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Discussiones Mathematicae, Differential Inclusions, Control and Optimization

2001 | 21 | 2 | 235-247
Tytuł artykułu

On relations among the generalized second-order directional derivatives

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In the paper, we deal with the relations among several generalized second-order directional derivatives. The results partially solve the problem which of the second-order optimality conditions is more useful.
Słowa kluczowe
EN
Kategorie tematyczne
Rocznik
Tom
Numer
Strony
235-247
Opis fizyczny
Daty
wydano
2001
otrzymano
2001-09-10
Twórcy
autor
• Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University, Tř. Svobody 26, 771 46 Olomouc, Czech Republic
Bibliografia
• [1] J.M. Borwein, M. Fabián, On generic second-order Gâteaux differentiability, Nonlinear Anal. T.M.A 20 (1993), 1373-1382.
• [2] J.M. Borwein and D. Noll, Second-order differentiability of convex functions in Banach spaces, Trans. Amer. Math. Soc. 342 (1994), 43-81.
• [3] A. Ben-Tal and J. Zowe, Directional derivatives in nonsmooth optimization, J. Optim. Theory Appl. 47 (1985), 483-490.
• [4] F.H. Clarke, Necessary conditions for nonsmooth problems in optimal control and the calculus of variations, Ph.D. thesis, Univ. of Washington 1973.
• [5] F.H. Clarke, Optimization and nonsmooth analysis, J. Wiley, New York 1983.
• [6] R. Cominetti, Equivalence between the classes of $C^{1,1}$ and twice locally Lipschitzian functions, Ph.D. thesis, Université Blaise Pascal 1989.
• [7] R. Cominetti and R. Correa, A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim. 28 (1990), 789-809.
• [8] W.L. Chan, L.R. Huang and K.F. Ng, On generalized second-order derivatives and Taylor expansions in nosmooth optimiyation, SIAM J. Control Optim. 32 (1994), 789-809.
• [9] P.G. Georgiev and N.P. Zlateva, Second-order subdifferentials of $C^{1,1}$ functions and optimality conditions, Set-Valued Analysis 4 (1996), 101-117.
• [10] J.B. Hiriart-Urruty and C. Lemarchal, Convex Analysis nad Minimization Algorithms, Springer Verlag, Berlin 1993.
• [11] L.R. Huang and K.F. Ng, On some relations between Chaney's generalized second-order directional derivative and that of Ben-Tal and Zowe, SIAM J. Control Optim. 34 (1996), 1220-1234.
• [12] V. Jeyakumar and X.Q. Yang, Approximate generalized Hessians and Taylor's expansions for continously Gâteaux differentiable functions, Nonlinear Anal. T.M.A 36 (1999), 353-368.
• [13] G. Lebourg, Generic differentiability of Lipschitz functions, Trans. Amer. Math. Soc. 256 (1979), 125-144.
• [14] Y. Maruyama, Second-order necessary conditions for nonlinear problems in Banach spases and their applications to optimal control problems, Math. Programming 41 (1988), 73-96.
• [15] P. Michel and J.P. Penot, Second-order moderate derivatives, Nonlinear Anal. T.M.A 22 (1994), 809-821.
• [16] K. Pastor, Generalized second-order directional derivatives for locally Lipschitz functions, Preprint no. 17/2001 of Palacký Univerzity, Olomouc.
• [17] R.T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497-509.
• [18] R.T. Rockafellar, Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. Oper. Res. 14 (1989), 462-484.
• [19] X.Q. Yang, On relations and applications of generalized second-order directional derivatives, Non. Anal. T.M.A 36 (1999), 595-614.
• [20] X.Q. Yang, On second-order directional derivatives, Non. Anal. T.M.A 26, 55-66.
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Bibliografia
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