Approximate subdifferentials and applications. I. The finite-dimensional theory

Ioffe, A. D.

doi:10.1090/S0002-9947-1984-0719677-1

Approximate subdifferentials and applications. I. The finite-dimensional theory
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Trans. Amer. Math. Soc. 281 (1984), 389-416 Request permission

Erratum: Trans. Amer. Math. Soc. 288 (1985), 429.

Abstract:

We introduce and study a new class of subdifferentials associated with arbitrary functions. Among the questions considered are: connection with other derivative-like objects (e.g. derivatives, convex subdifferentials, generalized gradients of Clarke and derivate containers of Warga), calculus of approximate subdifferentials and applications to analysis of set-valued maps and to optimization. It turns out that approximate subdifferentials are minimal (as sets) among other conceivable subdifferentials satisfying some natural requirements. This shows that certain results involving approximate subdifferentials are the best possible and, at the same time, marks certain limitations of nonsmooth analysis. Another important property of approximate subdifferentials is that, being essentially nonconvex, they admit a rich calculus that covers the calculus of convex subdifferentials and leads to more precise and sometimes new results for generalized gradients of Clarke.

References

Jean-Pierre Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 159–229. MR 634239

Lipschitz behavior of solutions to convex minimization problems

Frank H. Clarke, Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247–262. MR 367131, DOI 10.1090/S0002-9947-1975-0367131-6
Frank H. Clarke, Generalized gradients of Lipschitz functionals, Adv. in Math. 40 (1981), no. 1, 52–67. MR 616160, DOI 10.1016/0001-8708(81)90032-3
J.-B. Hiriart-Urruty, Refinements of necessary optimality conditions in nondifferentiable programming. I, Appl. Math. Optim. 5 (1979), no. 1, 63–82. MR 526428, DOI 10.1007/BF01442544
Alexander D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61–69. MR 531969, DOI 10.1090/S0002-9947-1979-0531969-6
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
Alexandre D. Ioffe, Sous-différentielles approchées de fonctions numériques, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 14, 675–678 (French, with English summary). MR 618883

Calculus of Dini subdifferentials

Approximate subdifferentials of non-convex functions

Aleksandr D. Ioffe, On subdifferentiability spaces, Fifth international conference on collective phenomena, Ann. New York Acad. Sci., vol. 410, New York Acad. Sci., New York, 1983, pp. 107–119. MR 775520, DOI 10.1111/j.1749-6632.1983.tb23308.x

Calculus of generalized derivatives

Generalized normals and derivatives and necessary conditions for an extremum in non-differentiable programming

B. Sh. Mordukhovich, Maximum principle in the problem of time optimal response with nonsmooth constraints, Prikl. Mat. Meh. 40 (1976), no. 6, 1014–1023 (Russian); English transl., J. Appl. Math. Mech. 40 (1976), no. 6, 960–969 (1977). MR 0487669, DOI 10.1016/0021-8928(76)90136-2
B. Š. Morduhovič, Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR 254 (1980), no. 5, 1072–1076 (Russian). MR 592682
Jean-Paul Penot, Calcul sous-différentiel et optimisation, J. Functional Analysis 27 (1978), no. 2, 248–276 (French). MR 481218, DOI 10.1016/0022-1236(78)90030-7

Convex analysis

R. T. Rockafellar, Directionally Lipschitzian functions and subdifferential calculus, Proc. London Math. Soc. (3) 39 (1979), no. 2, 331–355. MR 548983, DOI 10.1112/plms/s3-39.2.331
R. T. Rockafellar, Generalized directional derivatives and subgradients of nonconvex functions, Canadian J. Math. 32 (1980), no. 2, 257–280. MR 571922, DOI 10.4153/CJM-1980-020-7
Jack Warga, Derivative containers, inverse functions, and controllability, Calculus of variations and control theory (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975; dedicated to Laurence Chisholm Young on the occasion of his 70th birthday), Publ. Math. Res. Center Univ. Wisconsin, No. 36, Academic Press, New York, 1976, pp. 13–45; errata, p. 46. MR 0427561