Abstract
We present a survey of some uses of a remarkable convergence on families of sets or functions. We evoke some of its applications and stress some calculus rules. The main novelty lies in the use of a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. This notion yields criteria for the study of boundedness properties.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Key words
References
S. Adly, E. Ernst and M. Théra, Stability of the solution set of non-coercive variational inequalities, Commun. Contemp. Math. 4 (2002), 145–160.
A. Agadi and J.-P. Penot, New asymptotic cones and usual tangent cones, submitted.
H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, (1984).
H. Attouch, D. Azé and G. Beer, On some inverse stability problems for the epigraphical sum, Nonlinear Anal. 16 (1991), 241–254.
H. Attouch, R. Lucchetti and R. J.-B. Wets, The topology of the ρ-Hausdorff distance, Ann. Mat. Pura Appl. (4) 160 (1991), 303–320.
H. Attouch, A. Moudafi and H. Riahi, Quantitative stability analysis for maximal monotone operators and semi-groups of contractions, Nonlinear Anal. 21 (1993), 697–723.
H. Attouch, J. Ndoutoume and M. Théra, Epigraphical convergence of functions and convergence of their derivatives in Banach spaces, Sém. Anal. Convexe 20 (1990), Exp. No. 9, 45 pp.
H. Attouch and R. J.-B. Wets, Isometries for the Legendre-Fenchel transform, Trans. Amer. Math. Soc. 296 (1986), 33–60.
H. Attouch and R. J.-B. Wets, Epigraphical analysis, Ann. Inst. H. Poincaré Anal. Non Linéaire 6(suppl.) (1989), 73–100.
H. Attouch and R. J.-B. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695–729.
H. Attouch and R. J.-B. Wets, Quantitative stability of variational systems. II. A framework for nonlinear conditioning, SIAM J. Optim. 3 (1993), 359–381.
H. Attouch and R. J.-B. Wets, Quantitative stability of variational systems. III: ε-approximate solutions, Math. Programming 61A (1993), 197–214.
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhäuser, Basel, (1990).
A. Auslender, How to deal with the unboundedness in optimization: theory and algorithms, Math. Programming ser. B 31 (1997), 3–19.
A. Auslender and M. Teboulle, Asymptotic cones and functions in optimization and variational inequalities, Springer, New York, 2002.
D. Azé, An inversion theorem for set-valued maps, Bull. Aust. Math. Soc. 37, No.3 (1988), 411–414.
D. Azé and C.C. Chou, On a Newton type iterative method for solving inclusions, Math. Oper. Res. 20, No.4 (1995), 790–800.
D. Azé and J.-P. Penot, Recent quantitative results about the convergence of convex sets and functions, Functional analysis and approximation (Bagni di Lucca, 1988), Pitagora, Bologna, (1989), 90–110.
D. Azé and J.-P. Penot, Operations on convergent families of sets and functions, Optimization 21 (1990), 521–534.
D. Azé and J.-P. Penot, Qualitative results about the convergence of convex sets and convex functions, Optimization and nonlinear analysis (Haifa, 1990), Longman Sci. Tech., Harlow, (1992), 1–24.
D. Azé and J.-P. Penot, On the dependence of fixed point sets of pseudo-contractive multimappings. Applications to differential inclusions, submitted.
D. Azé and A. Rahmouni, On primal dual stability in convex optimization, J. Convex Anal. 3 (1996), 309–329.
G. Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117–126.
G. Beer, Topologies on Closed and Convex Sets, Kluwer, Dordrecht, (1993).
G. Beer and R. Lucchetti, Convex optimization and the epi-distance topology, Trans. Amer. Math. Soc. 327 (1991), 795–813.
G. Beer and R. Lucchetti, The epi-distance topology: continuity and stability results with applications to convex optimization problems, Math. Oper. Res. 17 (1992), 715–726.
L. Contesse and J.-P. Penot, Continuity of the Fenchel correspondence and continuity of polarities, J. Math. Anal. Appl. 156 (1991), 305–328.
J. Daneš and J. Durdill, A note on the geometric characterization of differentiability, Comm. Math. Univ. Carolin. 17 (1976), 195–204.
J.-P. Dedieu, Cône asymptote d’un ensemble non convexe. Application à l’optimisation, C. R. Acad. Sci. Paris 287 (1977), 501–503.
A. Dontchev and T. Zolezzi, Well-posed Optimization Problems, Lecture Notes in Maths 1543, Springer-Verlag, Berlin, (1993).
J. Durdill, On the geometric characterization of differentiability I, Comm. Math. Univ. Carolin. 15 (1974), 521–540; II, idem, 727–744.
A. Eberhard and R. Wenczel, Epi-distance convergence of parametrised sums of convex functions in non-reflexive spaces, J. Convex Anal. 7 (2000), 47–71.
M. Fabian, Theory of Fréchet cones, Casopis Pro Pěstivani Mat., 107 (1982), 37–58.
A. D. Ioffe, Regular points of Lipschitz functions, Trans. Amer. Math. Soc. 251 (1979), 61–69.
T. Kato, Perturbation theory for linear operators, Springer Verlag, Berlin (1966).
D. Klatte, On quantitative stability for non-isolated minima, Control Cybern. 23 (1994), 183–200.
M. A. Krasnoselskii, Positive solutions of operator equations, Noordhoff, Groningen (1964).
J. Lahrache, Stabilité et convergence dans les espaces non réflexifs, Sém. Anal. Convexe 21 (1991), Exp. No. 10, 50 pp.
D. T. Luc, Recession maps and applications, Optimization 27 (1993), 1–15.
D. T. Luc, Recessively compact sets: properties and uses, Set-Valued Anal. 10 (2002), 15–35.
D. T. Luc and J.-P. Penot, Convergence of asymptotic directions, Trans. Amer. Math. Soc. 353 (2001), 4095–4121.
R. Lucchetti and A. Pasquale, A new approach to a hyperspace theory, J. Convex Anal. 1 (1994), 173–193.
R. Lucchetti and A. Torre, Classical convergences and topologies, Set-Valued Anal. 2 (1994), 219–241.
L. McLinden and R. C. Bergstrom, Preservation of convergence of convex sets and functions in finite dimensions, Trans. Amer. Math. Soc. 268 (1981), 127–142.
F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Funct. Anal. 22 (1976), 130–185.
J.-J. Moreau, Intersection of moving convex sets in a normed space, Math. Scand. 36 (1975), 159–173.
U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3 (1969), 510–585.
S. B. Nadler, Multivalued contraction mappings, Pacific J. Math. 30 (1969), 475–488.
T. Pennanen, R. T. Rockafellar, M. Théra, Graphical convergence of sums of monotone mappings, Proc. Amer. Math. Soc. 130 (2002), 2261–2269.
J.-P. Penot, On regularity conditions in mathematical programming, Math. Prog. Study 19 (1982), 167–199.
J.-P. Penot, Compact nets, filters and relations, J. Math. Anal. Appl., 93 (1983), 400–417.
J.-P. Penot, Differentiability of relations and differential stability of perturbed optimization problems, SIAM J. Control Optim. 22 (1984), 529–551.
J.-P. Penot, Preservation of persistence and stability under intersections and operations, Part I: Persistence, J. Optim. Theory Appl. 79 (1993), 525–550; Part II: Stability, idem, 551–561.
J.-P. Penot, The cosmic Hausdorff topology, the bounded Hausdorff topology and continuity of polarity, Proc. Amer. Math. Soc, 113 (1991), 275–285.
J.-P. Penot, Topologies and convergences on the space of convex functions, Nonlinear Anal. 18 (1992), 905–916.
J.-P. Penot, On the convergence of subdifferentials of convex functions, Nonlinear Anal. 21 (1993), 87–101.
J.-P. Penot, Conditioning convex and nonconvex problems, J. Optim. Theory Appl. 90 (1996), 535–554.
J.-P. Penot, Metric estimates for the calculus of multimappings, Set-Valued Anal. 5 (1997), 291–308.
J.-P. Penot, What is quasiconvex analysis? Optimization 47 (2000), 35–110.
J.-P. Penot, A metric approach to asymptotic analysis, Bull. Sci. Maths,.
J.-P. Penot and C. Zălinescu, Approximation of functions and sets, in Approximation, Optimization and Mathematical Economics, M. Lassonde ed., Physica-Verlag, Heidelberg, (2001), 255–274.
J.-P. Penot and C. Zălinescu, Continuity of usual operations and variational convergences, Set-Valued Anal. 11 (2003), 225–256.
J.-P. Penot and C. Zălinescu, Persistence and stability of solutions to Hamilton-Jacobi equations, preprint, Univ. of Pau, June 2000.
J.-P. Penot and C. Zălinescu, Fenchel-Legendre transform and variational convergences, preprint, 2003.
H. Rådström, An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc. 3 (1952), 165–169.
S. M. Robinson, Stability theory for systems of inequalities, Part I: linear systems, SIAM J. Numer. Anal., 12 (1975), 754–769.
S. M. Robinson, Regularity and stability for convex multivalued functions, Math. Oper. Res. 1 (1976), 130–143.
R. T. Rockafellar and R. J.-B. Wets, Cosmic convergence, in: Optimization and Nonlinear Analysis, A. Ioffe et al. eds., Pitman Notes 244, Longman, Harlow, 1992, 249–272.
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin, 1997.
Y. Sonntag and C. Zălinescu, Set convergences. An attempt of classification, Trans. Amer. Math. Soc. 340 (1993), 199–226.
Y. Sonntag and C. Zălinescu, Set convergences: a survey and a classification, Set-Valued Analysis 2 (1994), 339–356.
T. Strömberg, The operation of infimal convolution, Dissert. Math. 352 (1996), 1–58.
S. Villa, A W-convergence and well-posedness of non convex functions, J. Convex Anal. (2003), to appear.
C. Zălinescu, On convex sets in general position, Linear Algebra Appl. 64 (1985), 191–198.
C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, in Nonsmooth Optimization and Related Fields, F.H. Clarke et al. eds., Plenum Press, London and New York (1989), 437–458.
C. Zălinescu, Recession cones and asymptotically compact sets, J. Optim. Theory Appl., 77 (1993), 209–220.
C. Zălinescu, Convex Analysis in General Vector Spaces, World Scientific, Singapore (2002).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Penot, JP., Zălinescu, C. (2005). Bounded (Hausdorff) Convergence: Basic Facts and Applications. In: Giannessi, F., Maugeri, A. (eds) Variational Analysis and Applications. Nonconvex Optimization and Its Applications, vol 79. Springer, Boston, MA. https://doi.org/10.1007/0-387-24276-7_49
Download citation
DOI: https://doi.org/10.1007/0-387-24276-7_49
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-24209-5
Online ISBN: 978-0-387-24276-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)