An extension of Attouch’s theorem and its application to second-order epi-differentiation of convexly composite functions
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- by René A. Poliquin PDF
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Abstract:
In 1977, Hedy Attouch established that a sequence of (closed proper) convex functions epi-converges to a convex function if and only if the graphs of the subdifferentials converge (in the Mosco sense) to the subdifferential of the limiting function and (roughly speaking) there is a condition that fixes the constant of integration. We show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping (in addition a constraint qualification is imposed). Using Attouch’s Theorem, Rockafellar showed that second-order epi-differentiation of a convex function and proto-differentiability of the subdifferential set-valued mapping are equivalent, moreover the subdifferential of one-half the second-order epi-derivative is the proto-derivative of the subdifferential mapping; we will extend this result to the convexly composite setting.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
- 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 —, Optimization and nonsmooth analysis, Centre de Recherches Mathématiques, Université de Montréal, 1983.
- 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
- Philip D. Loewen, The proximal normal formula in Hilbert space, Nonlinear Anal. 11 (1987), no. 9, 979–995. MR 907818, DOI 10.1016/0362-546X(87)90079-4
- 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 J. L. Ndoutoume, Conditions nécessaires d’optimalité du premier ordre pour des problèmes de contrôle optimal d’inéquations variationnelles, Publications AVAMAC, Université Perpignan, Mathématiques, 66025 Cedex, 1987.
- Umberto Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math. 3 (1969), 510–585. MR 298508, DOI 10.1016/0001-8708(69)90009-7
- René A. Poliquin, Subgradient monotonicity and convex functions, Nonlinear Anal. 14 (1990), no. 4, 305–317. MR 1040008, DOI 10.1016/0362-546X(90)90167-F
- 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, 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 —, Proto-differentiation and integration of proximal subgradients, Dissertation, Univ. of Washington, June 1988. R. Poliquin and R. T. Rockafellar, Proto-derivative of solution mappings in parametric optimization, manuscript. —, A calculus of epi-derivatives applicable to optimization, manuscript. —, Amenable functions in optimization, Proc. Internat. School of Math. G. Stampacchia, 10th course: Nonsmooth Optimization: Methods and Applications (to appear).
- 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. 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 —, Convex analysis, Princeton Univ. Press, 1970.
- R. Tyrrell Rockafellar, La théorie des sous-gradients et ses applications à l’optimisation, Collection “Chaire Aisenstadt”, Presses de l’Université de Montréal, Montreal, Que., 1979 (French). Fonctions convexes et non convexes; Translated from the English by Godeliève Vanderstraeten-Tilquin. MR 531033
- 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. 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 —, Perturbation of Kuhn-Tucker points in finite-dimensional optimization, Nonsmooth Optimization and Related Topics, (F. Giannessi, Ed.), Plenum, 1989.
- 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. 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
- Gabriella Salinetti and Roger J.-B. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), no. 1, 18–33. MR 516381, DOI 10.1137/1021002
- R. J.-B. Wets, Convergence of convex functions, variational inequalities and convex optimization problems, Variational inequalities and complementarity problems (Proc. Internat. School, Erice, 1978) Wiley, Chichester, 1980, pp. 375–403. MR 578760
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 332 (1992), 861-874
- MSC: Primary 49J52; Secondary 49J45, 58C06
- DOI: https://doi.org/10.1090/S0002-9947-1992-1145732-5
- MathSciNet review: 1145732