Annali di Matematica Pura ed Applicata

, Volume 130, Issue 1, pp 223–255 | Cite as

Tangency and differentiation: Some applications of convergence theory

  • Szymon Dolecki


We present a unified approach based on convergence theory to approximating cones and generalized derivatives.


Unify Approach Generalize Derivative Convergence Theory 
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  1. [1]
    G.Bouligand,Introduction à la géometrie infinitesimale directe, Vuibert, Paris, 1932.Google Scholar
  2. [2]
    G. Buttazzo,Su una definizione generale dei Γ-limiti, Boll. U.M.I., (5)14-B (1977), pp. 722–744.Google Scholar
  3. [3]
    G.Buttazzo - G.Dal Maso,Γ-convergence and optimal control problems, Scuola Normale Superiore, Pisa (preprint) (1981).Google Scholar
  4. [4]
    G. Choquet,Convergences, Annales Université de Grenoble,23 (1947–1948), pp. 55–112.Google Scholar
  5. [5]
    F. H. Clarke,Generalized gradients and applications, Trans. A.M.S.,205 (1975), pp. 247–262.Google Scholar
  6. [6]
    B. D. Craven -B. Mond,Lagrangean conditions for quasi differentiable optimisation, Survey of Math. Programming,12 (1980), pp. 177–192.Google Scholar
  7. [7]
    E. De Giorgi,Γ-convergenza, G-convergenza, Boll. U.M.I., (5)14-A (1977), pp. 213–220.Google Scholar
  8. [8]
    E. De Giorgi,Convergence problems for functionals and operators, in Proc. Internat. Meeting « Recent Methods in Nonlinear Analysis », Rome, 1978,E. De Giorgi -E. Magenes -U. Mosco (eds.), Pitagora, Bologna, 1979, pp. 131–188.Google Scholar
  9. [9]
    E. De Giorgi -T. Franzoni,Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei,58 (8) (1975), pp. 842–850.Google Scholar
  10. [10]
    Z.Denkowski,The convergence of generalized sequences of states and functions in locally convex spaces, Istitute of Math. Polish Academy of Sciences, Preprint 151.Google Scholar
  11. [11]
    S.Dolecki,Metrically upper semicontinuous multifunctions and their intersections, Math. Res. Center, Madison, Wis., Report 2035 (1980).Google Scholar
  12. [12]
    S.Dolecki,Hypertangent cones for a special class of sets, to appear in Optimization: Theory and Algorithms, Hiriart-Urruty, Oettli, Stoer (eds.), M. Dekker, New York.Google Scholar
  13. [13]
    S. Dolecki -S. Rolewicz,A characterization of semicontinuity-preserving multifunctions, J. Math. Anal. Appl.,65 (1978), pp. 26–31.Google Scholar
  14. [14]
    S.Dolecki - G.Salinetti - R. J.-B.Wets,Convergence of functions: equi-semicontinuity, to appear, Trans. Amer. Math. Soc.Google Scholar
  15. [15]
    A. Ya. Dubovitzkii -A. A. Milyutin,Extremal problems in presence of constraints, Z. Vichisl. Mat. i Mat. Fiz.,5 (1965), pp. 395–453 (in Russian).Google Scholar
  16. [16]
    J.-B.Hiriart-Urruty, Thèse, Université de Clermont II (1977).Google Scholar
  17. [17]
    J.-B. Hiriart-Urruty,Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math. Operations Res.,4 (1979), pp. 79–97.Google Scholar
  18. [18]
    A. D. Ioffe,Necessary and sufficient conditions for a local minimum. — II:Levitin-Milyutin-Osmolovskii approximation, SIAM J. Control Optim.,17 (1979), pp. 251–265.Google Scholar
  19. [19]
    J.-L. Joly,Une famille de topologies sur l'ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. pures et appl.,52 (1973), pp. 421–441.Google Scholar
  20. [20]
    C. Kuratowski,Topologie, Vol. I, Panstwowe Wydawnictwo Naukowe, Warszawa, 1958, 4th edition.Google Scholar
  21. [21]
    R. Mifflin,Semismooth and semiconvex functions in constrained optimization, SIAM J. Control,15 (1977), pp. 959:972.Google Scholar
  22. [22]
    S.Mirică,The contingent and the paratingent as generalized derivatives of vector-valued and set-valued mappings, INCREST, Preprint 31 (1981) (Math.), Bucuresti.Google Scholar
  23. [23]
    U. Mosco,Convergence of convex sets and of solutions of variational inequalities, Adv. in Math.,3, No. 4 (1969), pp. 510–585.Google Scholar
  24. [24]
    M. Z. Nashed,Differentiability and related properties of nonlinear operators: Some aspects of the role of differentials in nonlinear functional analysis, inNonlinear Functional Analysis and Applications, Proc. Adv. Seminar, Math. Res. Centr., Madison, Wis., 1970, Acad. Press, New York-London, 1971, pp. 103–309.Google Scholar
  25. [25]
    J.-P.Penot,The use of generalized subdifferential calculus in optimization theory, Proc. 3 Symp., Operations Res. Mannheim (1978); Methods of Operations Res., 31.Google Scholar
  26. [26]
    J.-P. Penot,A characterization of tangential regularity, Nonlinear Anal. Theory, Math. Appl.,5 (1981), pp. 625–663.Google Scholar
  27. [27]
    B. N. Pshenichnii,Necessary Conditions for an Extremum, M. Dekker, New York, 1971.Google Scholar
  28. [28]
    R. T.Rockafellar,The theory of subgradients and its applications to problems of optimization, Lecture notes, University of Montreal, 1978.Google Scholar
  29. [29]
    R. T. Rockafellar,Generalized directional derivatives and subgradients of nonconvex functions, Can. J. Math.,32 (1980), pp. 257–280.Google Scholar
  30. [30]
    R. T. Rockafellar,Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc.,39 (1979), pp. 331–355.Google Scholar
  31. [31]
    S. Rolewicz,On intersection of multifunctions, Math. Operationsforsch. Statistik, Optimization,11 (1980), pp. 3–11.Google Scholar
  32. [32]
    E. Sachs,Differentiability in optimization theory, Math. Operationsforsch. Statistik, Optim,9 (1978), pp. 497–513.Google Scholar
  33. [33]
    F. Severi,Sulla differenziabilità totale delle funzioni di più variabili reali, Annali Mat. Pura e Appl., Serie 4,13 (1935), pp. 1–35.Google Scholar
  34. [34]
    L.Thibault,Proprietés des sous-differentiels sur un espace de Banach séparable, Applications, Thèse, Université de Languedoc, Montpellier, 1976.Google Scholar
  35. [35]
    C.Ursescu,Tangent sets' calculus and necessary conditions for extremality, to appear.Google Scholar
  36. [36]
    M. Vlach,Approximation operators in optimization theory, Zeitschrift für Operations Research,25 (1981), pp. 15–23.Google Scholar
  37. [37]
    R. A. Wijsman,Convergence of sequences of convex sets, cones and functions, II, Trans. Amer. Math. Soc.,123 (1963), pp. 32–45.Google Scholar
  38. [38]
    J.-P.Penot,Compact filters, nets and relations, to appear in J. Math. Anal. Appl.Google Scholar

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© Fondazione Annali di Matematica Pura ed Applicata 1982

Authors and Affiliations

  • Szymon Dolecki
    • 1
  1. 1.WarszawaPoland

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