgebraic Sard theorem for generalized critical values, Journal of Differential Geometry, 2000, 56(1): 67–92.MathSciNetzbMATHGoogle Scholar
  55. [55]
    Jelonek Z, On the generalized critical values of a polynomial mapping, Manuscripta Mathematica, 2003, 110(2): 145–157.MathSciNetzbMATHGoogle Scholar
  56. [56]
    Jelonek Z, On asymptotic critical values and the Rabier theorem, Banach Center Publications, 2004, 1(65): 125–133.MathSciNetzbMATHGoogle Scholar
  57. [57]
    Cohen R L, The Topology of Fiber Bundles Lecture Notes, Standford University, San Francisco, 1998.Google Scholar
  58. [58]
    Cox D, Little J, and O’Shea D, Ideals, Varieties, and Algorithms, Springer, New York, 2007.zbMATHGoogle Scholar
  59. [59]
    Neuhaus R, Computation of real radicals of polynomial ideals II, Journal of Pure and Applied Algebra, 1998, 124(1): 261–280.MathSciNetzbMATHGoogle Scholar
  60. [60]
    Becker E and Neuhaus R, Computation of real radicals of polynomial ideals, Computational Algebraic Geometry, Springer, 1993, 1–20.zbMATHGoogle Scholar
  61. [61]
    Spang S J, On the Computation of the Real Radical, PhD thesis, Thesis, Technische Universität Kaiserslautern, 2007.Google Scholar
  62. [62]
    Spang S J, A zero-dimensional approach to compute real radicals, The Computer Science Journal of Moldova, 2008, 16(1): 64–92.MathSciNetzbMATHGoogle Scholar
  63. [63]
    Safey El Din M, Yang Z H, and Zhi L, On the complexity of computing real radicals of polynomial systems, Proceedings of the 2018 International Symposium on Symbolic and Algebraic Computation, ISSAC’18, (New York, USA), ACM, 2018, 351–358.Google Scholar
  64. [64]
    Aubry P, Rouillier F, and Safey El Din M, Real solving for positive dimensional systems, Journal of Symbolic Computation, 2002, 34(6): 543–560.MathSciNetzbMATHGoogle Scholar
  65. [65]
    Safey El Din M and Schost É, Properness defects of projections and computation of at least one point in each connected component of a real algebraic set, Discrete & Computational Geometry, 2004, 32(3): 417–430.MathSciNetzbMATHGoogle Scholar
  66. [66]
    Safey El Din M, Finding sampling points on real hypersurfaces is easier in singular situations, MEGA (Effective Methods in Algebraic Geometry) Electronic Proceedings, 2005.Google Scholar
  67. [67]
    Safey El Din M, Testing sign conditions on a multivariate polynomial and applications, Mathematics in Computer Science, 2007, 1(1): 177–207.MathSciNetzbMATHGoogle Scholar
  68. [68]
    Krick T and Logar A, An algorithm for the computation of the radical of an ideal in the ring of polynomials, International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, Springer, 1991, 195–205.Google Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Beijing Jinghang Computation and Communication Research InstituteBeijingChina
  2. 2.KLMM, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina
  3. 3.University of Chinese Academy of SciencesBeijingChina

Personalised recommendations