Geodesic
Support properties of a family of connected compact sets
Mathematica Bohemica, Tome 126 (2001) no. 1, pp. 67-79
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A problem of finding a system of proportionally located parallel supporting hyperplanes of a family of connected compact sets is analyzed. A special attention is paid to finding a common supporting halfspace. An existence theorem is proved and a method of solution is proposed.
A problem of finding a system of proportionally located parallel supporting hyperplanes of a family of connected compact sets is analyzed. A special attention is paid to finding a common supporting halfspace. An existence theorem is proved and a method of solution is proposed.
DOI : 10.21136/MB.2001.133924
Classification : 15A03, 15A39, 52B55, 52C35, 90C34
Keywords: set family; supporting hyperplane; lexicographic optimization; polyhedral approximation. 
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Nedoma, Josef. Support properties of a family of connected compact sets. Mathematica Bohemica, Tome 126 (2001) no. 1, pp. 67-79. doi: 10.21136/MB.2001.133924

[1] G. B. Dantzig: Linear programming and extensions. Princeton University Press, Princeton, 1973. | MR

[2] H. W. Kuhn, A. W. Tucker: Linear inequalities and related systems. Princeton University Press, Princeton, 1956.

[3] J. E. Falk: Exact solutions of inexact linear programs. Oper. Res. 24 (1976), 783–787. | DOI | MR | Zbl

[4] L. Grygarová: A calculation of all separating hyperplanes of two convex polytopes. Optimization 41 (1997), 57–69. | DOI | MR

[5] L. Grygarová: On a calculation of an arbitrary separating hyperplane of convex polyhedral sets. Optimization 43 (1997), 93–112. | MR

[6] L. Grygarová: Separating support syperplanes for a pair of convex polyhedral sets. Optimization 43 (1997), 113–143. | MR

[7] L. Grygarová: On a supporting hyperplane for two convex polyhedral sets. Optimization 43 (1997), 235–255. | MR

[8] V. Klee: Separation and support properties of convex sets—A survey. In: A. V. Balakrishnan (ed.): Control Theory and the Calculus of Variations. Academic Press, New York, 1969. | MR

[9] D. G. Luenberger: Introduction to linear and nonlinear programming. Addison-Wesley Publishing Comp., 1973. | Zbl

[10] R. Hettich, P. Zehncke: Numerische Methoden der Approximation und semi-infiniten Optimierung. Teubner, Stuttgart, 1982. | MR

[11] R. Hettich, K. O. Kortanek: Semi-infinite programming, theory, methods and applications. SIAM Review 35, 380–429. | MR

[12] J. Nedoma: Linear independence and total separation of set families. Ekonomicko-matematický obzor 14 (1978). | MR | Zbl

[13] J. Nedoma: Vague matrices in linear programming. Ann. Oper. Res. 47 (1993), 483–496. | DOI | MR | Zbl

[14] J. Nedoma: Inaccurate linear equation systems with a restricted-rank error matrix. Linear and Multilinear Algebra 44 (1998), 29–44. | DOI | MR

[15] J. Nedoma: Positively regular vague matrices. (to appear). | MR | Zbl

[16] W. Oettli: On the solution set of a linear system with inaccurate coefficients. SIAM J. Numer. Anal. 2 (1965), 115–118. | MR | Zbl

[17] W. Oettli, W. Prager: Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides. Numer. Math. 6 (1964), 405–409. | DOI | MR

[18] J. Ramík: Linear programming with inexact coefficients. Res. Report, Japan Adv. Inst. Sci. Techn., Hokuriku, 1997.

[19] R. T. Rockafellar: Convex analysis. Princeton University Press, Princeton, 1970. | MR | Zbl

[20] J. Rohn: Systems of linear interval equations. Linear Algebra Appl. 126 (1989), 39–78. | MR | Zbl

[21] J. Stoer, C. Witzgall: Convexity and optimization in finite dimensions I. Springer, Berlin, 1970. | MR

[22] J. Tichatschke, R. Hettich, G. Still: Connections between generalized, inexact and semi-infinite linear programming. ZOR-Methods Models Oper. Res. 33 (1989), 367–382. | MR

[23] D. J. Thuente: Duality theory for generalized linear programs with computational methods. Operations Research 28 (1980), 1005–1011. | DOI | MR | Zbl

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