Abstract
The paper considers upper semicontinuous behavior in distribution of sequences of random closed sets. Semiconvergence in distribution will be described via convergence in distribution of random variables with values in a suitable topological space. Convergence statements for suitable functions of random sets are proved and the results are employed to derive stability statements for random optimization problems where the objective function and the constraint set are approximated simultaneously.
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References
Artstein, Z. (1984). “Distributions of Random Sets and Random Selections.” Israel J. Mathematics 46, 313–324.
Attouch, H. (1984). Variational Convergence for Functions and Operators. London: Pitman.
Bank, B. et al. (1982). Non-Linear Parametric Optimization. Berlin: Akademie-Verlag.
Beer, G. (1993). Topologies on Closed and Closed Convex Sets. Kluwer Academic Publishers.
Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
Dentcheva, D. (2000). “Regular Castaing Representations with Applications to Stochastic Programming.” SIAM J. Optimization 10, 732–749.
Dentcheva, D. (2001). “Approximations, Extensions, and Univalued Representations of Multifunctions.” Nonlinear Analysis: Theory, Methods, & Applications 45, 85–108.
Dudley, R.M. (1989). Real Analysis and Probability. Pacific Grove, Ca.: Wadsworth & Brooks/Cole.
Dudley, R.M. (1998). “Consistency of M-estimators and One-sided Bracketing.” In E. Eberlein, M. Hahn, M. Talagrand (eds.), High Dimensional Probability. 151–189. Basel: Birkhäuser.
DupaČová, J. and R.J.-B. Wets. (1988). “Asymptotic Behavior of Statistical Estimators and of Optimal Solutions of Stochastic Problems.” Annals of Statistics 16, 1517–1549.
Embrechts, P., C. Klüppelberg, and T. Mikosch. (1997). Modelling Extremal Events. Springer-Verlag.
Francaviglia, S., A. Lechicki, and S. Levy. (1985). “Quasi-uniformization of Hyperspaces and Convergence of Nets of Semicontinuous Multifunctions.” J. Math. Anal. Appl. 112, 347–370.
Hoffmann-Jørgensen, J. (1998). “Convergence in Law of Random Elements and Random Sets.” In E. Eberlein, M. Hahn, M. Talagrand (eds.), High Dimensional Probability. 151-189. Basel: Birkhäuser.
Hu, S. and N.S. Papageorgiou. (1997). Handbook of Multivariate Analysis. Kluwer Academic Publishers.
King, A.J. (1989). “Generalized Delta Theorems for Multivalued Mappings and Measurable Selections.” Math. Oper. Res. 14 (4), 720–736.
King, A.J. and R.T. Rockafellar. (1993). “Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming.” Math. Oper. Res. 18, 148–162.
Kuratowski, K. (1966). Topology, Vol. I. London, New York: Academic Press.
Lachout, P. (2000). “Epi-Convergence Versus Topology: Splitting and Localization.” Research Report of the Academy of Sciences of the Czech Republic, UTIA No. 1987.
Loève, M. (1997). Probability Theory. 4th edition, Springer-Verlag.
Lucchetti, R. and A. Torre. (1994). “Classiscal Set Convergence and Topologies.” Set-Valued Analysis 2, 219–240.
Matheron, G. (1975). Random Sets and Integral Geometry. Wiley.
Pflug, G.C. (1992). “Asymptotic Dominance and Confidence for Solutions of Stochastic Programs.” Czechoslovak J. Oper. Res. 1 (1), 21–30.
Pflug, G.C. (1995). “Asymptotic Stochastic Programs.” Math. Oper. Res. 20, 769–789.
Pollard, D. (1984). Convergence of Stochastic Processes. Springer-Verlag.
Robinson, S.M. (1987). “Local Epi-Continuity and Local Optimization.” Math. Programming 37, 208–222.
Rockafellar, R.T. and R.J.-B. Wets. (1998). Variational Analysis. Springer-Verlag.
Salinetti, G. and R.J.-B. Wets. (1986). “On the Convergence in Distribution of Measurable Multifunctions (Random Sets), Normal Integrands, Stochastic Processes and Stochastic Infima.” Math. Oper. Res. 11, 385–419.
Shapiro, A. (1991). “Asymptotic Analysis of Stochastic Programs.” Annals of Operations Research 30, 169–186.
Shapiro, A. (2000). “Statistical Inference of Stochastic Optimization Problems.” In S. Uryasev (ed.), Probabilistic Constrained Optimization: Methodology and Applications. 282-304. Kluwer Academic Pubishers.
van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge University Press.
van der Vaart, A.W. and J.A. Wellner. (1996). Weak Convergence and Empirical Processes. Springer-Verlag.
Vogel, S. (1991). Stochastische Stabilitätskonzepte. Habilitation, TH Ilmenau.
Vogel, S. (1994). “A Stochastic Approach to Stability in Stochastic Programming.” J. Comput. and Appl. Mathematics, Series Appl. Analysis and Stochastics. 56, 65–96.
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The author is grateful to two anonymous referees for helpful suggestions.
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Vogel, S. Semiconvergence in distribution of random closed sets with application to random optimization problems. Ann Oper Res 142, 269–282 (2006). https://doi.org/10.1007/s10479-006-6172-0
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DOI: https://doi.org/10.1007/s10479-006-6172-0