Preservation of convergence of convex sets and functions in finite dimensions

McLinden, L.; Bergstrom, Roy C.

doi:10.1090/S0002-9947-1981-0628449-5

Preservation of convergence of convex sets and functions in finite dimensions
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by L. McLinden and Roy C. Bergstrom PDF

Trans. Amer. Math. Soc. 268 (1981), 127-142 Request permission

Abstract:

We study a convergence notion which has particular relevance for convex analysis and lends itself quite naturally to successive approximation schemes in a variety of areas. Motivated particularly by problems in optimization subject to constraints, we develop technical tools necessary for systematic use of this convergence in finite-dimensional settings. Simple conditions are established under which this convergence for sequences of sets, functions and subdifferentials is preserved under various basic operations, including, for example, those of addition and infimal convolution in the case of functions.

References

Hédy Attouch, Familles d’opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl. (4) 120 (1979), 35–111 (French, with English summary). MR 551062, DOI 10.1007/BF02411939
Hédy Attouch and Yoshio Konishi, Convergence d’opérateurs maximaux monotones et inéquations variationnelles, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), no. 9, Ai, A467–A469. MR 430877

Optimization, convergence, and duality

Convergent sequences of dual convex programs

Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert

Une famille de topologies et de convergences sur l’ensemble des fonctionelles convexes

J.-L. Joly, Une famille de topologies sur l’ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. (9) 52 (1973), 421–441 (1974) (French). MR 500129
M. Matzeu, Su un tipo di continuità dell’operatore subdifferenziale, Boll. Un. Mat. Ital. B (5) 14 (1977), no. 2, 480–490 (Italian, with English summary). MR 0461235

Successive approximation and linear stability involving convergent sequences of optimization problems

Convergent sequences of minimax problems and saddle functions

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
R. Tyrrell Rockafellar, Convex analysis, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Reprint of the 1970 original; Princeton Paperbacks. MR 1451876
R. Tyrrell Rockafellar, Conjugate duality and optimization, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Lectures given at the Johns Hopkins University, Baltimore, Md., June, 1973. MR 0373611
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
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
David W. Walkup and Roger J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229–235. MR 209806, DOI 10.1090/S0002-9939-1967-0209806-6
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
R. A. Wijsman, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Math. Soc. 70 (1964), 186–188. MR 157278, DOI 10.1090/S0002-9904-1964-11072-7
R. A. Wijsman, Convergence of sequences of convex sets, cones and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32–45. MR 196599, DOI 10.1090/S0002-9947-1966-0196599-8