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Towards a New Era in Subdifferential Analysis?

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Computational and Analytical Mathematics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 50))

Abstract

We give some new attention to the foundations of nonsmooth analysis. We endeavour to delineate the common features of usual subdifferentials. In particular, we stress calculus rules and properties linked with order. Our objective is to give the possibility of using subdifferentials without dealing with specific constructions.

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References

  1. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkhaüser, Boston (1990)

    MATH  Google Scholar 

  2. Aussel, D., Corvellec, J.-N., Lassonde, M.: Mean value property and subdifferential criteria for lower semicontinuous functions. Trans. Am. Math. Soc. 347, 4147–4161 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aussel, D., Corvellec, J.-N., Lassonde, M.: Nonsmooth constrained optimization and multidirectional mean value inequalities. SIAM J. Optim. 9, 690–706 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borwein, J., Preiss, D.: A smooth variational principle with applications to subdifferentiability and differentiability of convex functions. Trans. Am. Math. Soc. 303, 517–537 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borwein, J., Zhu, Q.J.: Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity. SIAM J. Control Optim. 34, 1568–1591 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borwein, J., Zhu, Q.J.: A survey of subdifferential calculus with applications. Nonlinear Anal. 38, 687–773 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. CMS Books in Mathematics, vol. 20. Springer, New York (2005)

    Google Scholar 

  8. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  9. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Optimal Control Theory. Springer, New York (1998)

    Google Scholar 

  10. Correa, R., Jofre, A., Thibault, L.: Subdifferential monotonicity as characterization of convex functions. Numer. Funct. Anal. Optim. 15(5–6), 531–535 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Correa, R., Jofre, A., Thibault, L.: Subdifferential characterization of convexity. In: Du, D.Z., Qi L., Womersley R.S. (eds.). Recent Advances in Nonsmooth Optimization, pp. 18–23. World Scientific, Singapore (1995)

    Chapter  Google Scholar 

  12. Fabian, M.: Subdifferentiability and trustworthiness in the light of the new variational principle of Borwein and Preiss. Acta Univ. Carol. Math. Phys. 30, 51–56 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Ioffe, A.: On subdifferentiability spaces. Ann. N.Y. Acad. Sci. 410, 107–119 (1983)

    Google Scholar 

  14. Ioffe, A.D.: Subdifferentiability spaces and nonsmooth analysis. Bull. Am. Math. Soc. 10, 87–89 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ioffe, A.D.: On the theory of subdifferential. Fermat Days 85: Mathematics for optimization (Toulouse, 1985), North-Holland Math. Stud., vol. 129, pp. 183–200. North-Holland, Amsterdam (1986)

    Google Scholar 

  16. Ioffe, A.D.: Calculus of Dini subdifferentials of functions and contingent coderivatives of set-valued maps. Nonlinear Anal. 8, 517–539 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ioffe, A.: Approximate subdifferentials and applications, I. Trans. Am. Math. Soc. 281, 389–416 (1984)

    MathSciNet  MATH  Google Scholar 

  18. Ioffe, A.D.: Approximate subdifferentials and applications, II. Mathematika 33, 111–128 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ioffe, A.D.: On the local surjection property. Nonlinear Anal. 11, 565–592 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ioffe, A.: Approximate subdifferentials and applications, III: The metric theory. Mathematika 36, 1–38 (1989)

    MathSciNet  MATH  Google Scholar 

  21. Ioffe, A.D.: Proximal analysis and approximate subdifferentials. J. London Math. Soc. 41(2), 175–192 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ioffe, A.: Separable reduction theorem for approximate subdifferentials. C. R. Acad. Sci. Paris 323(Série I), 107–112 (1996)

    Google Scholar 

  23. Ioffe, A.: Fuzzy principles and characterization of trustworthiness. Set-Valued Anal. 6, 265–276 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ioffe, A.D.: Codirectional compactness, metric regularity and subdifferential calculus. In: Thera M. (ed.). Constructive, Experimental and Nonlinear Analysis. CMS Conference proceeding, vol. 27, pp. 123–165 (2000)

    MathSciNet  Google Scholar 

  25. Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspehi Mat. Nauk 55(3), 103–162 (2000) (in Russian). English translation: Russian Math. Surveys, 55(3), 501–558 (2000)

    Google Scholar 

  26. Ioffe, A.D.: On the theory of subdifferentials. Adv. Nonlinear Anal. 1, 47–120 (2012)

    MathSciNet  Google Scholar 

  27. Ioffe, A.D., Penot, J.-P.: Subdifferentials of performance functions and calculus of coderivatives of set-valued mappings. Serdica Math. J. 22, 359–384 (1996)

    MathSciNet  MATH  Google Scholar 

  28. Jules, F.: Sur la somme de sous-différentiels de fonctions continues inférieurement. Dissertationes Mathematicae 423, Polska Akad. Nauk, Warsaw (2003)

    Google Scholar 

  29. Lassonde, M.: First-order rules for nonsmooth constrained optimization. Nonlinear Anal. Ser. A Theory Methods 44(8), 1031–1056 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mordukhovich, B.S.: Metric approximations and necessary conditions for optimality for general classes of nonsmooth optimization problems. Dokl. Acad. Nauk SSSR 254, 1072–1076 (1980)

    MathSciNet  Google Scholar 

  31. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. Springer, Berlin (2005)

    MATH  Google Scholar 

  32. Penot, J.-P.: Miscellaneous incidences of convergence theories in optimization and nonsmooth analysis, II: applications to nonsmooth analysis. In: Du, D.Z., Qi L., Womersley, R.S., (eds.). Recent Advances in Nonsmooth Optimization, pp. 289–321. World Scientific Publishers, Singapore (1995)

    Chapter  Google Scholar 

  33. Penot, J.-P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Penot, J.-P.: The compatibility with order of some subdifferentials. Positivity 6, 413–432 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Penot, J.-P.: Semigroups of relations and subdifferentials of the minimal time function and of the geodesic distance. SIAM J. Control Optim. 51(4), 2839–2868 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Penot, J.-P.: Calculus Without Derivatives. Graduate Texts in Mathematics. 266 Springer, New York (2013)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  38. Thibault, L.: A note on the Zagrodny mean value theorem. Optimization 35, 127–130 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhu, Q.J.: The equivalence of several basic theorems Please cite it for subdifferentials. Set-Valued Anal. 6, 171–185 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author gratefully acknowledges the influence of numerous exchanges with A.D. Ioffe during the last decades and the useful criticisms of an anonymous referee.

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France

    Jean-Paul Penot

Authors
  1. Jean-Paul Penot
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Corresponding author

Correspondence to Jean-Paul Penot .

Editor information

Editors and Affiliations

  1. Lawrence Berkeley National Laboratory, Berkeley, USA

    David H. Bailey

  2. Mathematics, University of British Columbia, Kelowna, British Columbia, Canada

    Heinz H. Bauschke

  3. Dept. Mathematics & Statistics, Simon Fraser University, Burnaby, British Columbia, Canada

    Peter Borwein

  4. Mathematics, University of Florida, Gainesville, Florida, USA

    Frank Garvan

  5. Mathematics and Computer Science, University of Limoges, Limoges, France

    Michel Théra

  6. Mathematics and Computer Science Department, La Sierra University, Riverside, California, USA

    Jon D. Vanderwerff

  7. Dept. Combinatorics & Optimization, University of Waterloo Faculty of Mathematics, Waterloo, Ontario, Canada

    Henry Wolkowicz

Additional information

Dedicated to Jon Borwein on the occasion of his 60th birthday

Communicated By Michel Théra.

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Penot, JP. (2013). Towards a New Era in Subdifferential Analysis?. In: Bailey, D., et al. Computational and Analytical Mathematics. Springer Proceedings in Mathematics & Statistics, vol 50. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7621-4_29

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