Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings

Levy, A. B.; Poliquin, R.; Thibault, L.

doi:10.1090/S0002-9947-1995-1290725-3

Partial extensions of Attouch’s theorem with applications to proto-derivatives of subgradient mappings
HTML articles powered by AMS MathViewer

by A. B. Levy, R. Poliquin and L. Thibault PDF

Trans. Amer. Math. Soc. 347 (1995), 1269-1294 Request permission

Abstract:

Attouch’s Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain difference quotient mappings, and Attouch’s Theorem has been used to relate these various generalized derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems. We prove in a Hilbert space several "partial extensions" of Attouch’s Theorem to functions more general than convex; these functions are called primal-lower-nice. Furthermore, we use our extensions to derive a relationship between the second-order epi-derivatives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.

References

H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850
H. Attouch and R. J.-B. Wets, Epigraphical analysis, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. suppl., 73–100. Analyse non linéaire (Perpignan, 1987). MR 1019109, DOI 10.1016/S0294-1449(17)30036-7
H. Attouch, J. L. Ndoutoume, and M. Théra, Epigraphical convergence of functions and convergence of their derivatives in Banach spaces, Sém. Anal. Convexe 20 (1990), Exp. No. 9, 45. MR 1114679
Hédy Attouch, Roberto Lucchetti, and Roger J.-B. Wets, The topology of the $\rho$-Hausdorff distance, Ann. Mat. Pura Appl. (4) 160 (1991), 303–320 (1992). MR 1163212, DOI 10.1007/BF01764131
Hédy Attouch and Gerald Beer, On the convergence of subdifferentials of convex functions, Arch. Math. (Basel) 60 (1993), no. 4, 389–400. MR 1206324, DOI 10.1007/BF01207197
Hédy Attouch and Roger J.-B. Wets, Quantitative stability of variational systems. I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), no. 2, 695–729. MR 1018570, DOI 10.1090/S0002-9947-1991-1018570-0
Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, Optimization 21 (1990), no. 4, 521–534. MR 1069660, DOI 10.1080/02331939008843576
Gerald Beer and Roberto Lucchetti, The epi-distance topology: continuity and stability results with applications to convex optimization problems, Math. Oper. Res. 17 (1992), no. 3, 715–726. MR 1177732, DOI 10.1287/moor.17.3.715
Gerald Beer and Roberto Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991), no. 2, 795–813. MR 1012526, DOI 10.1090/S0002-9947-1991-1012526-X
J. M. Borwein and J. R. Giles, The proximal normal formula in Banach space, Trans. Amer. Math. Soc. 302 (1987), no. 1, 371–381 (English, with French summary). MR 887515, DOI 10.1090/S0002-9947-1987-0887515-5
J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), no. 2, 517–527. MR 902782, DOI 10.1090/S0002-9947-1987-0902782-7
F. H. Clarke, Optimization and nonsmooth analysis, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1989. Reprint of the 1983 original. MR 1019086
Frank H. Clarke, Methods of dynamic and nonsmooth optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 57, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1085948, DOI 10.1137/1.9781611970142
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
Roberto Cominetti, On pseudo-differentiability, Trans. Amer. Math. Soc. 324 (1991), no. 2, 843–865. MR 992605, DOI 10.1090/S0002-9947-1991-0992605-3
R. Correa, A. Jofré, and L. Thibault, Characterization of lower semicontinuous convex functions, Proc. Amer. Math. Soc. 116 (1992), no. 1, 67–72. MR 1126193, DOI 10.1090/S0002-9939-1992-1126193-4
Chi Ngoc Do, Generalized second-order derivatives of convex functions in reflexive Banach spaces, Trans. Amer. Math. Soc. 334 (1992), no. 1, 281–301. MR 1088019, DOI 10.1090/S0002-9947-1992-1088019-1

Second-order variational analysis with applications to sensitivity in optimization

Philip D. Loewen, The proximal subgradient formula in Banach space, Canad. Math. Bull. 31 (1988), no. 3, 353–361. MR 956368, DOI 10.4153/CMB-1988-051-9
Jean-Paul Penot, On the convergence of subdifferentials of convex functions, Nonlinear Anal. 21 (1993), no. 2, 87–101. MR 1233335, DOI 10.1016/0362-546X(93)90040-Y
René A. Poliquin, Proto-differentiation of subgradient set-valued mappings, Canad. J. Math. 42 (1990), no. 3, 520–532. MR 1062743, DOI 10.4153/CJM-1990-027-2
René A. Poliquin, Integration of subdifferentials of nonconvex functions, Nonlinear Anal. 17 (1991), no. 4, 385–398. MR 1123210, DOI 10.1016/0362-546X(91)90078-F
René A. Poliquin, An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions, Trans. Amer. Math. Soc. 332 (1992), no. 2, 861–874. MR 1145732, DOI 10.1090/S0002-9947-1992-1145732-5
R. A. Poliquin and R. T. Rockafellar, Amenable functions in optimization, Nonsmooth optimization: methods and applications (Erice, 1991) Gordon and Breach, Montreux, 1992, pp. 338–353. MR 1263511
R. A. Poliquin and R. T. Rockafellar, A calculus of epi-derivatives applicable to optimization, Canad. J. Math. 45 (1993), no. 4, 879–896. MR 1227665, DOI 10.4153/CJM-1993-050-7
R. A. Poliquin and R. T. Rockafellar, Proto-derivative formulas for basic subgradient mappings in mathematical programming, Set-Valued Anal. 2 (1994), no. 1-2, 275–290. Set convergence in nonlinear analysis and optimization. MR 1285834, DOI 10.1007/BF01027106
René Poliquin, Jon Vanderwerff, and Václav Zizler, Convex composite representation of lower semicontinuous functions and renormings, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 6, 545–549 (English, with English and French summaries). MR 1240796

Convex analysis

R. T. Rockafellar, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), no. 3, 424–436. MR 629642, DOI 10.1287/moor.6.3.424
R. Tyrrell Rockafellar, Favorable classes of Lipschitz-continuous functions in subgradient optimization, Progress in nondifferentiable optimization, IIASA Collaborative Proc. Ser. CP-82, vol. 8, Internat. Inst. Appl. Systems Anal., Laxenburg, 1982, pp. 125–143. MR 704977
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
R. T. Rockafellar, Proto-differentiability of set-valued mappings and its applications in optimization, Ann. Inst. H. Poincaré C Anal. Non Linéaire 6 (1989), no. suppl., 449–482. Analyse non linéaire (Perpignan, 1987). MR 1019126, DOI 10.1016/S0294-1449(17)30034-3
R. T. Rockafellar, Perturbation of generalized Kuhn-Tucker points in finite-dimensional optimization, Nonsmooth optimization and related topics (Erice, 1988) Ettore Majorana Internat. Sci. Ser.: Phys. Sci., vol. 43, Plenum, New York, 1989, pp. 393–402. MR 1034071
R. Tyrrell Rockafellar, Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives, Math. Oper. Res. 14 (1989), no. 3, 462–484. MR 1008425, DOI 10.1287/moor.14.3.462
R. T. Rockafellar, Generalized second derivatives of convex functions and saddle functions, Trans. Amer. Math. Soc. 322 (1990), no. 1, 51–77. MR 1031242, DOI 10.1090/S0002-9947-1990-1031242-0
R. T. Rockafellar, Nonsmooth analysis and parametric optimization, Methods of nonconvex analysis (Varenna, 1989) Lecture Notes in Math., vol. 1446, Springer, Berlin, 1990, pp. 137–151. MR 1079762, DOI 10.1007/BFb0084934
R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
Lionel Thibault and Dariusz Zagrodny, Integration of subdifferentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995), no. 1, 33–58. MR 1312029, DOI 10.1006/jmaa.1995.1003
Tullio Zolezzi, Convergence of generalized gradients, Set-Valued Anal. 2 (1994), no. 1-2, 381–393. Set convergence in nonlinear analysis and optimization. MR 1285841, DOI 10.1007/BF01027113