Abstract
Results about the continuity of the optimal value of a linear program and of related polyhedral-valued multifunctions (determined by the constraints) are reviewed. A framework is provided for studying their interconnections.
Research supported in part by a Guggenheim Fellowship
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References
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-linear parametric optimization (Akademie-Verlag, Berlin, 1982).
B. Bereanu, “The continuity of the optimum in parametric programming and applications to sotchastic programming”, Journal of Optimization Theory and Applications 18 (1976) 319–333; see also: “On stochastic linear programming, IV”, Proceedings of the Fourth Conference on Probability Theory Brasov (Editura Academic Republici Socialiste, Bucharest, 1973).
C. Berge, Topological Spaces (Macmillan, New York, 1963).
V. Böhm, “Continuity of optimal policy set for linear programs”, SIAM Journal of Applied Mathematics 28 (1975) 303–306.
G. Dantzig, J. Folkman and N. Shapiro, “On the continuity of the minimum set of a continuous function”, Journal of Mathematical Analysis and Applications 17 (1967) 519–548.
W. Dinkelbach, Sensitivitätsanalysen und Parametrische Program (Springer-Verlag, Berlin, 1969).
W. Hogan, “Point-to-set maps in mathematical programming”, SIAM Review 15 (1973) 591–603.
D. Klatte, “Lineare Optimierungsprobleme mit Parameter in der Koeffizientenmatrix der Restriktionen” in: K. Lommatzsch, ed., Anwendungen der Linearen parametrischen Optimierung. (Akademie-Verlag, Berlin, 1979) pp. 23–53.
D.H. Martin, “On the continuity of the maximum in parametric linear programming”, Journal of Optimization Theory and Applications 17 (1975) 205–210.
S. Robinson, “A characterization of stability in linear programming”, Operations Research 25 (1977) 435–447.
S.M. Robinson, “Some continuity properties of polyhedral multifunctions”, Mathematical Programming Study 14 (1981) 206–214.
R.T. Rockafellar and R. Wets, “Variational systems, an introduction”, in: G. Salinetti, ed., Multifunctions and normal integrands: Stochastic analysis, approximation and optimization (Springer-Verlag Lecture Notes in Mathematics 1097, Berlin, 1984) pp. 1–54.
D. Walkup and R. Wets, “Continuity of some convex-cone valued mappings”, Proceedings of the American Mathematical Society 18 (1967) 229–235.
D. Walkup and R. Wets, “A note on decision rules for stochastic programs”, Journal of Computer and System Sciences 2 (1968) 305–311.
D. Walkup and R. Wets, “A Lipschitzian characterization of convex polyhedra”, Proceedings of the American Mathematical Society 23 (1969) 167–173.
D. Walkup and R. Wets, “Lifting projections of convex polyhedra”, Pacific Journal of Mathematics 28 (1969) 465–475.
R. Wets, “The distribution problem and its relation to other problems in stochastic programming” in: M. Dempster ed., Stochastic Programming, Oxford Conference 1974. (Academic Press, London, 1980) 245–262.
S. Zlobec, “Stable planning by linear and convex models”, Mathematische Operationsforschung und Statistik, ser. Optimization 14 (1983) 519–535.
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Dedicated to Professor George B. Dantzig on the occasion of his seventieth birthday.
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© 1985 The Mathematical Programming Society, Inc.
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Wets, R.JB. (1985). On the continuity of the value of a linear program and of related polyhedral-valued multifunctions. In: Cottle, R.W. (eds) Mathematical Programming Essays in Honor of George B. Dantzig Part I. Mathematical Programming Studies, vol 24. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121040
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DOI: https://doi.org/10.1007/BFb0121040
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