Abstract
A set is called Motzkin decomposable when it can be expressed as the Minkowski sum of a compact convex set with a closed convex cone. This paper analyzes the continuity properties of the set-valued mapping associating to each couple \(\left( C,D\right) \) formed by a compact convex set C and a closed convex cone D its Minkowski sum C + D. The continuity properties of other related mappings are also analyzed.
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Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer, Dordrecht (1993)
Brosowski, B.: Parametric semi-infinite linear programming I. Continuity of the feasible set and the optimal value. Math. Program. Study 21, 18–42 (1984)
Cánovas, M.J., López, M.A., Parra, J.: Upper semicontinuity of the feasible set mapping for linear inequality systems. Set-Valued Var. Anal. 10, 361–378 (2002)
Chuong, T.D.: Lower semi-continuity of the Pareto solution map in quasiconvex semi-infinite vector optimization. J. Math. Anal. Appl. 388, 443–450 (2012)
Cooper, W.W., Seiford, L.M., Tone, K.: Data Envelopment Analysis, 2nd edn. Springer, New York (2006)
Fischer, T.: Contributions to semi-infinite linear optimization. In: Brosowski, B., Martensen, E., (eds.) Approximation and Optimization in Mathematical Physics, pp. 175–199. Peter Lang, Frankfurt-am-Main (1983)
Goberna, M.A., González, E., Martínez-Legaz, J.E., Todorov, M.I.: Motzkin decomposition of closed convex sets. J. Math. Anal. Appl. 364, 209–221 (2010)
Goberna, M.A., Martínez-Legaz, J.E., Todorov, M.I.: On Motzkin decomposable sets and functions. J. Math. Anal. Appl. 372, 525–537 (2010)
Goberna, M.A., Iusem, A., Martínez-Legaz, J.E., Todorov, M.I.: Motzkin decomposition of closed convex sets via truncation. J. Math. Anal. Appl. 400, 35–47 (2013)
Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)
Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. A. 17, 730–743 (1996)
Greenberg, H.J., Pierskalla, W.P.: Stability theorems for infinitely constrained mathematical programs. J. Optim. Theory Appl. 16, 409–428 (1975)
Jess, A., Jongen, H.Th., Neralic, L., Stein, O.: A semi-infinite programming model in data envelopment analysis. Optimization 49, 369–385 (2001)
Lucchetti, R.: Convexity and Well-Posed Problems. Springer, New York (2006)
Michael, E.: Continuous selections (I). Ann. Math. (2nd Ser.) 63, 361–382 (1956)
Mira, J.A., Mora, G.: Stability of linear inequality systems measured by the Hausdorff metric. Set-Valued Var. Anal. 8, 253–266 (2000)
Motzkin, Th.: Beiträge zur Theorie der linearen Ungleichungen. Basel: Inaugural Dissertation 73 S. (1936)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)
Tannino, T., Sawaragi, Y.: Stability of nondominated solutions in multicriteria decision making. J. Optim. Theor. Appl. 30, 229–253 (1980)
Todorov, M.I.: Kuratowski convergence of the efficient sets in the parametric linear vector semi-infinite optimization. Eur. J. Oper. Res. 94, 610–617 (1996)
Van Tiel, J.: Convex Analysis: An Introductory Text. Wiley, New York (1984)
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M. I. Todorov is on leave from IMI-BAS, Sofia, Bulgaria.
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Goberna, M.A., Todorov, M.I. On the Stability of the Motzkin Representation of Closed Convex Sets. Set-Valued Var. Anal 21, 635–647 (2013). https://doi.org/10.1007/s11228-013-0251-6
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DOI: https://doi.org/10.1007/s11228-013-0251-6