Convergence of asymptotic directions

The Luc, Dinh; Penot, Jean-Paul

doi:10.1090/S0002-9947-01-02664-2

Convergence of asymptotic directions
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by Dinh The Luc and Jean-Paul Penot PDF

Trans. Amer. Math. Soc. 353 (2001), 4095-4121 Request permission

Abstract:

We study convergence properties of asymptotic directions of unbounded sets in normed spaces. The links between the continuity of a set-valued map and the convergence of asymptotic directions are examined. The results are applied to investigate continuity properties of marginal functions and asymptotic directions of level sets.

References

Samir Adly, Daniel Goeleven, and Michel Théra, Recession mappings and noncoercive variational inequalities, Nonlinear Anal. 26 (1996), no. 9, 1573–1603. MR 1377476, DOI 10.1016/0362-546X(94)00364-N
A. AGADI and J.-P. PENOT, New asymptotic cones and usual tangent cones, submitted.
A. AGADI and J.-P. PENOT, Asymptotic approximation of sets with ap= plication in mathematical programming, preprint, Univ. of Pau, February 1996.
A. Auslender, How to deal with the unbounded in optimization: theory and algorithms, Math. Programming 79 (1997), no. 1-3, Ser. B, 3–18. Lectures on mathematical programming (ismp97) (Lausanne, 1997). MR 1464758, DOI 10.1016/S0025-5610(97)00053-1
Alfred Auslender, Noncoercive optimization problems, Math. Oper. Res. 21 (1996), no. 4, 769–782. MR 1419901, DOI 10.1287/moor.21.4.769
J.-P. AUBIN and H. FRANKOWSKA, “Set-valued Analysis”, Birkhäuser, Basel, 1990.
Claudio Baiocchi, Giuseppe Buttazzo, Fabio Gastaldi, and Franco Tomarelli, General existence theorems for unilateral problems in continuum mechanics, Arch. Rational Mech. Anal. 100 (1988), no. 2, 149–189. MR 913962, DOI 10.1007/BF00282202
Claude Berge, Espaces topologiques: Fonctions multivoques, Collection Universitaire de Mathématiques, Vol. III, Dunod, Paris, 1959 (French). MR 0105663
Gustave Choquet, Ensembles et cônes convexes faiblement complets, C. R. Acad. Sci. Paris 254 (1962), 2123–2125 (French). MR 132987
L. Contesse and J.-P. Penot, Continuity of the Fenchel correspondence and continuity of polarities, J. Math. Anal. Appl. 156 (1991), no. 2, 305–328. MR 1103014, DOI 10.1016/0022-247X(91)90399-K
Jean-Pierre Crouzeix, Pseudomonotone variational inequality problems: existence of solutions, Math. Programming 78 (1997), no. 3, Ser. A, 305–314. MR 1466134, DOI 10.1016/S0025-5610(96)00074-3
A. DANILIIDIS and N. HADJISSAVAS, Coercivity conditions and variational inequalities, preprint, Univ. of the Aegean, October 1997.
George B. Dantzig, Jon Folkman, and Norman Shapiro, On the continuity of the minimum sets of a continuous function, J. Math. Anal. Appl. 17 (1967), 519–548. MR 207426, DOI 10.1016/0022-247X(67)90139-4
G. DEBREU, “Theory of value", New Haven and London, Yale University Press, 1975.
Jean-Pierre Dedieu, Cône asymptote d’un ensemble non convexe. Application à l’optimisation, C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 7, A501–A503 (French, with English summary). MR 458117
Jean-Pierre Dedieu, L’image de la limite supérieure d’une famille d’ensembles est-elle égale à la limite supérieure de la famille des images?, Ann. Fac. Sci. Toulouse Math. (5) 11 (1990), no. 2, 91–103 (French, with English summary). MR 1191713
J. DURDILL, On the geometric characterization of differentiability II, Com. Math. University Carolinae 15 (1974), 727–744.
Marián Fabián, Theory of Fréchet cones, Časopis Pěst. Mat. 107 (1982), no. 1, 37–58, 91 (English, with Czech summary). With a loose Russian summary. MR 651081
W. FENCHEL, “Convex cones, sets and functions", mimeographed lecture notes, Princeton University, 1951.
Victor Klee, Asymptotes and projections of convex sets, Math. Scand. 8 (1960), 356–362. MR 158309, DOI 10.7146/math.scand.a-10617
Erwin Klein and Anthony C. Thompson, Theory of correspondences, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1984. Including applications to mathematical economics; A Wiley-Interscience Publication. MR 752692
D. Klatte, On the lower semicontinuity of optimal sets in convex parametric optimization, Math. Programming Stud. 10 (1979), 104–109. Point-to-set maps and mathematical programming. MR 527061, DOI 10.1007/bfb0120847
Issoufou Kouada, Upper-semi-continuity and cone-concavity of multi-valued vector functions in a duality theory for vector optimization, Math. Methods Oper. Res. 46 (1997), no. 2, 169–192. MR 1481929, DOI 10.1007/BF01217689
B. Kummer, A note on the continuity of the solution set of special dual optimization problems, Math. Programming Stud. 10 (1979), 110–114. Point-to-set maps and mathematical programming. MR 527062, DOI 10.1007/bfb0120848
J. P. Evans and F. J. Gould, Stability in nonlinear programming, Operations Res. 18 (1970), 107–118. MR 264984, DOI 10.1287/opre.18.1.107
Đinh Thế Lục, Theory of vector optimization, Lecture Notes in Economics and Mathematical Systems, vol. 319, Springer-Verlag, Berlin, 1989. MR 1116766
Đinh Thế Lục, Recession cones and the domination property in vector optimization, Math. Programming 49 (1990/91), no. 1, (Ser. A), 113–122. MR 1077747, DOI 10.1007/BF01588781
D. T. Luc, Recession maps and applications, Optimization 27 (1993), no. 1-2, 1–15. MR 1275544, DOI 10.1080/02331939308843868
D. T. LUC, Recessively compact sets: uses and properties, Set-Valued Analysis (to appear).
D. T. LUC, Existence results for densely pseudomonotone variational inequalities, Journal of Mathematical Analysis and Applications 254 (2001), 291–308.
Dinh The Luc and Pham Huy Dien, Differentiable selection of optimal solutions in parametric linear programming, Proc. Amer. Math. Soc. 125 (1997), no. 3, 883–892. MR 1301514, DOI 10.1090/S0002-9939-97-03090-6
Dinh The Luc and Michel Théra, Derivatives with support and applications, Math. Oper. Res. 19 (1994), no. 3, 659–675. MR 1288892, DOI 10.1287/moor.19.3.659
D. T. Luc and M. Volle, Levels sets infimal convolution and level addition, J. Optim. Theory Appl. 94 (1997), no. 3, 695–714. MR 1470889, DOI 10.1023/A:1022657102069
Paulo K. Monteiro, Frank H. Page Jr., and Myrna Holtz Wooders, Arbitrage, equilibrium, and gains from trade: a counterexample, J. Math. Econom. 28 (1997), no. 4, 481–501. MR 1486487, DOI 10.1016/S0304-4068(96)00774-4
Frank H. Page Jr. and Myrna Holtz Wooders, A necessary and sufficient condition for the compactness of individually rational and feasible outcomes and the existence of an equilibrium, Econom. Lett. 52 (1996), no. 2, 153–162. MR 1418024, DOI 10.1016/S0165-1765(96)00855-5
Jean-Paul Penot, Continuity properties of performance functions, Optimization: theory and algorithms (Confolant, 1981) Lecture Notes in Pure and Appl. Math., vol. 86, Dekker, New York, 1983, pp. 77–90. MR 716358
Jean-Paul Penot, Compact nets, filters, and relations, J. Math. Anal. Appl. 93 (1983), no. 2, 400–417. MR 700155, DOI 10.1016/0022-247X(83)90184-1
J.-P. Penot, Preservation of persistence and stability under intersections and operations. I. Persistence, J. Optim. Theory Appl. 79 (1993), no. 3, 525–550. MR 1255285, DOI 10.1007/BF00940557
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
Stephen M. Robinson, Stability theory for systems of inequalities. I. Linear systems, SIAM J. Numer. Anal. 12 (1975), no. 5, 754–769. MR 410521, DOI 10.1137/0712056
R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
R. T. Rockafellar and R. J.-B. Wets, Cosmic convergence, Optimization and nonlinear analysis (Haifa, 1990) Pitman Res. Notes Math. Ser., vol. 244, Longman Sci. Tech., Harlow, 1992, pp. 249–272. MR 1184648
R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
J. E. Spingarn, Fixed and variable constraints in sensitivity analysis, SIAM J. Control Optim. 18 (1980), no. 3, 297–310. MR 569019, DOI 10.1137/0318021
R. E. STEINITZ, Bedingt konvergente Reihen und konvexe Systeme I,II,III, J. Math. 143 (1913), 128–175; 144 (1914), 1-40; 146 (1916), 1–52.
C. Zălinescu, Stability for a class of nonlinear optimization problems and applications, Nonsmooth optimization and related topics (Erice, 1988) Ettore Majorana Internat. Sci. Ser.: Phys. Sci., vol. 43, Plenum, New York, 1989, pp. 437–458. MR 1034075
C. Zălinescu, Recession cones and asymptotically compact sets, J. Optim. Theory Appl. 77 (1993), no. 1, 209–220. MR 1222792, DOI 10.1007/BF00940787