A characterization of epi-convergence in terms of convergence of level sets

Beer, Gerald; Rockafellar, R. T.; Wets, Roger J.-B.

doi:10.1090/S0002-9939-1992-1119262-6

A characterization of epi-convergence in terms of convergence of level sets
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by Gerald Beer, R. T. Rockafellar and Roger J.-B. Wets PDF

Proc. Amer. Math. Soc. 116 (1992), 753-761 Request permission

Abstract:

Let $\operatorname {LSC} (X)$ denote the extended real-valued lower semicontinuous functions on a separable metrizable space $X$. We show that a sequence $\left \langle {{f_n}} \right \rangle$ in $\operatorname {LSC} (X)$ is epi-convergent to $f \in \operatorname {LSC} (X)$ if and only for each real $\alpha$, the level set of height $\alpha$ of $f$ can be recovered as the Painlevé-Kuratowski limit of an appropriately chosen sequence of level sets of the ${f_n}$ at heights ${\alpha _n}$ approaching $\alpha$. Assuming the continuum hypothesis, this result fails without separability. An analogous result holds for weakly lower semicontinuous functions defined on a separable Banach space with respect to Mosco epi-convergence.

References

Jean-Pierre Aubin and Hélène Frankowska, Set-valued analysis, Systems & Control: Foundations & Applications, vol. 2, Birkhäuser Boston, Inc., Boston, MA, 1990. MR 1048347
H. Attouch, Variational convergence for functions and operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 773850

The topology of the

-Hausdorff distance

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
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
J. Baxter, G. Dal Maso, and U. Mosco, Stopping times and $\Gamma$-convergence, Trans. Amer. Math. Soc. 303 (1987), no. 1, 1–38. MR 896006, DOI 10.1090/S0002-9947-1987-0896006-7
Gerald Beer, Metric spaces with nice closed balls and distance functions for closed sets, Bull. Austral. Math. Soc. 35 (1987), no. 1, 81–96. MR 875510, DOI 10.1017/S000497270001306X
Gerald Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), no. 2, 239–253. MR 969914, DOI 10.1017/S0004972700027519
Gerald Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), no. 1, 117–126. MR 982400, DOI 10.1090/S0002-9939-1990-0982400-8
Gerald Beer and Jonathan M. Borwein, Mosco convergence and reflexivity, Proc. Amer. Math. Soc. 109 (1990), no. 2, 427–436. MR 1012924, DOI 10.1090/S0002-9939-1990-1012924-9
G. Beer and R. Lucchetti, Minima of quasi-convex functions, Optimization 20 (1989), no. 5, 581–596. MR 1015430, DOI 10.1080/02331938908843480
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
Gerald Beer and Devidas Pai, On convergence of convex sets and relative Chebyshev centers, J. Approx. Theory 62 (1990), no. 2, 147–169. MR 1068425, DOI 10.1016/0021-9045(90)90029-P
Jonathan M. Borwein and Simon Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), no. 3, 843–851. MR 969313, DOI 10.1090/S0002-9939-1989-0969313-4
J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472–476. MR 139135, DOI 10.1090/S0002-9939-1962-0139135-6
Sebastiano Francaviglia, Alojzy Lechicki, and Sandro Levi, Quasiuniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), no. 2, 347–370. MR 813603, DOI 10.1016/0022-247X(85)90246-X
Ennio De Giorgi, Convergence problems for functionals and operators, Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978) Pitagora, Bologna, 1979, pp. 131–188. MR 533166
K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
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, 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
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
Jean-Paul Penot, The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity, Proc. Amer. Math. Soc. 113 (1991), no. 1, 275–285. MR 1068129, DOI 10.1090/S0002-9939-1991-1068129-X
R. T. Rockafellar and Roger J.-B. Wets, Variational systems, an introduction, Multifunctions and integrands (Catania, 1983) Lecture Notes in Math., vol. 1091, Springer, Berlin, 1984, pp. 1–54. MR 785574, DOI 10.1007/BFb0098800
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

Convergence au sens de Mosco; théorie et applications à l’approximation des solutions d’inéquations

Epi-convergence et convergence des sections

Makoto Tsukada, Convergence of best approximations in a smooth Banach space, J. Approx. Theory 40 (1984), no. 4, 301–309. MR 740641, DOI 10.1016/0021-9045(84)90003-0

Contributions à la dualité en optimisation et à l’epi-convergence

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
Roger J.-B. Wets, A formula for the level sets of epi-limits and some applications, Mathematical theories of optimization (Genova, 1981) Lecture Notes in Math., vol. 979, Springer, Berlin-New York, 1983, pp. 256–268. MR 713816