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Mathematical Programming

, Volume 116, Issue 1–2, pp 37–56 | Cite as

Weak sharp minima revisited, Part III: error bounds for differentiable convex inclusions

  • James V. BurkeEmail author
  • Sien Deng
FULL LENGTH PAPER

Abstract

The notion of weak sharp minima unifies a number of important ideas in optimization. Part I of this work provides the foundation for the theory of weak sharp minima in the infinite-dimensional setting. Part II discusses applications of these results to linear regularity and error bounds for nondifferentiable convex inequalities. This work applies the results of Part I to error bounds for differentiable convex inclusions. A number of standard constraint qualifications for such inclusions are also examined.

Keywords

Weak sharp minima Convex inclusion Affine convex inclusion Constraint qualification Error bounds Calmness 

Mathematics Subject Classification (2000)

90C25 90C31 49J52 

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References

  1. 1.
    Abadie J.: On the Kuhn–Tucker Theorem. In: Abadie J. (ed) Nonlinear Programming, pp. 21–36. North Holland (1967)Google Scholar
  2. 2.
    Aubin J.-P. and Ekeland I. (1984). Applied Nonlinear Analysis. Wiley–Interscience, New York zbMATHGoogle Scholar
  3. 3.
    Auslender A. and Crouzeix J.-P. (1988). Global regularity theorem. Math. Oper. Res. 13: 243–253 zbMATHMathSciNetGoogle Scholar
  4. 4.
    Auslender A. and Teboulle M. (2003). Asymptotic Cones and Functions in Optimization and Variational Inequalities. Springer, Heidelberg zbMATHGoogle Scholar
  5. 5.
    Bauschke, H.: Projection algorithms and monotone operators. Ph.D. Thesis, Simon Fraser University, Department of Mathematics, Burnaby, British Columbia, V5A 1S6, Canada (1996)Google Scholar
  6. 6.
    Bauschke H., Borwein J. and Li W. (1999). Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Prog. 86: 135–160 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Borwein J. and Lewis A. (2000). Convex Analysis and Nonlinear Optimization, Theory and Examples CMS Books in Mathematics. Springer, New York Google Scholar
  8. 8.
    Burke J.V. (1991). An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29: 968–998 zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Burke J.V. (1991). Calmness and exact penalization. SIAM J. Control Optim. 29: 493–497 zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Burke J.V. and Deng S. (2002). Weak sharp minima revisited, part I: Basic theory. Control Cybernetics 31: 439–469 zbMATHMathSciNetGoogle Scholar
  11. 11.
    Burke J.V. and Deng S. (2005). Weak sharp minima revisited, part II: Application to linear regularity and error bounds. Math. Program. Ser. B 104: 235–261 zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Burke J.V. and Ferris M.C. (1993). Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31: 1340–1359 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Burke J.V. and Ferris M.C. (1995). A Gauss–Newton method for convex composite optimization. Math. Prog. 71: 179–194 MathSciNetGoogle Scholar
  14. 14.
    Burke J.V. and Moré J.J. (1988). On the identification of active constraints. SIAM J. Numer. Anal. 25: 1197–1211 zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Burke J.V. and Tseng P. (1996). A unified analysis of Hoffman’s bound via Fenchel duality. SIAM J. Optim. 6: 265–282 zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Clarke F.H. (1976). A new approach to Lagrange multipliers. Math. Oper. Res. 2: 165–174 Google Scholar
  17. 17.
    Deng S. (1998). Global error bounds for convex inequality systems in Banach spaces. SIAM J. Control Optim. 36: 1240–1249 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Dontchev A.L. and Rockafellar R.T. (2004). Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12: 79–109 zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ekeland, I., Temam, R.: Convex analysis and variational problems. North Holland (1976)Google Scholar
  20. 20.
    Ferris, M.C.: Weak sharp minima and penalty functions in mathematical programming. Tech. Report 779, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin (1988)Google Scholar
  21. 21.
    Henrion R and Jourani A. (2002). Subdifferential conditions for calmness of convex constraints. SIAM J. Optim. 13: 520–534 zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Henrion R. and Outrata J. (2001). A subdifferential criterion for calmness of multifunctions. J. Math. Anal. Appl. 258: 110–130 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Henrion R. and Outrata J. (2005). Calmness of constraint systems with applications. Math. Prog. Ser. B. 104: 437–464 zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Hiriart-Urruty J.-B. and Lemaréchal C. (1993). Convex Analysis and Minimization Algorithms I, volume 306 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg Google Scholar
  25. 25.
    Hoffman A.J. (1952). On approximate solutions to systems of linear inequalities. J. Res. Nat. Bur. Stand. 49: 263–265 Google Scholar
  26. 26.
    Jourani A. (2000). Hoffman’s error bounds, local controllability and sensitivity analysis. SIAM J. Control Optim. 38: 947–970 zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Klatte D. (1997). Hoffman’s error bound for systems of convex inequalities. In: Fiacco, A.V. (eds) Mathematical Programming with Data Perturbations, pp 185–199. Marcel Dekker Publ., Moscow Google Scholar
  28. 28.
    Klatte D. and Li W. (1999). Asymptotic constraint qualifications and global error bounds for convex inequalities. Math. Prog. 84: 137–160 zbMATHMathSciNetGoogle Scholar
  29. 29.
    Lewis A.S. and Pang J.-S. (1998). Error bounds for convex inequality systems. In: Crouzeix, J.P., Martinez- Legaz, J.-E. and Volle, M. (eds) Proceedings of the Fifth International Symposium on Generalized Convexity held in Luminy June 17-21, 1996, pp 75–101. Kluwer Academic Publishers, Dordrecht Google Scholar
  30. 30.
    Li W. (1997). Abadie’s constraint qualification, metric regularity and error bounds for differentiable convex inequalities. SIAM J. Optim. 7: 966–978 zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Luo X.D. and Luo Z.Q. (1994). Extension of hoffman’s error bound to polynomial systems. SIAM J. Optim. 4: 383–392 zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Maguregui, J.: Regular multivalued functions and algorithmic applications. Ph.D. Thesis, University of Wisconsin at Madison, Madison, WI (1977)Google Scholar
  33. 33.
    Ng K.F. and Yang W.H. (2004). Regularities and their relations to error bounds. Math. Prog. Ser. A 99: 521–538 zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Ngai H.V. and Thera M. (2005). Error bounds for convex differentiable inequality systems in Banach spaces. Math. Program. Ser. B 104: 465–482 zbMATHCrossRefGoogle Scholar
  35. 35.
    Pang J.-S. (1997). Error bounds in mathematical programming. Math. Prog. 79: 299–333 Google Scholar
  36. 36.
    Polyak, B.T.: Sharp minima. In: Proceedings of the IIASA Workshop on Generalized Lagrangians and Their Applications: Laxenburg, Austria. Institute of Control Sciences Lecture Notes, Moscow (1979)Google Scholar
  37. 37.
    Robinson S.M. (1972). Normed convex processes. Trans. Amer. Math. Soc. 174: 127–140 CrossRefMathSciNetGoogle Scholar
  38. 38.
    Robinson S.M. (1975). An application of error bounds for convex programming in a linear space. SIAM J. Control 13: 271–273 zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Robinson S.M. (1976). Regularity and stability for convex multivalued functions. Math. Oper. Res. 1: 130–143 zbMATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    Rockafellar R.T. (1970). Convex Analysis. Princeton University Press, Princeton zbMATHGoogle Scholar
  41. 41.
    Rockafellar R.T. (1974). Conjugate Duality and Optimization. SIAM, Philadelphia zbMATHGoogle Scholar
  42. 42.
    Rockafellar R.T. and Wets R.J.-B. (1998). Variational Analysis. Springer, Heidelberg zbMATHGoogle Scholar
  43. 43.
    Ursescu C. (1975). Multifunctions with closed convex graph. Czech. Math. J. 25: 438–441 MathSciNetGoogle Scholar
  44. 44.
    Yosida K. (1980). Functional Analysis. Springer, Heidelberg zbMATHGoogle Scholar
  45. 45.
    Zălinescu, C.: Weak sharp minima, well-behaving functions and global error bounds for convex inequalities in Banach spaces. In: Bulatov, V., Baturin, V. (ed.) Proceedings of the 12th Baikal International Conference on Optimization Methods and their Applications, pp. 272–284, Institute of System Dynamics and Control Theory of SB RAS, Irkutsk (2001)Google Scholar
  46. 46.
    Zheng X.Y. and Ng K.F. (2004). Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14: 757–772 zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Zheng X.Y. and Ng K.F. (2004). Error moduli for conic convex systems on Banach spaces. Math. Oper. Res. 29: 213–228 zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of Mathematical SciencesNorthern Illinois UniversityDeKalbUSA

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