Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Convex optimization and the epi-distance topology


Authors: Gerald Beer and Roberto Lucchetti
Journal: Trans. Amer. Math. Soc. 327 (1991), 795-813
MSC: Primary 49J45; Secondary 41A50, 90C48
DOI: https://doi.org/10.1090/S0002-9947-1991-1012526-X
MathSciNet review: 1012526
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Gamma (X)$ denote the proper, lower semicontinuous, convex functions on a Banach space $ X$, equipped with the completely metrizable topology $ \tau $ of uniform convergence of distance functions on bounded sets. A function $ f$ in $ \Gamma (X)$ is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of $ f \in \Gamma (X)$ is the minimal condition that guarantees strong convergence of approximate minima of $ \tau $-approximating functions to the minimum of $ f$. Moreover, we show that most functions in $ \langle \Gamma (X),{\tau _{aw}}\rangle $ are well-posed, and that this fails if $ \Gamma (X)$ is topologized by the weaker topology of Mosco convergence, whenever $ X$ is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.


References [Enhancements On Off] (What's this?)

  • [1] E. Asplund and R. T. Rockafellar, Gradients of convex functions, Trans. Amer. Math. Soc. 139 (1969), 443-467. MR 0240621 (39:1968)
  • [2] H. Attouch, Variational convergence for functions and operators, Pitman, Boston, Mass., 1984. MR 773850 (86f:49002)
  • [3] H. Attouch, R. Lucchetti, and R. Wets, The topology of the $ \rho $-Hausdorff distance, Ann. Mat. Pura Appl. (to appear). MR 1163212 (93d:54022)
  • [4] H. Attouch and R. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. (to appear). MR 1018570 (92c:90111)
  • [5] -, Quantitative stability of variational systems; III. Stability of $ \varepsilon $-minimizers, Working paper IIASA, Laxenburg, Austria, 1988.
  • [6] D. Azé and J-P. Penot, Operations on convergent families of sets and functions, Optimization 21 (1990), 521-534. MR 1069660 (92b:49022)
  • [7] G. Beer, On Mosco convergence of convex sets, Bull. Austral. Math. Soc. 38 (1988), 239-253. MR 969914 (90a:46026)
  • [8] -, On the Young-Fenchel transform for convex functions, Proc. Amer. Math. Soc. 104 (1988), 1115-1123. MR 937844 (89i:49006)
  • [9] -, Convergence of continuous linear functionals and their level sets, Arch. Math. 52 (1989), 482-491. MR 998621 (90i:46018)
  • [10] -, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117-126. MR 982400 (90f:46018)
  • [11] -, Metric projections and the Fell topology, Boll. Un. Mat. Ital. (7) 3-B (1989), 925-937. MR 1032618 (90k:54026)
  • [12] -, Three characterizations of the topology of Mosco convergence for convex functions, Arch. Math. 55 (1990), 285-292. MR 1075054 (91k:49013)
  • [13] G. Beer and R. Lucchetti, Minima of quasi-convex functions, Optimization 20 (1989), 581-596. MR 1015430 (90i:49010)
  • [14] J. Borwein and S. Fitzpatrick, Mosco convergence and the Kadec property, Proc. Amer. Math. Soc. 106 (1989), 843-850. MR 969313 (90i:46025)
  • [15] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Math., vol. 580, Springer-Verlag, Berlin, 1975. MR 0467310 (57:7169)
  • [16] L. Contesse and J.-P. Penot, Continuity of polarity and conjugacy for the epi-distance topology, preprint.
  • [17] F. DeBlasi and J. Myjak, On the minimum distance to a closed convex set in a Banach space, Bull. Acad. Sci. Polon. 29 (1981), 373-376. MR 640331 (82m:41027)
  • [18] -, Some generic properties in convex and nonconvex optimization theory, Ann. Soc. Math. Polon. 24 (1984), 1-14. MR 759049 (86e:49032)
  • [19] N. Efimov and S. Stechkin, Approximative compactness and Chebyshev sets, Soviet Math. Dokl. 2 (1961), 1226-1228.
  • [20] S. Francaviglia, A. Lechicki, and S. Levi, Quasi-uniformization of hyperspaces and convergence of nets of semicontinuous multifunctions, J. Math. Anal. Appl. 112 (1985), 347-370. MR 813603 (87e:54025)
  • [21] M. Furi and A. Vignoli, About well-posed optimization problems for functionals in metric spaces, J. Optim. Theory Appl. (1970), 225-229. MR 0264482 (41:9075)
  • [22] P. G. Georgiev, The strong Ekeland variational principle, the strong drop theorem, and applications, J. Math. Anal. Appl. 131 (1988), 1-21. MR 934428 (89c:46019)
  • [23] J. Giles, Convex analysis with application in differentiation of convex functions, Research Notes in Math., no. 58, Pitman, London, 1982. MR 650456 (83g:46001)
  • [24] R. Holmes, A course in optimization and best approximation, Lecture Notes in Math., vol. 257, Springer-Verlag, New York, 1972. MR 0420367 (54:8381)
  • [25] J. Joly, Une famille de topologies sur l'ensemble des fonctions convexes pour lesquelles la polarité est bicontinue, J. Math. Pures Appl. 52 (1973), 421-441. MR 0500129 (58:17826)
  • [26] V. Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51-63. MR 0115076 (22:5879)
  • [27] K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [28] R. Lucchetti, Some aspects of the connections between Hadamard and Tyhonov well-posedness of convex programs, Boll. Un. Mat. Ital. Anal. 1 (1982), 337-344. MR 696278 (84j:49030)
  • [29] -, Hadamard and Tyhonov well-posedness in optimal control, Methods Oper. Res. 45 (1983), 113-125.
  • [30] R. Lucchetti and F. Patrone, Sulla densità e genericità di alcuni problemi di minimo ben posti, Boll. Un. Mat. Ital. 15B (1978), 225-240. MR 0494933 (58:13711)
  • [31] R. Lucchetti, G. Salinetti, and R. Wets, Uniform convergence of probability measures: topological criteria; preprint. MR 1321297 (96c:60007)
  • [32] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. in Math. 3 (1969), 510-585. MR 0298508 (45:7560)
  • [33] -, On the continuity of the Young-Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518-535. MR 0283586 (44:817)
  • [34] F. Patrone, Most convex functions are nice, Numer. Funct. Anal. Optim. 9 (1987), 359-369. MR 887074 (88e:90073)
  • [35] J. Revalski, Generic well-posedness in some classes of optimization problems, Acta Univ. Carolin. Math. Phys. 28 (1987), 117-125. MR 932748 (89d:90238)
  • [36] R. T. Rockafellar, Convex analysis, Princeton Univ. Press, Princeton, N.J., 1970.
  • [37] R. T. Rockafellar and R. Wets, Variational systems, an introduction, Multifunctions and Integrands, G. Salinetti, Ed., Lecture Notes in Math., vol. 1091, Springer-Verlag, Berlin, 1984. MR 785574 (86h:90116)
  • [38] G. Salinetti and R. Wets, On the convergence of sequences of convex sets in finite dimensions, SIAM Rev. 21 (1979), 18-33. MR 516381 (80h:52007)
  • [39] Y. Sonntag, Convergence au sens de Mosco; théorie et applications à l'approximation des solutions d'inéquations, Thèse d'Etat, Université de Provence, Marseille, 1982.
  • [40] A. Tyhonov, On the stability of the functional optimization problem, U.S.S.R. Comput. Math. and Math. Phys. 6 (1966), 28-33.
  • [41] R. Wijsman, Convergence of sequences of convex sets, cones, and functions II, Trans. Amer. Math. Soc. 123 (1966), 32-45. MR 0196599 (33:4786)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 49J45, 41A50, 90C48

Retrieve articles in all journals with MSC: 49J45, 41A50, 90C48


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1012526-X
Keywords: Convex optimization, convex function, well-posed minimization problem, epi-distance topology, Mosco convergence, metric projection, approximative compactness, Chebyshev set
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society