Set-Valued and Variational Analysis

, Volume 18, Issue 3–4, pp 569–600 | Cite as

Primal-Lower-Nice Property of Value Functions in Optimization and Control Problems

Article

Abstract

The paper studies value functions associated with optimization problems and with Mayer-type control problems. Using methods belonging to proximal analysis and control theory, we establish new results for the primal-lower-nice (pln) property of the value functions for these problems.

Keywords

Primal-lower/upper-nice property Value function 

Mathematics Subject Classifications (2010)

49J15 49J52 49J53 58C06 58C20 93C10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Attouch, H., Wets, R.J.-B.: Epigraphical analysis. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 6, 73–100 (1989)MathSciNetGoogle Scholar
  2. 2.
    Aussel, D., Daniilidis, A., Thibault, L.: Subsmooth sets: functional characterizations and related concepts. Trans. Am. Math. Soc. 357, 1275–1301 (2005)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. In: Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997)Google Scholar
  4. 4.
    Barles, G.: Solutions de viscosité des équations de Hamilton–Jacobi (Viscosity solutions of Hamilton–Jacobi equations). In: Mathematiques & Applications (Paris), vol. 17. Springer, Paris (1994)Google Scholar
  5. 5.
    Bernard, F., Thibault, L.: Prox-regular functions in Hilbert spaces. J. Math. Anal. Appl. 303(1), 1–14 (2005)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bernard, F., Thibault, L.: Uniform prox-regularity of functions and epigraphs in Hilbert spaces. Nonlinear Anal. 60, 187–207 (2005)MATHMathSciNetGoogle Scholar
  7. 7.
    Bernard, F., Thibault, L., Zagrodny, D.: Integration of primal lower nice functions in Hilbert spaces. J. Optim. Theory Appl. 124(3), 561–579 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cannarsa, P., Frankowska, H.: Some characterizations of optimal trajectories in control theory. SIAM J. Control Optim. 29(6), 1322–1347 (1991)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cannarsa, P., Frankowska, H.: Interior sphere property of attainable sets and time optimal control problems. ESAIM, Control Optim. Calc. Var. 12, 350–370 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton–Jacobi equations, and optimal control. In: Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkhäuser, Boston (2004)Google Scholar
  11. 11.
    Combari, C., Elhilali Alaoui, A., Levy, A.B., Poliquin, R.A., Thibault, L.: Convex composite functions in baanach spaces and the primal lower-nice property. Proc. Am. Math. Soc. 126(12), 3701–3708 (1998)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Combari, C., Poliquin, R., Thibault, L.: Convergence of subdifferentials of convexly composite functions. Can. J. Math. 51(2), 250–265 (1999)MATHMathSciNetGoogle Scholar
  13. 13.
    Correa, R., Jofre, A., Thibault, L.: Characterization of lower semicontinuous convex functions. Proc. Am. Math. Soc. 116, 67–72 (1992)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Correa, R., Jofre, A., Thibault, L.: Subdifferential characterization of convexity. In: Du, D.-Z., et al. (eds.) Recent Advances in Nonsmooth Optimization, pp. 18–23. World Scientific, Singapore (1994)Google Scholar
  15. 15.
    Degiovanni, M., Marino, A., Tosques, M.: Evolution equations with lack of convexity. Nonlinear Anal. 9(12), 1401–1443 (1985)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19. AMS, Providence (1998)Google Scholar
  17. 17.
    Hare, W., Sagastizábal, C.: Computing proximal points of nonconvex functions. Math. Program. 116(1–2(B)), 221–258 (2009)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13(2), 603–618 (2002)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Ioffe, A.D.: Metric regularity and subdifferential calculus. Russ. Math. Surv. 55, 501–558 (2000)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Janin, R.: Sur une classe de fonctions sous-linéarisables. C. R. Acad. Sci. (Paris) 277, 265–267 (1973)MATHMathSciNetGoogle Scholar
  21. 21.
    Jourani, A., Thibault, L.: Metric regularity for strongly compactly lipschitzian mappings. Nonlinear Anal. 24(2), 229–240 (1995)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Levy, A.B., Poliquin, R.A., Thibault, L.: Partial extensions of Attouch’s theorem with applications to proto- derivatives of subgradient mappings. Trans. Am. Math. Soc. 347(4), 1269–1294 (1995)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Lions, P.L.: Generalized Solutions of Hamilton–Jacobi Equations. Pitman Advanced Publishing Program, Boston (1982)MATHGoogle Scholar
  24. 24.
    Marcellin, S., Thibault, L.: Evolution problems associated with primal lower nice functions. J. Convex Anal. 13(2), 385–421 (2006)MATHMathSciNetGoogle Scholar
  25. 25.
    Mordukhovich, B.S.: Metric approximations and necessary optimality conditions for general classes of extremal problems. Sov. Math. Dokl. 22, 526–530 (1980)MATHGoogle Scholar
  26. 26.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. II: Applications. Springer (2005)Google Scholar
  27. 27.
    Poliquin, R.A.: Subgradient monotonicity and convex functions. Nonlinear Anal. 14(4), 305–317 (1990)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Poliquin, R.A.: Integration of subdifferentials of nonconvex functions. Nonlinear Anal. 17(4), 385–398 (1991)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Poliquin, R.A.: An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions. Trans. Am. Math. Soc. 332(2), 861–874 (1992)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Poliquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Am. Math. Soc. 348(5), 1805–1838 (1996)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Poliquin, R.A., Rockafellar, R.T., Thibault, L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352(11), 5231–5249 (2000)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)MATHCrossRefGoogle Scholar
  33. 33.
    Serea, O.S.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559–575 (2003)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Thibault, L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193(1), 1–26 (2003)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Thibault, L., Zagrodny, D.: Integration of subdifferentials of lower semicontinuous functions on Banach spaces. J. Math. Anal. Appl. 189(1), 33–58 (1995)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Vial, J.-P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.CMAPEcole PolytechniquePalaiseau cedexFrance
  2. 2.Département de MathématiquesUniversité Montpellier IIMontpellier cedex 5France

Personalised recommendations