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

Authors:
A. B. Levy, R. Poliquin and L. Thibault

Journal:
Trans. Amer. Math. Soc. **347** (1995), 1269-1294

MSC:
Primary 49J52; Secondary 58C20

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290725-3

MathSciNet review:
1290725

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**H. Attouch,*Variational convergence for functions and operators*, Pitman, 1984. MR**773850 (86f:49002)****[2]**H. Attouch, and R. J.-B. Wets,*Epigraphical analysis*, Analyse Non Linéaire**6**(1989), 73-100. MR**1019109 (91g:90161)****[3]**H. Attouch, J.L. Ndoutoume, and M. Thera,*Epigraphical convergence of functions and convergence of their derivatives in Banach spaces*, Exp. No. 9, Sem. Anal. Convexe (Montpellier)**20**(1990), 9.1-9.45. MR**1114679 (92i:46050)****[4]**H. Attouch, R. Lucchetti and R. J.-B. Wets,*The topology of the*-*Hausdorff distance*, Ann. Mat. Pura Appl.**160**(1991), 303-320. MR**1163212 (93d:54022)****[5]**H. Attouch and G. Beer,*On the convergence of subdifferentials of convex functions*, Arch. Math.**60**(1993), 389-400. MR**1206324 (94b:49018)****[6]**H. Attouch, and R. J.-B. Wets,*Quantitative stability of variational systems*: I.*The epigraphical distance*, Trans. Amer. Math. Soc.**328**(1991), 695-729. MR**1018570 (92c:90111)****[7]**J. P. Aubin and H. Frankowska,*Set-valued analysis*, Birkhäuser, 1990. MR**1048347 (91d:49001)****[8]**D. Azé and J.P. Penot,*Operations on convergent families of sets and functions*, Optimization**21**(1990), 521-534. MR**1069660 (92b:49022)****[9]**G. Beer and R. Lucchetti,*The epi-distance topology: continuity and stability results with applications to convex optimization problems*, Math. Oper. Res.**17**(1992), 715-726. MR**1177732 (93k:49011)****[10]**G. Beer and R. Lucchetti,*Convex optimization and the epi-distance topology*, Trans. Amer. Math. Soc.**327**(1991), 795-813. MR**1012526 (92a:49018)****[11]**J.M. Borwein and J.R. Gilles,*The proximal normal formula in Banach space*, Trans. Amer. Math. Soc.**302**(1987), 371-381. MR**887515 (88m:49013)****[12]**J.M. Borwein and D. Preiss,*A smooth variational principle with applications to subdifferentials and to differentiability of convex functions*, Trans. Amer. Math. Soc.**303**(1987), 517-527. MR**902782 (88k:49013)****[13]**F.H. Clarke,*Optimization and nonsmooth analysis*, Centre de Recherches Mathématiques, Université de Montréal (C.P. 6128 "A", Montrál, Québec, Canada, H3C 3J7), 1989. MR**1019086 (90g:49011)****[14]**-,*Methods of dynamic and nonsmooth optimization*, CBMS-NSF Regional Conference Series in Applied Mathematics, no. 57, 1989. MR**1085948 (91j:49001)****[15]**R. Cominetti and R. Correa,*A generalized second order derivative in nonsmooth optimization*, SIAM J. Control Optim.**28**(1990), 789-809. MR**1051624 (91h:49017)****[16]**R. Cominetti,*On pseudo-differentiability*, Trans. Amer. Math. Soc.**324**(1991), 843-865. MR**992605 (91h:26009)****[17]**R. Correa, A. Joffre and L. Thibault,*Characterization of lower semicontinuous convex functions*, Proc. Amer. Math. Soc.**116**(1992), 61-72. MR**1126193 (92k:49027)****[18]**C. Do,*Generalized second derivatives of convex functions in reflexive Banach spaces*, Trans. Amer. Math. Soc.**334**(1992), 281-301. MR**1088019 (93a:49011)****[19]**A. Levy,*Second-order variational analysis with applications to sensitivity in optimization*, Ph.D. Thesis, University of Washington, 1994.**[20]**P.D. Loewen,*The proximal subgradient formula in Banach space*, Canad. Math. Bull.**31**(1988), 353-361. MR**956368 (89k:49010)****[21]**J.P. Penot,*On the convergence of subdifferentials of convex functions*, Nonlinear Anal. Th. Meth. Appl.**21**(1993), 87-101. MR**1233335 (94m:49024)****[22]**R.A. Poliquin,*Proto-differentiation of subgradient set-valued mappings*, Canad. J. Math.**42**(1990), 520-532. MR**1062743 (91g:49007)****[23]**-,*Integration of subdifferentials of nonconvex functions*, Nonlinear Anal. Th. Meth. Appl.**17**(1991), 385-398. MR**1123210 (92j:49008)****[24]**-,*An extension of Attouch's Theorem and its application to second-order epi-differentiation of convexly composite functions*, Trans. Amer. Math. Soc.**332**(1992), 861-874. MR**1145732 (93a:49013)****[25]**R.A. Poliquin and R.T. Rockafellar,*Amenable functions in optimization*, Nonsmooth Optimization Methods and Applications (F. Giannessi, ed.), Gordon & Breach, Philadelphia, PA, 1992, pp. 338-353. MR**1263511 (95d:49033)****[26]**-,*A calculus of epi-derivatives applicable to optimization*, Canad. J. Math.**45**(1993), 879-896. MR**1227665 (94d:49023)****[27]**-,*Proto-derivative formulas for basic subgradient mappings in mathematical programming*, Set-Valued Analysis**2**(1994), 275-290. MR**1285834 (95c:49024)****[28]**R.A. Poliquin, J.Vanderwerff and V. Zizler,*Convex composite representation of lower semicontinuous functions and renormings*, C.R. Acad. Sci. Paris Sér. I**317**(1993), 545-549. MR**1240796 (94i:46102)****[29]**R.T. Rockafellar,*Convex analysis*, Princeton Univ. Press, Princeton, NJ, 1970.**[30]**-,*Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization*, Math. Oper. Res.**6**(1981), 424-436. MR**629642 (83m:90088)****[31]**-,*Favorable classes of Lipschitz-continuous functions in subgradient optimization*, Progress in Nondifferentiable Optimization (E.Nurminski, ed.), IIASA Collaborative Proceedings Series, International Institute of Applied Systems Analysis, Laxenburg, Austria, 1982, pp. 125-144. MR**704977 (85e:90069)****[32]**-,*First- and second-order epi-differentiability in nonlinear programming*, Trans. Amer. Math. Soc.**307**(1988), 75-107. MR**936806 (90a:90216)****[33]**-,*Proto-differentiability of set-valued mappings and its applications in optimization*, Analyse Non Linéaire (H. Attouch et al., eds.), Gauthier-Villars, Paris, 1989, pp. 449-482. MR**1019126 (90k:90140)****[34]**-,*Perturbation of generalized Kuhn-Tucker points in finite dimensional optimization*, Nonsmooth Optimization and Related Topics (F.H. Clarke et al., eds.), Plenum Press, 1989, pp. 393-402. MR**1034071 (91e:90113)****[35]**-,*Second-order optimality conditions in nonlinear programming obtained by way of epi-derivatives*, Math. Oper. Res.**14**(1989), 462-484. MR**1008425 (91b:49022)****[36]**-,*Generalized second derivatives of convex functions and saddle functions*, Trans. Amer. Math. Soc.**320**(1990), 810-822. MR**1031242 (91b:90190)****[37]**-,*Nonsmooth analysis and parametric optimization*, Methods of Nonconvex Analysis (A. Cellina, ed.), Lecture Notes in Math., vol. 1446, Springer-Verlag, 1990, pp. 137-151. MR**1079762 (91i:49016)****[38]**R.T. Rockafellar and R. J.-B. Wets,*Variational analysis*(to appear). MR**1491362 (98m:49001)****[39]**L. Thibault and D. Zagrodny,*Integration of subdifferentials of lower semicontinuous functions*, J. Math. Anal. Appl (to appear). MR**1312029 (95i:49032)****[40]**T. Zolezzi,*Convergence of generalized gradients*, preprint. MR**1285841 (95f:49016)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1290725-3

Keywords:
Attouch's Theorem,
epi-derivatives,
Mosco epi-convergence,
primal lower-nice functions,
Attouch-Wets convergence,
graph convergence,
nonsmooth analysis,
sensitivity analysis,
optimization,
proto-derivatives,
Painlevé-Kuratowski convergence,
set-valued analysis

Article copyright:
© Copyright 1995
American Mathematical Society