Comparative Properties of Three Metrics in the Space of Compact Convex Sets
Tóm tắt
Along with the Hausdorff metric, we consider two other metrics on the space of convex sets, namely, the metric induced by the Demyanov difference of convex sets and the Bartels–Pallaschke metric. We describe the hierarchy of these three metrics and of the corresponding norms in the space of differences of sublinear functions. The completeness of corresponding metric spaces is demonstrated. Conditions of differentiability of convex-valued maps of one variable with respect to these metrics are proved for some special cases. Applications to the theory of convex fuzzy sets are given.
Tài liệu tham khảo
Attouch, H.: Variational Convergence for Functions and Operators, Appl. Math. Series, Pitman, Boston, 1984.
Aubin, J. P. and Ekeland, I.: Applied Nonlinear Analysis, Pure Appl. Math., Wiley, New York, 1984.
Aubin, J. P. and Frankowska, H.: Set-Valued Analysis, Birkhauser-Verlag, Boston, 1990.
Banks, H. T. and Jacobs, M. Q.: A differential calculus for multifunctions, J. Math. Anal. Appl. 29 (1970), 246–272.
Bartels, S. G. and Pallaschke, D.: Some remarks on the space of differences of sublinear functions, Appl. Math. (Warsaw) 22(3) (1994), 419–426.
de Blasi, F. S.: On the differentiability of multifunctions, Pacific J. Math. 66 (1976), 67–81.
Demyanov, V. F.: The relation between the Clarke subdifferential and a quasidifferential, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. 3 (1980), 18–24 (in Russian).
Demyanov, V. F. and Rubinov, A. M.: Constructive Nonsmooth Analysis, Volume 7 of Approximation & Optimization, Peter Lang, Frankfurt am Main, 1995.
Diamond, P. and Kloeden, P.: Metric Spaces of Fuzzy Sets. Theory and Applications, World Scientific, River Edge, NJ, 1994.
Edwards, R. E.: Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965.
Hiriart-Urruty, J. B.: Miscellanies on Nonsmooth Analysis and Optimization, Lecture Notes in Econom. and Math. Systems 255, Springer, New York, 1985.
Hukuhara, M.: Intégration des applications measurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 205–223 (in French).
Kloeden, P. E.: Compact supported sendographs and fuzzy sets, Fuzzy Sets and Systems 4 (1980), 193–201.
Minchenko, L. I., Borisenko, O. F. and Gritsai, S. P.: Set-valued Analysis and Perturbed Nonlinear Programming Problems, Navuka i Tekhnika, Minsk, 1993 (in Russian).
Mosco, U.: Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3 (1969), 510–585.
Mosco, U.: On the continuity of the Young–Fenchel transform, J. Math. Anal. Appl. 35 (1971), 518–535.
Pecherskaya, N. A.: Quasidifferentiable mappings and the differentiability of maximum functions, Math. Programming Study 29 (1986), 145–159.
Penot, J.-P. and Volle, M.: On quasiconvex duality, Math. Oper. Res. 15 (1990), 597–625.
Pontryagin, L. S.: On linear differential games II, Dokl. Akad. Nauk SSSR 175 (1967), 764–766 (in Russian).
Rockafellar, R. T.: Convex Analysis, Princeton Math. Ser. 28, Princeton University Press, Princeton, NJ, 1970.
Rubinov, A. M.: The conjugate derivative of a multivalued mapping and differentiability of the maximum under connected constraints, Sibirsk. Mat. Zh. 26(3) (1985), 147–155 (in Russian).
Rubinov, A. M.: Differences of convex compact sets and their applications in nonsmooth analysis, in: F. Giannesi (ed.), Nonsmooth Optimization. Methods and Applications, Gordon and Breach, Amsterdam, 1992, pp. 366–378.
Rubinov, A. M. and Akhundov, I. S.: Difference of compact sets in the sense of Demyanov and its application to nonsmooth analysis, Optimization 23(3) (1992), 179–188.
Rubinov, A. M., Borisov, K. Y., Desnitskaya, V. N. and Matveenko, V. D.: Optimal Control in Aggregated Models in Economics, Nauka, Leningrad, 1991 (in Russian).
Rubinov, A. M. and Simsek, B.: Conjugate quasiconvex nonnegative functions, Optimization 35 (1995), 1–22.
Rubinov, A. M. and Simsek, B.: Dual problems of quasiconvex optimization, Bull. Austral. Math. Soc. 51 (1995), 139–144.
Sonntag, Y. and Zalinescu, C.: Set convergences: a survey and a classification, Set-Valued Anal. 2 (1994) 339–356.
Straszewicz, S.: Uber exponierte Punkte abgeschlossener Punktmengen, Fund. Math. 24 (1935), 139–143.
Tyurin, Y. N.: A mathematical formulation of a simplified model of industrial planning, Ekonomika i Matematicheskie Metody 1 (1965), 391–409 (in Russian).