Skip to main content
SpringerLink
Log in
Book cover

Set-valued Optimization pp 109–211Cite as

Tangent Cones and Tangent Sets

  • Chapter
  • First Online:

  • 2651 Accesses

Part of the book series: Vector Optimization ((VECTOROPT))

Abstract

Tangent cones of first-order and tangent cones and tangent sets of higher-order play a very important role in set-valued optimization. For instance, derivatives and epiderivatives of set-valued maps are commonly defined by taking tangent cones and tangent sets of graphs and epigraphs of set-valued maps. Moreover, properties of tangent cones and tangent sets are quite decisive in giving calculus rules for derivatives and epiderivatives of set-valued maps. Furthermore, optimality conditions in set-valued optimization are also most conveniently expressed by using tangent cones and tangent sets. Sensitivity analysis, constraints qualifications, and many other issues in set-valued optimization heavily rely on tangent cones and tangent sets.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   59.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Abadie, J.: On the Kuhn-Tucker theorem. In: Nonlinear Programming (NATO Summer School, Menton, 1964), pp. 19–36. North-Holland, Amsterdam (1967)

    Google Scholar 

  2. Anh, N.L., Khanh, P.Q., Tung, L.T.: Variational sets: calculus and applications to nonsmooth vector optimization. Nonlinear Anal. 74(6), 2358–2379 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aragón Artacho, F.J., Mordukhovich, B.S.: Enhanced metric regularity and Lipschitzian properties of variational systems. J. Global Optim. 50(1), 145–167 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aubin, J.P., Cellina, A.: Differential inclusions. Set-Valued Maps and Viability Theory [Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences)], vol. 264. Springer, Berlin (1984)

    Google Scholar 

  5. Aubin, J.P., Ekeland, I.: Applied nonlinear analysis. Pure and Applied Mathematics (New York). Wiley, New York (1984)

    Google Scholar 

  6. Aubin, J.P., Frankowska, H.: Set-valued analysis. Systems & Control: Foundations & Applications, vol. 2. Birkhäuser Boston, Boston (1990)

    Google Scholar 

  7. Avakov, E.R.: Necessary conditions for an extremum for smooth abnormal problems with constraints of equality and inequality type. Mat. Zametki 45(6), 3–11, 110 (1989)

    Google Scholar 

  8. Bazaraa, M., Goode, J., Shetty, C.: Constraint qualifications revisited. Manag. Sci. 18, 567–573 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bazaraa, M.S., Goode, J.J., Nashed, M.Z.: On the cones of tangents with applications to mathematical programming. J. Optim. Theory Appl. 13, 389–426 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bazaraa, M.S., Shetty, C.M., Goode, J.J., Nashed, M.Z.: Nonlinear programming without differentiability in Banach spaces: necessary and sufficient constraint qualification. Applicable Anal. 5(3), 165–173 (1975/76)

    Google Scholar 

  11. Bazarra, M., Shetty, C.: Foundations of Optimization. Springer, New York (1976)

    Book  Google Scholar 

  12. Ben-Israel, A., Ben-Tal, A., Zlobec, S.: Optimality in Nonlinear Programming: A Feasible Directions Approach. Wiley, New York (1981)

    MATH  Google Scholar 

  13. Ben-Tal, A., Zowe, J.: A unified theory of first and second order conditions for extremum problems in topological vector spaces. Math. Program. Stud. (19), 39–76 (1982)

    Article  MathSciNet  Google Scholar 

  14. Bigolin, F., Greco, G.H.: Geometric characterizations of C 1 manifolds in Euclidean spaces by tangent cones. J. Math. Anal. Appl. 396(1), 145–163 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. Springer Series in Operations Research. Springer, New York (2000)

    Book  MATH  Google Scholar 

  16. Borwein, J.M.: Tangent cones, starshape and convexity. Internat. J. Math. Math. Sci. 1(4), 459–477 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  17. Borwein, J.M.: Weak tangent cones and optimization in a Banach space. SIAM J. Control Optim. 16(3), 512–522 (1978)

    Article  MATH  Google Scholar 

  18. Borwein, J.M.: Stability and regular points of inequality systems. J. Optim. Theory Appl. 48(1), 9–52 (1986)

    MathSciNet  MATH  Google Scholar 

  19. Borwein, J.M.: Epi-Lipschitz-like sets in Banach space: theorems and examples. Nonlinear Anal. 11(10), 1207–1217 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  20. Borwein, J.M., Lucet, Y., Mordukhovich, B.: Compactly epi-Lipschitzian convex sets and functions in normed spaces. J. Convex Anal. 7(2), 375–393 (2000)

    MathSciNet  MATH  Google Scholar 

  21. Borwein, J.M., O’Brien, R.: Tangent cones and convexity. Can. Math. Bull. 19(3), 257–261 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. Borwein, J.M., Strójwas, H.M.: The hypertangent cone. Nonlinear Anal. 13(2), 125–144 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  23. Borwein, J.M., Strójwas, H.M.: Tangential approximations. Nonlinear Anal. 9(12), 1347–1366 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bouligand, G.: Sur l’existence des demi-tangentes à une courbe de Jordan. Fundamenta Math. 15, 215–218 (1030)

    Google Scholar 

  25. Bouligand, G.: Introduction á la géométrie Infinitseminale Directe. Gauthier-Villars, Paris (1932)

    Google Scholar 

  26. Cambini, A., Martein, L., Vlach, M.: Second order tangent sets and optimality conditions. Math. Jpn. 49(3), 451–461 (1999)

    MathSciNet  MATH  Google Scholar 

  27. Canon, M.D., Cullum Jr., C.D., Polak, E.: Theory of Optimal Control and Mathematical Programming. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  28. Castellani, M.: A necessary second-order optimality condition in nonsmooth mathematical programming. Oper. Res. Lett. 19(2), 79–86 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  29. Castellani, M., Pappalardo, M.: First-order cone approximations and necessary optimality conditions. Optimization 35(2), 113–126 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Castellani, M., Pappalardo, M.: Local second-order approximations and applications in optimization. Optimization 37(4), 305–321 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Clarke, F.H.: Optimization and nonsmooth analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1983)

    MATH  Google Scholar 

  32. Cominetti, R.: Metric regularity, tangent sets, and second-order optimality conditions. Appl. Math. Optim. 21(3), 265–287 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  33. Constantin, E.: Higher order necessary and sufficient conditions for optimality. Panamer. Math. J. 14(3), 1–25 (2004)

    MathSciNet  MATH  Google Scholar 

  34. Constantin, E.: Higher order necessary conditions in smooth constrained optimization. In: Communicating Mathematics, Contemp. Math., vol. 479, pp. 41–49. Amer. Math. Soc., Providence, RI (2009)

    Google Scholar 

  35. Constantin, E.: Second-order optimality conditions for problems with locally Lipschitz data via tangential directions. Comm. Appl. Nonlinear Anal. 18(2), 75–84 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Cornet, B.: Regularity properties of open tangent cones. Math. Program. Stud. (30), 17–33 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  37. Dancs, S.: Generalized tangent cone and an optimization problem in a normed space. J. Optim. Theory Appl. 67(1), 43–55 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  38. Demyanov, V.F., Rubinov, A.M.: Constructive nonsmooth analysis. Approximation & Optimization, vol. 7. Peter Lang, Frankfurt am Main (1995)

    Google Scholar 

  39. Di, S.: Classical optimality conditions under weaker assumptions. SIAM J. Optim. 6(1), 178–197 (1996)

    Article  MathSciNet  Google Scholar 

  40. Di, S., Poliquin, R.: Contingent cone to a set defined by equality and inequality constraints at a Fréchet differentiable point. J. Optim. Theory Appl. 81(3), 469–478 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  41. Di, S., Poliquin, R.A.: Optimality conditions for max-type nonsmooth minimization problems. Optimization 41(3), 219–236 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  42. Dolecki, S.: Tangency and differentiation: some applications of convergence theory. Ann. Mat. Pura Appl. (4) 130, 223–255 (1982)

    Google Scholar 

  43. Dolecki, S.: Tangency and differentiation: marginal functions. Adv. Appl. Math. 11(4), 389–411 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  44. Dolecki, S., Greco, G.H.: Towards historical roots of necessary conditions of optimality: regula of Peano. Control Cybern. 36(3), 491–518 (2007)

    MathSciNet  MATH  Google Scholar 

  45. Dolecki, S., Greco, G.H.: Tangency vis-à-vis differentiability by Peano, Severi and Guareschi. J. Convex Anal. 18(2), 301–339 (2011)

    MathSciNet  MATH  Google Scholar 

  46. Dontchev, A.L., Rockafellar, R.T.: Implicit functions and solution mappings. Springer Monographs in Mathematics. Springer, Dordrecht (2009)

    Book  MATH  Google Scholar 

  47. Durea, M., Strugariu, R.: Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions. J. Global Optim. 56(2), 587–603 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Elster, K.H., Thierfelder, J.: On cone approximation of sets. In: Mathematical Methods in Operations Research, Lectures Presented at the Summer School of Opearetions Research, Primorsko, pp. 33–59 (1984)

    Google Scholar 

  49. Elster, K.H., Thierfelder, J.: The general concept of cone approximations in nondifferentiable optimization. In: Nondifferentiable Optimization: Motivations and Applications (Sopron, 1984), Lecture Notes in Econom. and Math. Systems, vol. 255, pp. 170–189. Springer, Berlin (1985)

    Google Scholar 

  50. Elster, K.H., Thierfelder, J.: On cone approximations and generalized directional derivatives. In: Nonsmooth Optimization and Related Topics (Erice, 1988), Ettore Majorana Internat. Sci. Ser. Phys. Sci., vol. 43, pp. 133–154. Plenum, New York (1989)

    Google Scholar 

  51. Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  52. Flett, T.M.: Differential Analysis. Cambridge University Press, Cambridge (1980)

    Book  MATH  Google Scholar 

  53. Furukawa, N., Yoshinaga, Y.: Higher-order variational sets, variational derivatives and higher-order necessary conditions in abstract mathematical programming. Bull. Inform. Cybern. 23(1-2), 9–40 (1988)

    MathSciNet  MATH  Google Scholar 

  54. Ginchev, I., Guerraggio, A., Rocca, M.: Geoffrion type characterization of higher-order properly efficient points in vector optimization. J. Math. Anal. Appl. 328(2), 780–788 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. Ginchev, I., Guerraggio, A., Rocca, M.: Higher order properly efficient points in vector optimization. In: Generalized Convexity and Related Topics, Lecture Notes in Econom. and Math. Systems, vol. 583, pp. 227–245. Springer, Berlin (2007)

    Google Scholar 

  56. Giorgi, G., Guerraggio, A.: On a characterization of Clarke’s tangent cone. J. Optim. Theory Appl. 74(2), 369–372 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  57. Giorgi, G., Guerraggio, A.: On the notion of tangent cone in mathematical programming. Optimization 25(1), 11–23 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  58. Giorgi, G., Jiménez, B., Novo, V.: An overview of second order tangent sets and their application to vector optimization. Bol. Soc. Esp. Mat. Apl. SeMA (52), 73–96 (2010)

    MATH  Google Scholar 

  59. Hestenes, M.R.: Calculus of Variations and Optimal Control Theory. Wiley, New York (1966)

    MATH  Google Scholar 

  60. Hestenes, M.R.: Optimization Theory. Wiley-Interscience, New York (1975)

    MATH  Google Scholar 

  61. Hiriart-Urruty, J.B.: New concepts in nondifferentiable programming. Bull. Soc. Math. France Mém 60, 57–85 (1979)

    MathSciNet  MATH  Google Scholar 

  62. Hiriart-Urruty, J.B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4(1), 79–97 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  63. Hiriart-Urruty, J.B.: A note on the mean value theorem for convex functions. Boll. Un. Mat. Ital. B (5) 17(2), 765–775 (1980)

    Google Scholar 

  64. Hoffmann, K.H., Kornstaedt, H.J.: Higher-order necessary conditions in abstract mathematical programming. J. Optim. Theory Appl. 26(4), 533–568 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  65. Hu, S., Papageorgiou, N.S.: Handbook of multivalued analysis. Vol. I, Mathematics and Its Applications, vol. 419. Kluwer Academic, Dordrecht (1997)

    Google Scholar 

  66. Hu, S., Papageorgiou, N.S.: Handbook of multivalued analysis. Vol. II, Mathematics and Its Applications, vol. 500. Kluwer Academic, Dordrecht (2000)

    Google Scholar 

  67. Ioffe, A.D.: Approximate subdifferentials and applications. II. Mathematika 33(1), 111–128 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  68. Jiménez, B., Novo, V.: Second order necessary conditions in set constrained differentiable vector optimization. Math. Methods Oper. Res. 58(2), 299–317 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  69. Jiménez, B., Novo, V.: Optimality conditions in differentiable vector optimization via second-order tangent sets. Appl. Math. Optim. 49(2), 123–144 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  70. Jiménez, B., Novo, V.: Characterization of the cone of attainable directions. J. Optim. Theory Appl. 131(3), 493–499 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  71. Jofré, A., Penot, J.P.: Comparing new notions of tangent cones. J. Lond. Math. Soc. (2) 40(2), 280–290 (1989)

    Google Scholar 

  72. Jourani, A.: Regularity and strong sufficient optimality conditions in differentiable optimization problems. Numer. Funct. Anal. Optim. 14(1-2), 69–87 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  73. Jourani, A.: Metric regularity and second-order necessary optimality conditions for minimization problems under inclusion constraints. J. Optim. Theory Appl. 81(1), 97–120 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  74. Kawasaki, H.: An envelope-like effect of infinitely many inequality constraints on second-order necessary conditions for minimization problems. Math. Program. 41(1, (Ser. A)), 73–96 (1988)

    Google Scholar 

  75. Krabs, W.: Optimization and Approximation. Wiley, Chichester (1979)

    MATH  Google Scholar 

  76. Křivan, V.: On the intersection of contingent cones. J. Optim. Theory Appl. 70(2), 397–404 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  77. Lasiecka, I.: Conical approximations of sets in optimization theory. Control Cybern. 4(3-4), 39–58 (1975)

    MathSciNet  MATH  Google Scholar 

  78. Laurent, P.J.: Approximation et Optimisation, Collection Enseignement des Sciences, No. 13. Hermann, Paris (1972)

    Google Scholar 

  79. Ledzewicz, U., Schaettler, H.: Second-order conditions for extremum problems with nonregular equality constraints. J. Optim. Theory Appl. 86(1), 113–144 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  80. Ledzewicz, U., Schättler, H.: A second-order Dubovitskii-Milyutin theory and applications to control. In: Advances in Nonlinear Dynamics. Stability Control Theory Methods Appl., vol. 5, pp. 179–192. Gordon and Breach, Amsterdam (1997)

    Google Scholar 

  81. Ledzewicz, U., Walczak, S.: On the Lusternik theorem for nonsmooth operators. Nonlinear Anal. 22(2), 121–128 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  82. Ledzewicz, U., Walczak, S.: Generalizations of the Lusternik theorem and applications to nonsmooth and abnormal problems in optimization and optimal control. In: World Congress of Nonlinear Analysts ’92, Vol. I–IV (Tampa, FL, 1992), pp. 2583–2594. de Gruyter, Berlin (1996)

    Google Scholar 

  83. Li, S.J., Sun, X.K., Zhu, S.K.: Higher-order optimality conditions for strict minimality in set-valued optimization. J. Nonlinear Convex Anal. 13(2), 281–291 (2012)

    MathSciNet  MATH  Google Scholar 

  84. Martin, D.H., Gardner, R.J., Watkins, G.G.: Indicating cones and the intersection principle for tangential approximants in abstract multiplier rules. J. Optim. Theory Appl. 33(4), 515–537 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  85. Martin, D.H., Watkins, G.G.: Cores of tangent cones and Clarke’s tangent cone. Math. Oper. Res. 10(4), 565–575 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  86. Michel, P., Penot, J.P.: Calcul sous-différentiel pour des fonctions lipschitziennes et non lipschitziennes. C. R. Acad. Sci. Paris Sér. I Math. 298(12), 269–272 (1984)

    MathSciNet  MATH  Google Scholar 

  87. Motreanu, D., Pavel, N.: Tangency, flow invariance for differential equations, and optimization problems. Monographs and Textbooks in Pure and Applied Mathematics, vol. 219. Marcel Dekker, New York (1999)

    Google Scholar 

  88. Motreanu, D., Pavel, N.H.: Quasitangent vectors in flow-invariance and optimization problems on Banach manifolds. J. Math. Anal. Appl. 88(1), 116–132 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  89. Palata, J.: A survey of conical approximations used in the optimization. Optimization 20, 147–161 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  90. Pallaschke, D., Rolewicz, S.: Foundations of mathematical optimization. Mathematics and Its Applications, vol. 388. Kluwer Academic, Dordrecht (1997)

    Google Scholar 

  91. Pavel, N.: Second order differential equations on closed subsets of a Banach space. Boll. Un. Mat. Ital. (4) 12(3), 348–353 (1975)

    Google Scholar 

  92. Pavel, N.H., Huang, Y.K., Kim, J.K.: Higher order necessary conditions for optimization. Libertas Math. 14, 41–50 (1994)

    MathSciNet  MATH  Google Scholar 

  93. Pavel, N.H., Ursescu, C.: Flow-invariant sets for autonomous second order differential equations and applications in mechanics. Nonlinear Anal. 6(1), 35–74 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  94. Penot, J.P.: A characterization of tangential regularity. Nonlinear Anal. 5(6), 625–643 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  95. Penot, J.P.: Variations on the theme of nonsmooth analysis: another subdifferential. In: Nondifferentiable Optimization: Motivations and Applications (Sopron, 1984). Lecture Notes in Econom. and Math. Systems, vol. 255, pp. 41–54. Springer, Berlin (1985)

    Google Scholar 

  96. Penot, J.P.: Second-order generalized derivatives: comparisons of two types of epi-derivatives. In: Advances in Optimization (Lambrecht, 1991), Lecture Notes in Econom. and Math. Systems, vol. 382, pp. 52–76. Springer, Berlin (1992)

    Google Scholar 

  97. Penot, J.P.: Second-order generalized derivatives: comparisons of two types of epi-derivatives. In: Advances in Optimization (Lambrecht, 1991), Lecture Notes in Econom. and Math. Systems, vol. 382, pp. 52–76. Springer, Berlin (1992)

    Google Scholar 

  98. Penot, J.P.: Optimality conditions in mathematical programming and composite optimization. Math. Program. 67(2, Ser. A), 225–245 (1994)

    Google Scholar 

  99. Penot, J.P.: Second-order conditions for optimization problems with constraints. SIAM J. Control Optim. 37(1), 303–318 (electronic) (1999)

    Google Scholar 

  100. Penot, J.P.: Recent advances on second-order optimality conditions. In: Optimization (Namur, 1998), Lecture Notes in Econom. and Math. Systems, vol. 481, pp. 357–380. Springer, Berlin (2000)

    Google Scholar 

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

    Google Scholar 

  102. Rockafellar, R.T.: Clarke’s tangent cones and the boundaries of closed sets in \(\mathbb{R}^{n}\). Nonlinear Anal. Theory Methods Appl. 3, 145–154 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  103. Rockafellar, R.T.: Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32(2), 257–280 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  104. Rockafellar, R.T.: The Theory of Subgradients and Its Applications to Problems of Optimization, R & E, vol. 1. Heldermann Verlag, Berlin (1981)

    Google Scholar 

  105. Rockafellar, R.T.: First- and second-order epi-differentiability in nonlinear programming. Trans. Am. Math. Soc. 307(1), 75–108 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  106. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317. Springer, Berlin (1998)

    Google Scholar 

  107. Rodríguez-Marín, L., Sama, M.: (Λ, C)-contingent derivatives of set-valued maps. J. Math. Anal. Appl. 335(2), 974–989 (2007)

    Google Scholar 

  108. Sama, M.: Some remarks on the existence and computation of contingent epiderivatives. Nonlinear Anal. 71(7-8), 2997–3007 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  109. Schirotzek, W.: Nonsmooth Analysis. Universitext. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  110. Severi, F.: Sulla differenziabilità totale delle funzioni di più variabili reali. Ann. Mat. Pura Appl. 13(1), 1–35 (1934)

    Article  MathSciNet  Google Scholar 

  111. Treiman, J.S.: Shrinking generalized gradients. Nonlinear Anal. 12,no.(12), 1429–1450 (1988)

    Google Scholar 

  112. Taa, A.: Set-valued derivatives of multifunctions and optimality conditions. Numer. Funct. Anal. Optim. 19(1-2), 121–140 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  113. Thibault, L.: Tangent cones and quasi-interiorly tangent cones to multifunctions. Trans. Am. Math. Soc. 277(2), 601–621 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  114. Treiman, J.S.: Characterization of Clarke’s tangent and normal cones in finite and infinite dimensions. Nonlinear Anal. 7(7), 771–783 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  115. Ursescu, C.: Tangent sets’ calculus and necessary conditions for extremality. SIAM J. Control Optim. 20(4), 563–574 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  116. Ursescu, C.: Adjacent and tangent regularity. Optimization 30(2), 105–118 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  117. Varaiya, P.: Nonlinear programming in Banach spaces. SIAM J. Appl. Math. 15, 284–293 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  118. Vlach, M.: Approximation operators in optimization theory. Zeitschrift Oper. Res. 25, 15–23 (1981)

    MathSciNet  MATH  Google Scholar 

  119. Wang, Q., Li, S., Teo, K.: Higher-order optimality conditions for weakly efficient solutions in nonconvex set-valued optimization. Optim. Lett. 4, 425–437 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  120. Wang, Y.T.: A characterization of paratingent cone and P-subderivative with applications in nonsmooth analysis. Acta Math. Sinica (N.S.) 7(2), 181–192 (1991)

    Google Scholar 

  121. Ward, D.: Convex subcones of the contingent cone in nonsmooth calculus and optimization. Trans. Am. Math. Soc. 302(2), 661–682 (1987)

    Article  MATH  Google Scholar 

  122. Ward, D.: Isotone tangent cones and nonsmooth optimization. Optimization 18(6), 769–783 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  123. Ward, D.: The quantificational tangent cones. Can. J. Math. 40(3), 666–694 (1988)

    Article  Google Scholar 

  124. Ward, D.: Which subgradients have sum formulas? Nonlinear Anal. 12(11), 1231–1243 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  125. Ward, D.: Calculus for parabolic second-order derivatives. Set Valued Anal. 1(3), 213–246 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  126. Ward, D.: A chain rule for parabolic second-order epiderivatives. Optimization 28(3-4), 223–236 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  127. Ward, D.E.: A chain rule for first- and second-order epiderivatives and hypoderivatives. J. Math. Anal. Appl. 348(1), 324–336 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  128. Ward, D.E., Borwein, J.M.: Nonsmooth calculus in finite dimensions. SIAM J. Control Optim. 25, 1312–1340 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  129. Zeidler, E.: Nonlinear Functional Analysis and its Applications. Springer, New York (1990)

    Book  Google Scholar 

  130. Zheng, X.Y., Ng, K.F.: Metric subregularity and calmness for nonconvex generalized equations in Banach spaces. SIAM J. Optim. 20(5), 2119–2136 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  131. Zowe, J.: A remark on a regularity assumption for the mathematical programming problem in Banach spaces. J. Optim. Theory Appl. 25(3), 375–414 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rochester Institute of Technology School of Mathematical Sciences, Rochester, New York, USA

    Akhtar A. Khan

  2. Halle, Germany

    Christiane Tammer

  3. Faculty of Mathematics, University “Al. I. Cuza” Iasi, Iasi, Romania

    Constantin Zălinescu

Authors
  1. Akhtar A. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christiane Tammer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Constantin Zălinescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Khan, A.A., Tammer, C., Zălinescu, C. (2015). Tangent Cones and Tangent Sets. In: Set-valued Optimization. Vector Optimization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54265-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation