Berge’s maximum theorem is an important statement in set-valued analysis that has significant applications in mathematical economics, operations research, and control. For a minimization problem for a continuous function of two variables and continuous compact-valued set-valued mapping defining feasible sets, this theorem states continuity of the value function and upper semi-continuity of the solution multifunction. One of the main limitations of this theorem is that the set-valued mapping is compact-valued. Recently the authors of this paper and their coauthors generalized Berge’s maximum theorem to set-valued mappings that may not be compact-valued. Here we formulate and prove the local Berge’s maximum theorem for possibly noncompact feasible sets and show that it is more general than the recently established Berge’s maximum theorem for possibly noncompact feasible sets and than the known formulations of the local Berge’s maximum theorem.
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Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 3rd edn., p xxii+704. Springer, Berlin (2006)
Berge, C., Spaces, Topological: xiii+270. Macmillan, New York (1963)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems, p xviii+601. Springer, New York (2000)
Engelking, R.: General Topology, revised ed., p 540. Helderman Verlag, Berlin (1989)
Feinberg, E.A., Kasyanov, P.O., Voorneveld, M.: Berge’s maximum theorem for noncompact image sets. J. Math. Anal. Appl. 413, 1040–1046 (2014)
Feinberg, E.A., Kasyanov, P.O., Zadoianchuk, N.V.: Average cost Markov decision processes with weakly continuous transition probabilities. Math. Oper. Res. 37(4), 591–607 (2012)
Feinberg, E.A., Kasyanov, P.O., Zadoianchuk, N.V.: Berge’s theorem for noncompact image sets. J. Math. Anal. Appl. 397(1), 255–259 (2013)
Feinberg, E.A., Lewis, M. E.: Optimality inequalities for average cost Markov decision processes and the stochastic cash balance problem. Math. Oper. Res. 32(4), 769–783 (2007)
Hu, Sh., Papageorgiou, N.S.: Handbook of Multivalued Analysis, volume I: Theory, p xxx+968. Kluwer, Dordrecht (1997)
Kelley, J.L.: General topology, p 298. Van Nostrand, New York (1955)
Munkres, J.R.: Topology, 2nd ed., p 537. Prentice Hall, New York (2000)
Ok, E.A.: Applications, Real Analysis with Economics. Princeton University Press (2007). ISBN 0-691-11768-3
Zgurovsky, M.Z., Mel’nik, V.S., Kasyanov, P.O.: Inclusions, Evolution Variation Inequalities for Earth Data Processing I, p xxx+250. Springer, Berlin (2011)
This research was partially supported by NSF grant CMMI-1335296 and by the Ukrainian State Fund for Fundamental Research under grant GP/F50/049.
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Feinberg, E.A., Kasyanov, P.O. Continuity of Minima: Local Results. Set-Valued Var. Anal 23, 485–499 (2015). https://doi.org/10.1007/s11228-015-0318-7
- Berge’s maximum theorem
- Set-valued mapping
Mathematics Subject Classification (2010)