Skip to main content
SpringerLink
Log in

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

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arutyunov, A.V., Izmailov, A.F.: Directional stability theorem and directional metric regularity. Math. Oper. Res. 31, 526–543 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bolte, J., Daniilidis, A., Lewis, A.S.: Tame functions are semismooth. Math. Program. 117, 5–19 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burke, J.V., Deng, S.: Weak sharp minima revisited. II: application to linear regularity and error bounds. Math. Program. 104, 235–261 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burke, J.V., Deng, S.: Weak sharp minima revisited, III: Error bounds for differentiable convex inclusions. Math. Program. 116, 37–56 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Danskin, J.: The theory of max-min, with applications. SIAM J. Appl. Math. 14, 641–664 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ferris, M.C., Pang, J.S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39, 669–713 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gwoździewicz, J.: The Łojasiewicz exponent of an analytic function at an isolated zero. Comment. Math. Helv. 74, 364–375 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hà, H.V.: Global Hölderian error bound for non-degenerate polynomials. SIAM J. Optim. 23, 917–933 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hoffman, A.J.: On approximare solution to systems of linear inequalities. J. Nat. Bur. Stand. 49, 253–265 (1952)

    Article  Google Scholar 

  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)

  20. Klatte, D., Li, W.: Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Program. 84, 137–160 (1999)

    MathSciNet  MATH  Google Scholar 

  21. Kollár, J.: An effective Lojasiewicz inequality for real polynomials. Periodica Mathematica Hungarica 38, 213–221 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kurdyka, K., Spodzieja, S.: Separation of real algebraic sets and the Łojasiewicz exponent. Proc. Am. Math. Soc. 142, 3089–3102 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Levy, A.B., Poliquin, R.A., Rockafellar, R.T.: Full stability of locally optimal solutions. SIAM J. Optim. 10, 580–604 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  26. Li, C., Mordukhovich, B.S., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21, 1523–1560 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, G.: On the asymptotic well behaved functions and global error bound for convex polynomials. SIAM J. Optim. 20, 1923–1943 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, G.: Global error bounds for piecewise convex polynomials. Math. Program. 137, 37–64 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, G., Mordukhovich, B.S.: Hölder metric subregularity with applications to proximal point method. SIAM J. Optim. 22, 1655–1684 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, W.: Error bounds for piecewise convex quadratic programs and applications. SIAM J. Control Optim. 33, 1510–1529 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Łojasiewicz, M.S.: Sur la probléme de la division. Studia Math. 18, 87–136 (1959)

    MathSciNet  MATH  Google Scholar 

  34. Łojasiewicz, M.S.: Ensembles semi-analytiques. Publ. Math. I.H.E.S, Bures-sur-Yvette (1965)

  35. Luo, X.D., Luo, Z.Q.: Extension of Hoffman’s error bound to polynomial systems. SIAM J. Optim. 4, 383–392 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Luo, Z.Q., Pang, J.S.: Error bounds for analytic systems and their applications. Math. Program. 67, 1–28 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  37. Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  38. Luo, Z.Q., Sturm, J.F.: Error bounds for quadratic systems. High performance optimization. Appl. Optim. 33, 383–404 (2000)

    Article  MathSciNet  Google Scholar 

  39. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  40. Mordukhovich, B.S.: Complete characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions. Trans. Am. Math. Soc. 340, 1–35 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mordukhovich, B.S., Rockafellar, R.T., Sarabi, M.E.: Characterizations of full stability in constrained optimization. SIAM J. Optim. 23, 1810–1849 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  44. Ng, K.F., Zheng, X.Y.: Global error bounds with fractional exponents. Math. Program. 88, 357–370 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  46. Ngai, H.V., Théra, M.: Error bounds for systems of lower semicontinuous functions in Asplund spaces. Math. Program. 116, 397–427 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ni, Q., Qi, L., Wang, F.: An eigenvalue method for the positive definiteness identification problem. IEEE Trans. Autom. Control 53, 1096–1107 (2008)

    Article  MathSciNet  Google Scholar 

  48. Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)

    MATH  Google Scholar 

  49. Phạm, T.S.: An explicit bound for the Łojasiewicz exponent of real polynomials. Kodai Math. J. 35, 311–319 (2012)

    Article  MathSciNet  Google Scholar 

  50. Poliqiun, R.A., Rockafellar, R.T.: Tilt stability of a local minimim. SIAM J. Optim. 8, 287–299 (1998)

    Article  MathSciNet  Google Scholar 

  51. Qi, L.: Eigenvalues of a real symmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MATH  Google Scholar 

  52. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 78, 353–368 (1993)

    Article  MathSciNet  Google Scholar 

  53. Qi, L., Yu, G., Wu, E.X.: Higher order positive semi-definite diffusion tensor imaging. SIAM J. Imaging Sci. 3, 416–433 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  54. Robinson, S.M.: An application of error bounds for convex programming in a linear space. SIAM J. Control 13, 271–273 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  55. Robinson, S.M.: Generalized equations and their solutions, I: basic theory. Math. Program. Study 10, 128–141 (1979)

    Article  MATH  Google Scholar 

  56. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  57. Sun, D., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27, 150–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  58. Wu, Z.L., Ye, J.J.: On error bounds for lower semicontinuous functions. Math. Program. 92, 301–314 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  59. Yang, W.H.: Error bounds for convex polynomials. SIAM J. Optim. 19, 1633–1647 (2008)

    Article  MathSciNet  Google Scholar 

  60. Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, University of New South Wales, Sydney, 2052, Australia

    G. Li

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    B. S. Mordukhovich

  3. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia

    B. S. Mordukhovich

  4. Center of Research and Development, Duy Tan University, K7/25, Danang, Quang Trung, Vietnam

    T. S. Phạm

  5. Department of Mathematics, University of Dalat, 1, Phu Dong Thien Vuong, Dalat, Vietnam

    T. S. Phạm

Authors
  1. G. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. S. Mordukhovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. S. Phạm
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. S. Mordukhovich.

Additional information

This research was partially supported by the Australian Research Council under Grant DP-12092508. Research of G. Li was also partially supported by the Australian Research Council Future Fellowship FT130100038. Research of B. S. Mordukhovich was also partially supported by the USA National Science Foundation under Grant DMS-1007132 and by the Portuguese Foundation of Science and Technologies under Grant MAT/11109. Research of T. S. Phạm was also partially supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 101.04–2013.07.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, G., Mordukhovich, B.S. & Phạm, T.S. New fractional error bounds for polynomial systems with applications to Hölderian stability in optimization and spectral theory of tensors. Math. Program. 153, 333–362 (2015). https://doi.org/10.1007/s10107-014-0806-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-014-0806-9

Keywords

Mathematics Subject Classification

Search

Navigation