Mathematical Programming

, Volume 153, Issue 2, pp 333–362 | Cite as

New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors

Full Length Paper Series A


In this paper we derive new fractional error bounds for polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a general polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed techniques are largely based on variational analysis and generalized differentiation, which allow us to establish, e.g., a nonsmooth extension of the seminal Łojasiewicz’s gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Hölderian stability of solution maps for parameterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.


Error bounds Polynomials Variational analysis   Generalized differentiation Łojasiewicz’s inequality  Hölderian stability Polynomial optimization and complementarity 

Mathematics Subject Classification

90C26 90C31 49J52 49J53 26D10 



The authors are gratefully indebted to the referees and the handling Associate Editor for their helpful remarks, which allowed us to significantly improved the original presentation.


  1. 1.
    Arutyunov, A.V., Izmailov, A.F.: Directional stability theorem and directional metric regularity. Math. Oper. Res. 31, 526–543 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bauschke, H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program. 86, 135–160 (1999)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bolte, J., Daniilidis, A., Lewis, A.S.: The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17, 1205–1223 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bolte, J., Daniilidis, A., Lewis, A.S.: Tame functions are semismooth. Math. Program. 117, 5–19 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Borwein, J.M., Li, G., Yao, L.: Analysis of the convergence rate for the cyclic projection algorithm applied to basic semi-algebraic convex sets. SIAM J. Optim. 24, 498–527 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Burke, J.V., Deng, S.: Weak sharp minima revisited. II: application to linear regularity and error bounds. Math. Program. 104, 235–261 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Burke, J.V., Deng, S.: Weak sharp minima revisited, III: Error bounds for differentiable convex inclusions. Math. Program. 116, 37–56 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    D’Acunto, D., Kurdyka, K.: Explicit bounds for the Łojasiewicz exponent in the gradient inequality for polynomials. Ann. Polon. Math. 87, 51–61 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Danskin, J.: The theory of max-min, with applications. SIAM J. Appl. Math. 14, 641–664 (1966)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Deng, S.: Pertubation analysis of a condition number for convex inequality systems and global error bounds for analytic systems. Math. Program. 83, 263–276 (1998)CrossRefMATHGoogle Scholar
  11. 11.
    Dinh, S.T., Hà, H.V., Thao, N.T.: Łojasiewicz inequality for polynomial functions on non compact domains. Int. J. Math. 23, 125–153 (2012)Google Scholar
  12. 12.
    Dinh, S.T., Hà, H.V., Phạm, T.S., Thao, N.T.: Global Łojasiewicz-type inequality for non-degenerate polynomial maps. J. Math. Anal. Appl. 410, 541–560 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fabian, M.J., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set-Valued Var. Anal. 18, 121–149 (2010)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gwoździewicz, J.: The Łojasiewicz exponent of an analytic function at an isolated zero. Comment. Math. Helv. 74, 364–375 (1999)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hà, H.V.: Global Hölderian error bound for non-degenerate polynomials. SIAM J. Optim. 23, 917–933 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Henrion, R., Mordukhovich, B.S., Nam, N.M.: Second-order analysis of polyhedral systems in finite and infinite dimensions with applications to robust stability of variational inequalities. SIAM J. Optim. 20, 2199–2227 (2010)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hoffman, A.J.: On approximare solution to systems of linear inequalities. J. Nat. Bur. Stand. 49, 253–265 (1952)CrossRefGoogle Scholar
  19. 19.
    Klatte, D.: Hoffman’s error bound for systems of convex inequalities. In: Mathematical Programming with Data Perturbations, Lecture Notes in Pure and Applied Mathematics, vol. 195, pp. 185–199. Marcel Dekker, New York (1998)Google Scholar
  20. 20.
    Klatte, D., Li, W.: Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Program. 84, 137–160 (1999)MathSciNetMATHGoogle Scholar
  21. 21.
    Kollár, J.: An effective Lojasiewicz inequality for real polynomials. Periodica Mathematica Hungarica 38, 213–221 (1999)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kruger, A.Y., Ngai, H.V., Théra, M.: Stability of error bounds for convex constraint systems in Banach spaces. SIAM J. Optim. 20, 3280–3296 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kurdyka, K., Spodzieja, S.: Separation of real algebraic sets and the Łojasiewicz exponent. Proc. Am. Math. Soc. 142, 3089–3102 (2014)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Full stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lewis, A.S., Pang, J.S.: Error bounds for convex inequality systems. In: Crouzeix, J.P., Martinez-Legaz, J.E., Volle, M. (eds.) Generalized Convexity and Generalized Monotonicity: Recent Results, pp. 75–110. Kluwer, Dordrecht (1998)Google Scholar
  26. 26.
    Li, C., Mordukhovich, B.S., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21, 1523–1560 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Li, G.: On the asymptotic well behaved functions and global error bound for convex polynomials. SIAM J. Optim. 20, 1923–1943 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Li, G.: Global error bounds for piecewise convex polynomials. Math. Program. 137, 37–64 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Li, G., Mordukhovich, B.S.: Hölder metric subregularity with applications to proximal point method. SIAM J. Optim. 22, 1655–1684 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Li, G., Ng, K.F.: Error bounds of generalized D-gap functions for nonsmooth and nonmonotone variational inequality problems. SIAM J. Optim. 20, 667–690 (2009)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Li, G., Qi, L., Yu, G.: Semismoothness of the maximum eigenvalue function of a symmetric tensor and its application. Linear Algebra Appl. 438, 813–833 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Li, W.: Error bounds for piecewise convex quadratic programs and applications. SIAM J. Control Optim. 33, 1510–1529 (1995)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Łojasiewicz, M.S.: Sur la probléme de la division. Studia Math. 18, 87–136 (1959)MathSciNetMATHGoogle Scholar
  34. 34.
    Łojasiewicz, M.S.: Ensembles semi-analytiques. Publ. Math. I.H.E.S, Bures-sur-Yvette (1965)Google Scholar
  35. 35.
    Luo, X.D., Luo, Z.Q.: Extension of Hoffman’s error bound to polynomial systems. SIAM J. Optim. 4, 383–392 (1994)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Luo, Z.Q., Pang, J.S.: Error bounds for analytic systems and their applications. Math. Program. 67, 1–28 (1994)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  38. 38.
    Luo, Z.Q., Sturm, J.F.: Error bounds for quadratic systems. High performance optimization. Appl. Optim. 33, 383–404 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Mordukhovich, B.S.: Complete characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–35 (1993)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)Google Scholar
  42. 42.
    Mordukhovich, B.S., Rockafellar, R.T.: Second-order subdifferential calculus with applications to tilt stability in optimization. SIAM J. Optim. 22, 953–986 (2012)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Ng, K.F., Zheng, X.Y.: Global error bounds with fractional exponents. Math. Program. 88, 357–370 (2000)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Ng, K.F., Zheng, X.Y.: Error bounds of constrained quadratic functions and piecewise affine inequality systems. J. Optim. Theory Appl. 118, 584–607 (2003)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Ngai, H.V., Théra, M.: Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math. Program. 116, 397–427 (2009)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Ni, Q., Qi, L., Wang, F.: An eigenvalue method for the positive definiteness identification problem. IEEE Trans. Autom. Control 53, 1096–1107 (2008)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)MATHGoogle Scholar
  49. 49.
    Phạm, T.S.: An explicit bound for the Łojasiewicz exponent of real polynomials. Kodai Math. J. 35, 311–319 (2012)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Poliqiun, R.A., Rockafellar, R.T.: Tilt stability of a local minimim. SIAM J. Optim. 8, 287–299 (1998)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Qi, L.: Eigenvalues of a real symmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)CrossRefMATHGoogle Scholar
  52. 52.
    Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 78, 353–368 (1993)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Qi, L., Yu, G., Wu, E.X.: Higher order positive semi-definite diffusion tensor imaging. SIAM J. Imaging Sci. 3, 416–433 (2010)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Robinson, S.M.: An application of error bounds for convex programming in a linear space. SIAM J. Control 13, 271–273 (1975)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Robinson, S.M.: Generalized equations and their solutions, I: basic theory. Math. Program. Study 10, 128–141 (1979)CrossRefMATHGoogle Scholar
  56. 56.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  57. 57.
    Sun, D., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150–169 (2002)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Wu, Z.L., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program. 92, 301–314 (2002)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Yang, W.H.: Error bounds for convex polynomials. SIAM J. Optim. 19, 1633–1647 (2008)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of New South WalesSydneyAustralia
  2. 2.Department of MathematicsWayne State UniversityDetroitUSA
  3. 3.Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
  4. 4.Center of Research and DevelopmentDuy Tan UniversityDanangVietnam
  5. 5.Department of MathematicsUniversity of DalatDalatVietnam

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