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Transactions of the American Mathematical Society

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Preservation of convergence of convex sets and functions in finite dimensions


Authors: L. McLinden and Roy C. Bergstrom
Journal: Trans. Amer. Math. Soc. 268 (1981), 127-142
MSC: Primary 26B25; Secondary 65K10
DOI: https://doi.org/10.1090/S0002-9947-1981-0628449-5
MathSciNet review: 628449
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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.


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  • [1] H. Attouch, Familles d'opérateurs maximaux monotones et mesurabilité, Ann. Mat. Pura Appl. 120 (1979), 35-111. MR 551062 (81h:47046)
  • [2] H. Attouch and Y. Konishi, Convergence d'opérateurs maximaux monotones et inéquations variationelles, C. R. Acad. Sci. Paris Sér. A-B 282 (1976), A467-A469. MR 0430877 (55:3882)
  • [3] R. C. Bergstrom, Optimization, convergence, and duality, Thesis, Univ. of Illinois at Urbana-Champaign, 1980.
  • [4] R. C. Bergstrom and L. McLinden, Convergent sequences of dual convex programs (submitted).
  • [5] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.
  • [6] J. L. Joly, Une famille de topologies et de convergences sur l'ensemble des fonctionelles convexes, Thesis, Univ. Scientifique et Médicale de Grenoble, Grenoble, 1970.
  • [7] -, Une famille de topologies sur l'ensembles des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. 52 (1973), 421-441. MR 0500129 (58:17826)
  • [8] M. Matzeu, Su un tipo de continuata' dell' operatore subdifferenziale, Boll. Un. Mat. Ital. B(5) 14 (1977), 480-490. MR 0461235 (57:1220)
  • [9] L. McLinden, Successive approximation and linear stability involving convergent sequences of optimization problems (submitted).
  • [10] -, Convergent sequences of minimax problems and saddle functions (in preparation).
  • [11] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969), 510-585. MR 0298508 (45:7560)
  • [12] -, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518-535. MR 0283586 (44:817)
  • [13] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N. J., 1970. MR 1451876 (97m:49001)
  • [14] -, Conjugate duality and optimization, Regional Conf. Ser. Appl. Math., no. 16, SIAM, Philadelphia, Pa., 1974. MR 0373611 (51:9811)
  • [15] G. Salinetti and R. J.-B. Wets, On the relations between two types of convergence for convex functions, J. Math. Anal. Appl. 60 (1977), 211-226. MR 0479398 (57:18828)
  • [16] -, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), 18-33. MR 516381 (80h:52007)
  • [17] D. W. Walkup and R. J.-B. Wets, Continuity of some convex-cone-valued mappings, Proc. Amer. Math. Soc. 18 (1967), 229-235. MR 0209806 (35:702)
  • [18] R. J.-B. Wets, Convergence of convex functions, variational inequalities, and convex optimization problems, Variational Inequalities and Complementarity Problems (R. W. Cottle, F. Giannessi and J.-L. Lions, eds.), Wiley, London, 1980, pp. 375-403. MR 578760 (83a:90140)
  • [19] R. A. Wijsman, Convergence of sequences of convex sets, cones, and functions, Bull. Amer. Math. Soc. 70 (1964), 186-188. MR 0157278 (28:514)
  • [20] -, Convergence of sequences of convex sets, cones, and functions. II, Trans. Amer. Math. Soc. 123 (1966), 32-45. MR 0196599 (33:4786)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0628449-5
Keywords: Successive approximation, constrained optimization, convergence, convex analysis, dual operations, subdifferentials, infimal convolution, conjugate duality, separable functions
Article copyright: © Copyright 1981 American Mathematical Society

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