On pseudo-differentiability

Cominetti, Roberto

doi:10.1090/S0002-9947-1991-0992605-3

On pseudo-differentiability

Author: Roberto Cominetti
Journal: Trans. Amer. Math. Soc. 324 (1991), 843-865
MSC: Primary 26B05; Secondary 49J45, 49J52, 58C20, 90C30
DOI: https://doi.org/10.1090/S0002-9947-1991-0992605-3
MathSciNet review: 992605
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Abstract | References | Similar Articles | Additional Information

Abstract: We present some new relations between the pseudo-derivatives and parabolic epiderivatives recently introduced by Rockafellar, and also its infinite dimensional counterparts. Significant extensions of the most important known results are proven, which further clarify the range of applicability of this new theory.

[1] H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
[2] Jean-Pierre Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), no. 1, 87–111. MR 736641, https://doi.org/10.1287/moor.9.1.87
[3] Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
[4] Alfred Auslender, Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim. 22 (1984), no. 2, 239–254. MR 732426, https://doi.org/10.1137/0322017
[5] A. Ben-Tal and J. Zowe, Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems, Math. Programming 24 (1982), no. 1, 70–91. MR 667940, https://doi.org/10.1007/BF01585095
[6] A. Ben-Tal and J. Zowe, A unified theory of first and second order conditions for extremum problems in topological vector spaces, Math. Programming Stud. 19 (1982), 39–76. Optimality and stability in mathematical programming. MR 669725, https://doi.org/10.1007/bfb0120982
[7] J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), no. 1, 9–52. MR 825383, https://doi.org/10.1007/BF00938588
[8] R. W. Chaney, Second-order sufficiency conditions for nondifferentiable programming problems, SIAM J. Control Optim. 20 (1982), no. 1, 20–33. MR 642177, https://doi.org/10.1137/0320004
[9] R. W. Chaney, On sufficient conditions in nonsmooth optimization, Math. Oper. Res. 7 (1982), no. 3, 463–475. MR 667935, https://doi.org/10.1287/moor.7.3.463
[10] Roberto Cominetti and Rafael Correa, Sur une dérivée du second ordre en analyse non différentiable, C. R. Acad. Sci. Paris Sér. I Math. 303 (1986), no. 17, 861–864 (French, with English summary). MR 870907
[11] R. Cominetti and R. Correa, A generalized second-order derivative in nonsmooth optimization, SIAM J. Control Optim. 28 (1990), no. 4, 789–809. MR 1051624, https://doi.org/10.1137/0328045
[12] Szymon Dolecki, Tangency and differentiation: some applications of convergence theory, Ann. Mat. Pura Appl. (4) 130 (1982), 223–255. MR 663973, https://doi.org/10.1007/BF01761497
[13] J.-B. Hiriart-Urruty, A new set-valued second order derivative for convex functions, FERMAT days 85: mathematics for optimization (Toulouse, 1985) North-Holland Math. Stud., vol. 129, North-Holland, Amsterdam, 1986, pp. 157–182. MR 874365, https://doi.org/10.1016/S0304-0208(08)72398-3
[14] Jean-Baptiste Hiriart-Urruty, Jean-Jacques Strodiot, and V. Hien Nguyen, Generalized Hessian matrix and second-order optimality conditions for problems with 𝐶^{1,1} data, Appl. Math. Optim. 11 (1984), no. 1, 43–56. MR 726975, https://doi.org/10.1007/BF01442169
[15] A. D. Ioffe, Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), no. 1, 1–56. MR 613784, https://doi.org/10.1090/S0002-9947-1981-0613784-7
[16] L. McLinden and Roy C. Bergstrom, Preservation of convergence of convex sets and functions in finite dimensions, Trans. Amer. Math. Soc. 268 (1981), no. 1, 127–142. MR 628449, https://doi.org/10.1090/S0002-9947-1981-0628449-5
[17] Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. MR 283586, https://doi.org/10.1016/0022-247X(71)90200-9
[18] J. L. Ndoutume, Epi-convergence en calcul différentiel généralisé, Résultats de convergence et approximation, Thesis, Université de Perpignan, 1987.
[19] Jean-Paul Penot, On regularity conditions in mathematical programming, Math. Programming Stud. 19 (1982), 167–199. Optimality and stability in mathematical programming. MR 669731, https://doi.org/10.1007/bfb0120988
[20] -, Generalized higher order derivatives and higher order optimality conditions, preprint, 1985.
[21] Stephen M. Robinson, Stability theory for systems of inequalities. II. Differentiable nonlinear systems, SIAM J. Numer. Anal. 13 (1976), no. 4, 497–513. MR 410522, https://doi.org/10.1137/0713043
[22] Stephen M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), no. 2, 130–143. MR 430181, https://doi.org/10.1287/moor.1.2.130
[23] R. T. Rockafellar, Maximal monotone relations and the second derivatives of nonsmooth functions, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 167–184 (English, with French summary). MR 797269
[24] R. T. Rockafellar, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), no. 1, 75–108. MR 936806, https://doi.org/10.1090/S0002-9947-1988-0936806-9
[25] Gabriella Salinetti and Roger J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), no. 1, 211–226. MR 479398, https://doi.org/10.1016/0022-247X(77)90060-9
[26] A. Seeger, Analyse du second ordre de problèmes non différentiables, Thesis, Université Paul Sabatier, 1986.