Error Bounds for Quadratic Systems

  • Zhi-Quan Luo
  • Jos F. Sturm
Part of the Applied Optimization book series (APOP, volume 33)


In this paper we consider the problem of estimating the distance from a given point to the solution set of a quadratic inequality system. We show, among other things, that a local error bound of order 1/2 holds for a system defined by linear inequalities and a single (nonconvex) quadratic equality. We also give a sharpening of Lojasiewicz’ error bound for piecewise quadratic functions. In contrast, the early results for this problem further require either a convexity or a nonnegativity assumption.


Error Bound SIAM Journal Residual Function Quadratic System Inequality System 
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  1. [1]
    Auslender, A., and J.P. Crouzeix. “Global regularity theorems”, Mathematics of Operations Research 13, 243–253, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Auslender, A., and J.P. Crouzeix. “Well behaved asymptotical convex functions”, Analyse Non-linéare 101–122, 1989.Google Scholar
  3. [3]
    Bergthaller, C., and I. Singer. “The distance to a polyhedron,” Linear Algebra and its Applications 169, 111–129, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Bierstone E., and P.D. Milman. “Semianalytic and subanalytic sets,” Institut des Hautes Etudes Scientifiques, Publications Mathématiques 67, 5–42, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Borwein, J. M. “Stability and regular points of inequality systems”, Journal of Optimization Theory and Applications 48, 9–52, 1986.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Boyd, S., L. El Ghaoui, E. Feron and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory, Philadelphia: SIAM, 1994.zbMATHCrossRefGoogle Scholar
  7. [7]
    Burke, J. V. “On the identification of active constraints II: The nonconvex case”, SIAM Journal on Numerical Analysis 27, 1081–1102, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Burke, J. V. “An exact penalization viewpoint of constrained optimization problem”, SIAM Journal on Control and Optimization 29, 968–998, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Burke, J. V., and M.C. Ferris. “Weak sharp minima in mathematical programming”, SIAM Journal on Control and Optimization 31, 1340–1359, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Burke, J.V., and J.J. Moré. “Exposing constraints”, SIAM Journal on Optimization 4, 573–595, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Chou, C. C., K.F. Ng, and J.S. Pang. “Minimizing and stationary sequences of optimization problems”, SIAM Journal on Control and Optimization, revision under review.Google Scholar
  12. [12]
    Dedieu, J. P. “Penalty functions in subanalytic optimization,” Optimization 26, 27–32, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Dontchev, A. L., and R.T. Rockafellar. “Characterizations of strong regularity for variational inequalities over polyhedral convex sets”, SIAM Journal on Optimization 6, 1087–1105, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    Facchinei, F., A. Fischer, and C. Kanzow. “On the accurate identification of active constraints”, manuscript, Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza” (Roma 1996).Google Scholar
  15. [15]
    Ferris, M.C. “Weak sharp minima and penalty functions in mathematical programming”, Technical report 779, Computer Science Department, University of Wisconsin (Madison June 1988).Google Scholar
  16. [16]
    Ferris, M.C. “Finite termination of the proximal point algorithm”, Mathematical Programming 50, 359–366, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Ferris, M.C., and O.L. Mangasarian. “Minimum principle sufficiency”, Mathematical Programming 57, 1–14, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    Fu, M., Z.-Q. Luo and Y. Ye. “Approximation algorithms for quadratic programming,” Journal of Combinatorial Optimization, 1997; forthcoming.Google Scholar
  19. [19]
    Fukushima, M., and J.S. Pang. “Minimizing and stationary sequences of merit functions for complementarity problems and variational inequalities”, in M.C. Ferris and J.S. Pang, eds., Variational and Complementarity Problems: State of the Art, refereed Proceedings of the International Conference on Complementarity Problems 1995, SIAM Publications (Philadelphia 1997), 91–104.Google Scholar
  20. [20]
    Gowda, M.S. “An analysis of zero set and global error bound properties of a piecewise affine function via its recession function”, SIAM Journal on Matrix Analysis 17, 594–609, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    Güler, O., A.J. Hoffman and U.G. Rothblum. “Approximations to solutions to systems of linear inequalities,” SIAM Journal on Matrix Analysis and Applications 16, 688–696, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    Hoffman, A.J. “On approximate solutions of systems of linear inequalities” , Journal of Research of the National Bureau of Standards 49, 263–265, 1952.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Klatte, D., and W. Li. “Asymptotic constraint qualifications and global error bounds for convex inequalities”, manuscript, Department of Mathematics and Statistics, Old Dominion University (Norfolk, October 1996).Google Scholar
  24. [24]
    Lewis, A.S., and J.S. Pang. “Error bounds for convex inequality systems”, in J.P. Crouzeix, ed., Proceedings of the Fifth Symposium on Generalized Convexity Luminy-Marseille 1996, forthcoming.Google Scholar
  25. [25]
    Li, W. “The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program,” Linear Algebra and its Applications 187, 15–40, 1993.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Li, Wu. “Error bounds for piecewise convex quadratic programs and applications”, SIAM Journal on Control and Optimization 33, 1510–1529, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    Li, W. “Abadie’s constraint qualification, metric regularity, and error bounds for differentible convex inequalities”, SIAM Journal on Optimization 7, 1997.Google Scholar
  28. [28]
    Lojasiewicz, M.S. “Sur le problème de la division,” Studia Mathematica 18, 87–136, 1959.MathSciNetzbMATHGoogle Scholar
  29. [29]
    Luo, X.D., and Z.-Q. Luo. “Extensions of Hoffman’s error bound to polynomial systems”, SIAM Journal on Optimization 4, 383–392, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Luo, Z.-Q., and J.S. Pang. “Error bounds for analytic systems and their applications”, Mathematical Programming 67, 1–28, 1994 .MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Luo, Z.-Q., J.S. Pang, and D. Ralph. Mathematical Programs with Equilibrium Constraints, Cambridge University Press (Cambridge 1996).CrossRefGoogle Scholar
  32. [32]
    Luo, Z.-Q., J.S. Pang, D. Ralph, and S.Q. Wu. “Exact penalization and stationarity conditions of mathematical programs with equilibrium”, Mathematical Programming 75, 19–76, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [33]
    Luo, Z.-Q., J.F. Sturm, and S.Z. Zhang. “Superlinear convergence of a symmetric primal-dual path following algorithm for semidef-inite programming”, manuscript, Econometric Institute, Erasmus University, Rotterdam (January 1996).Google Scholar
  34. [34]
    Luo, Z.-Q., and P. Tseng. “On the convergence of a matrix splitting algorithm for the symmetric monotone linear complementarity problem,” SIAM J. Contr. & Optim., 29, 1037–1060, 1991.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [35]
    Luo, Z.-Q., and P. Tseng. “On the linear convergence of descent methods for convex essentially smooth minimization”, SIAM Journal on Control and Optimization 30, 408–425, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Luo, Z.-Q., and P. Tseng. “Error bound and the convergence analysis of matrix splitting algorithms for the affine variational inequality problem”, SIAM Journal on Optimization 2, 43–54, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Luo, Z.-Q., and P. Tseng. “Error bounds and the convergence analysis of feasible descent methods: a general approach”, Annals of Operations Research 46, 157–178, 1993.MathSciNetCrossRefGoogle Scholar
  38. [38]
    Luo, Z.-Q., and P. Tseng. “On the convergence rate of dual ascent methods for linearly constrained convex minimization”, Mathematics of Operations Research 846–867, 1993.Google Scholar
  39. [39]
    Mangasarian, O.L. “A condition number of linear inequalities and equalities,” in G. Bamber and O. Optiz, eds., Methods of Operations Research 43, Proceedings of Sixth Symposium über Operations Research, Universität Augsburg, September 7–9, 1981, Verlagsgruppe Athennäum/Hain/Scriptor/Hanstein (Konigstein 1981), pp. 3–15.Google Scholar
  40. [40]
    Mangasarian, O.L., and J.S. Pang. “Exact penalty functions for mathematical programs with linear complementarity constraints”, Optimization, forthcoming.Google Scholar
  41. [41]
    Mangasarian, O.L., and T.-H. Shiau. “Error bounds for monotone linear complementarity problems”, Mathematical Programming 36, 81–89, 1986.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    Pang, J.S. private communication.Google Scholar
  43. [43]
    Pang, J.S. “Error Bounds in Mathemtical Programming,” Mathemtaical Programming, Series B 79, 299–332, 1997.zbMATHGoogle Scholar
  44. [44]
    Robinson, S. M. “An application of error bounds for convex programming in a linear space”, SIAM Journal on Control 13, 271–273, 1975.zbMATHCrossRefGoogle Scholar
  45. [45]
    Robinson, S.M. “Regularity and stability for convex multivalued functions”, Mathematics of Operations Research 1, 130–143, 1976.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [46]
    Robinson, S.M. “Some continuity properties of polyhedral multifunction”, Mathematical Programming Study 14, 206–214, 1981.zbMATHCrossRefGoogle Scholar
  47. [47]
    Sturm, J.F. “Superlinear convergence of an algorithm for monotone linear complementarity problems when no strictly complementary solution exists”, manuscript, Econometric Institute, Erasmus University Rotterdam (Rotterdam, September 1996).Google Scholar
  48. [48]
    Vandenberghe, L., and S. Boyd. “Semidefinite programming,” SIAM Review 38, 49–95, 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [49]
    Yakubovich, V. A. “S-procedure in nonlinear control theory” Vestnik Leningradskovo Universiteta, Seriya Matematika 62–77, 1971. (English translation in Vertnik Leningrad University 4, 73–93, 1977.)Google Scholar
  50. [50]
    Walkup, D.W., and R.J.B. Wets. “A Lipschitzian characterization of convex polyhedra”, Proceedings of the American Mathematical Society 20, 167–173, 1969.MathSciNetCrossRefGoogle Scholar
  51. [51]
    Wang, T., and J.S. Pang. “Global error bounds for convex quadratic inequality systems”, Optimization 31, 1–12, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [52]
    Warga, J. “A necessary and sufficient condition for a constrained minimum,” SIAM Journal on Optimization 2, 665–667, 1992.MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Zhi-Quan Luo
  • Jos F. Sturm

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