Skip to main content
SpringerLink
Log in

Revisiting some Rules of Convex Analysis

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

Convex analysis is devoted to the study and the use of four notions: conjugate functions, normal cones, subdifferentials, support functions under convexity assumptions. These notions are closely related and the calculus rules for one of them imply calculus rules for the other ones. But for none of these notions such rules are always valid without additional assumptions. Thus, much effort has been devoted to these “qualification conditions” (Attouch-Brézis, Boţ and his co-authors, Burachik-Jeyakumar, Rockafellar...). We introduce a new condition that requires a pointwise outer semicontinuity (or closedness) property of an appropriate multifunction. For subdifferentials (resp. normal cones), the multifunction is defined in terms of subdifferentials (resp. normal cones). We relate the new condition with previous ones from the literature.

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. Adly, S., Ernst, E., Théra, M.: On the closedness of the algebraic difference of closed convex sets. J. Math. Pures Appl. 82(9), 1219–1249 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Attouch, H., Brézis, H.: Duality for the sum of convex functions in general banach spaces. In: Barroso, J. (ed.) Aspects of mathematics and its applications, North Holland, pp. 125–133 (1986)

  3. Aubin, J.-P., Frankowska, H.: Set-valued analysis. Birkhäuser, Boston (1990)

    MATH  Google Scholar 

  4. Bauschke, H.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. Prog. 86, 135–169 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauschke, H.H., Borwein, J.M., Tseng, P.: Bounded linear regularity, strong CHIP, and CHIP are distinct properties. J. Convex Anal. 7(2), 395–412 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Bauschke, H.H., Combettes, P.-L.: Convex analysis and monotone operator theory in hilbert spaces, CMS books in mathematics. Springer, Berlin (2010)

    Google Scholar 

  7. Borwein, J.M., Burachik, R.S., Yao, L.: Conditions for zero duality gap in convex programming. J. Nonlinear Convex Anal. 15(1), 167–190 (2014)

    MathSciNet  MATH  Google Scholar 

  8. Borwein, J., Vanderwerff, J.D.: Convex functions: constructions, characterizations and counterexamples. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  9. Borwein, J.M., Zhu, Q.J.: Variational methods in convex analysis. J. Glob. Optim. 35(2), 197–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Borwein, J.M., Zhu, Q.J.: Limiting convex examples for nonconvex subdifferential calculus. J. Convex Anal. 5(2), 221–235 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Boţ, R.I., Grad, S.-M.: Lower semicontinuous type regularity conditions for subdifferential calculus. Optim. Methods and Software 25, 37–48 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Boţ, R.I., Grad, S.-M., Wanka, G.: A new constraint qualification for the formula of the subdifferential of composed convex functions in infinite dimensional spaces. Math. Nachr. 281, 1088–1107 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Boţ, R.I., Grad, S.-M., Wanka, G.: Generalized Moreau-Rockafellar results for composed convex functions. Optimization 58(7), 917–933 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Boţ, R.I., Grad, S.-M., Wanka, G.: New regularity conditions for Lagrange and Fenchel-Lagrange duality in infinite-dimensional spaces. Math. Inequal. Appl. 12 (1), 171–189 (2009)

    MathSciNet  MATH  Google Scholar 

  15. Boţ, R.I., Grad, S.-M., Wanka, G.: Duality in vector optimization. Springer, Berlin (2009)

    MATH  Google Scholar 

  16. Boţ, R.I., Wanka, G.: An alternative formulation for a new closed convex cone constraint qualification. Nonlinear Anal. Theory Methods Appl. 64(6), 1367–1381 (2006)

    Article  MATH  Google Scholar 

  17. Boţ, R.I., Wanka, G.: A weaker regularity condition for subdifferential calculus and Fenchel’s duality in infinite dimensional spaces. Nonlinear Anal. Theory Methods Appl. 64, 2787–2804 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brøndsted, A.: On the subdifferential of the maximum of two convex functions. Math. Scand. 31, 225–230 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Burachik, R.S., Jeyakumar, V.: A simple closure condition for the normal cone intersection formula. Proc. Amer. Math. Soc. 133(6), 1741–1748 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Burachik, R.S., Jeyakumar, V.: A dual condition for the convex subdifferential sum formula with applications. J. Convex Anal. 12(2), 279–290 (2005). MR2197287

    MathSciNet  MATH  Google Scholar 

  21. Burachik, R.S., Jeyakumar, V.: A new geometric condition for Fenchel’s duality in infinite dimensional spaces. Math. Program. 104(B), 229–233 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Burachik, R.S., Jeyakumar, V., Wu, Z.Y.: Necessary and sufficient conditions for stable conjugate duality. Nonlinear Anal Theory Meth. Appl. 64, 1998–2016 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Castaing, C. h., Valadier, M.: Convex Analysis and Measurable Multifunctions Lecture Notes in Mathematics, vol. 580. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  24. Combari, C., Laghdir, M., Thibault, L.: Sous-différentiels de fonctions convexes composées. Ann. Sci. Math. Québec 18(2), 119–148 (1994)

    MathSciNet  MATH  Google Scholar 

  25. Combari, C., Laghdir, M., Thibault, L.: On subdifferential calculus for convex functions defined on locally convex spaces. Ann. Sci. Math. Qué,bec 23(1), 23–36 (1999)

    MathSciNet  MATH  Google Scholar 

  26. Correa, R., Hantoute, A., Lòpez, M.A.: Towards supremum-sum subdifferential calculus free of qualification conditions. SIAM J. Optim 26(4), 2219–2234 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Correa, R., Hantoute, A., Lòpez-Cerda, M.A.: Weaker conditions for subdifferential calculus of convex functions. J. Funct. Anal. 271(5), 1177–1212 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Deutsch, F., Li, W., Swetits, J.: Fenchel duality and the strong conical hull intersection property. J. Optim. Th. Appl. 102, 681–695 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dinh, N., Goberna, M.A., Lòpez, M.A., Son, T.Q.: New Farkas-type constraint qualifications in convex infinite programming. ESAIM Control Optim Calc. Var. 13, 580–597 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dinh, N., Goberna, M.A., Lòpez, M.A.: From linear to convex systems: consistency, Farkas’ lemma and applications. J. Convex Anal. 13, 113–133 (2006)

    MathSciNet  MATH  Google Scholar 

  31. Dinh, N., Jeyakumar, V., Lee, G.M.: Sequential Lagrangian conditions for convex programs with applications to semi-definite programming. J. Optim. Theory Appl. 125, 85–112 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dinh, N., Vallet, G., Nghia, T.T.A.: Generalized Farkas lemmas for systems involving convex and DC functions and its applications. J. Convex Anal. 15(2), 235–262 (2008)

    MathSciNet  MATH  Google Scholar 

  33. Dinh, N., Nghia, T.T.A., Vallet, G.: A closedness condition and its applications to DC programs with convex constraints. Optimization 59(3-4), 541–560 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Fenchel, W.: Convexity through the ages, Copenhagen (1973)

  35. Fitzpatrick, S.P., Simons, S.: The conjugates, compositions and marginals of convex functions. J. Convex Anal. 8, 423–446 (2001)

    MathSciNet  MATH  Google Scholar 

  36. Goberna, M.A., López-Cerdá, M.A.: New Glimpses on Convex Infinite Optimization Duality, Revista Real Acad. Ciencias Exactas, Fisicas Natur. A Math (2015)

  37. Gowda, M.S., Teboulle, M.: A comparison of constraint qualifications in infinite dimensional convex programming. SIAM J. Control Optim. 28, 925–935 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. Hantoute, A., López, M.A., Zălinescu, C.: Subdifferential calculus rules in convex analysis: a unifying approach via pointwise supremum functions. SIAM J. Optim. 19(2), 863–882 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Hiriart-Urruty, J.B., Moussaoui, M., Seeger, A., Volle, M.: Subdifferential calculus without qualification conditions, using approximate subdifferential: a survey. Nonlinear Anal. 24, 1724–1754 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hiriart-Urruty, J.-B., Phelps, R.R.: Subdifferential calculus using epsilon-subdifferentials. J. Funct. Anal. 118, 154–166 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ioffe, A.D.: Convexity and variational analysis, computational and analytical mathematics, 411-444, Springer Proc. Math Stat, vol. 50. Springer, New York (2013)

    Google Scholar 

  42. Ioffe, A.D., Levin, V.L.: Subdifferentials of convex functions. Trans Moscow Math. Soc. 26, 3–73 (1977)

    MathSciNet  MATH  Google Scholar 

  43. Jeroslov, R.G.: Uniform duality in semi-infinite convex optimization. Math. Program. 27(2), 144–154 (1983)

    Article  MathSciNet  Google Scholar 

  44. Jeyakumar, V.: Constraint qualifications characterizing Lagrangian duality in convex optimization. J. Optim. Theory Appl. 136, 31–41 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  45. Jeyakumar, V., Wu, Z.Y.: A qualification free sequential Pshenichnyi-Rockafellar lemma and convex semidefinite programming. J. Convex Anal. 13(3-4), 773–784 (2006)

    MathSciNet  MATH  Google Scholar 

  46. Li, S.J., Yang, X.Q., Teo, K.L.: Duality for semi-definite and semi-infinite programming. Optimization 52(4-5), 507–528 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  47. Pataki, G.: On the closedness of the linear image of a closed convex cone. Math. Oper. Res. 32(2), 395–412 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  48. Penot, J.-P.: On regularity conditions in mathematical programming. Math. Program. Study 10, 167–199 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  49. Penot, J.-P.: Subdifferential calculus without qualification conditions. J. Convex Anal. 3(2), 1–13 (1996)

    MathSciNet  MATH  Google Scholar 

  50. Penot, J.-P.: Calculus without derivatives, Graduate Texts in Mathematics, vol. 266. Springer, New York (2013)

    Book  Google Scholar 

  51. Penot, J.-P.: Analysis, from concepts to applications. Universitext, Springer, London (2016)

    MATH  Google Scholar 

  52. Ramana, M.V., Tunçel, L., Wolkowicz, H.: Strong duality for semi-definite programming. SIAM J. Optim. 7(3), 644–662 (1997)

    Article  MATH  Google Scholar 

  53. Rockafellar, R.T.: Convex analysis, Princeton Math. Ser., vol. 28. Princeton University Press, Princeton (1970)

    Google Scholar 

  54. Rockafellar, R.T.: Conjugate duality and optimization CBMS-NSF regional conference series in applied mathematics, vol. 16. SIAM, Philadelphia (1974)

    Google Scholar 

  55. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis, Grundlehren Der Mathematishen Wissenschaften, vol. 317. Springer, Berlin (2002)

    Google Scholar 

  56. Strömberg, T.h.: The operation of infimal convolution. Diss. Math. 352, 1–61 (1996)

    MathSciNet  MATH  Google Scholar 

  57. Sun, X., Long, X.-J., Zen, J.: Constraint qualifications characterizing Fenchel duality in composed convex optimization. J. Nonlin. Convex Anal. 17(2), 325–347 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Thibault, L.: A generalized sequential formula for subdifferentials of sums of convex functions defined on Banach spaces. In: Duriez, R., Michelot, C. (eds.) Recent developments in optimization, Lecture Notes in Economics and Mathematical Systems 429, pp. 340–345. Springer, Berlin (1995)

  59. Thibault, L.: Sequential convex subdifferential calculus and sequential Lagrange multipliers. SIAM J. Control. Optim. 35(4), 1434–1444 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  60. Thibault, L.: Limiting convex subdifferential calculus with applications to integration and maximal monotonicity of subdifferential. In: Théra, M. (ed.) Constructive, experimental, and nonlinear analysis, CMS conference proceedings 27, pp. 279–289. American Math. Soc., Providence (2000)

  61. Zălinescu, C.: A note on d-stability of convex programs and limiting Lagrangians. Math. Program. 53, 267–277 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  62. Zălinescu, C.: A comparison of constraint qualifications in infinite-dimensional convex programming revisited. J. Australian Math. Soc., Ser. B 40, 353–378 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  63. Zălinescu, C.: Convex analysis in general vector spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  64. Zălinescu, C.: On duality gap in linear conic problems. Opt. Lett. 6(3), 393–402 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  65. Zălinescu, C.: On zero duality gap and the Farkas lemma for conic programming. Math. Oper. Res. 33, 991–1001 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sorbonne Universités, UPMC, Univ Paris 06 CNRS UMR 7598 Laboratoire J.-L. Lions, 75005, Paris, France

    Jean-Paul Penot

Authors
  1. Jean-Paul Penot
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Paul Penot.

Additional information

Dedicated to Michel Théra on the occasion of his seventieth birthday

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Penot, JP. Revisiting some Rules of Convex Analysis. Set-Valued Var. Anal 25, 773–788 (2017). https://doi.org/10.1007/s11228-017-0462-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-017-0462-3

Keywords

Mathematics Subject Classification (2010)

Search

Navigation