On pseudo-differentiability

Cominetti, Roberto

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

On pseudo-differentiability
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Trans. Amer. Math. Soc. 324 (1991), 843-865 Request permission

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.

References

H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
Jean-Pierre Aubin, Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), no. 1, 87–111. MR 736641, DOI 10.1287/moor.9.1.87
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
Alfred Auslender, Stability in mathematical programming with nondifferentiable data, SIAM J. Control Optim. 22 (1984), no. 2, 239–254. MR 732426, DOI 10.1137/0322017
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, DOI 10.1007/BF01585095
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, DOI 10.1007/bfb0120982
J. M. Borwein, Stability and regular points of inequality systems, J. Optim. Theory Appl. 48 (1986), no. 1, 9–52. MR 825383, DOI 10.1007/BF00938588
R. W. Chaney, Second-order sufficiency conditions for nondifferentiable programming problems, SIAM J. Control Optim. 20 (1982), no. 1, 20–33. MR 642177, DOI 10.1137/0320004
R. W. Chaney, On sufficient conditions in nonsmooth optimization, Math. Oper. Res. 7 (1982), no. 3, 463–475. MR 667935, DOI 10.1287/moor.7.3.463
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
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, DOI 10.1137/0328045
Szymon Dolecki, Tangency and differentiation: some applications of convergence theory, Ann. Mat. Pura Appl. (4) 130 (1982), 223–255. MR 663973, DOI 10.1007/BF01761497
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, DOI 10.1016/S0304-0208(08)72398-3
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, DOI 10.1007/BF01442169
A. D. Ioffe, Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), no. 1, 1–56. MR 613784, DOI 10.1090/S0002-9947-1981-0613784-7
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, DOI 10.1090/S0002-9947-1981-0628449-5
Umberto Mosco, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535. MR 283586, DOI 10.1016/0022-247X(71)90200-9

Epi-convergence en calcul différentiel généralisé, Résultats de convergence et approximation

Jean-Paul Penot, On regularity conditions in mathematical programming, Math. Programming Stud. 19 (1982), 167–199. Optimality and stability in mathematical programming. MR 669731, DOI 10.1007/bfb0120988

Generalized higher order derivatives and higher order optimality conditions

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, DOI 10.1137/0713043
Stephen M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), no. 2, 130–143. MR 430181, DOI 10.1287/moor.1.2.130
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
R. T. Rockafellar, First- and second-order epi-differentiability in nonlinear programming, Trans. Amer. Math. Soc. 307 (1988), no. 1, 75–108. MR 936806, DOI 10.1090/S0002-9947-1988-0936806-9
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, DOI 10.1016/0022-247X(77)90060-9

Analyse du second ordre de problèmes non différentiables