Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
set family; supporting hyperplane; lexicographic optimization; polyhedral approximation.
set family; supporting hyperplane; lexicographic optimization; polyhedral approximation.
Summary:
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.
References:
[1] G. B. Dantzig: Linear programming and extensions. Princeton University Press, Princeton, 1973. MR 1658673
[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 10.1287/opre.24.4.783 | MR 0437008 | Zbl 0335.90035
[4] L. Grygarová: A calculation of all separating hyperplanes of two convex polytopes. Optimization 41 (1997), 57–69. DOI 10.1080/02331939708844325 | MR 1460220
[5] L. Grygarová: On a calculation of an arbitrary separating hyperplane of convex polyhedral sets. Optimization 43 (1997), 93–112. MR 1638843
[6] L. Grygarová: Separating support syperplanes for a pair of convex polyhedral sets. Optimization 43 (1997), 113–143. MR 1638847
[7] L. Grygarová: On a supporting hyperplane for two convex polyhedral sets. Optimization 43 (1997), 235–255. MR 1774340
[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 0394357
[9] D. G. Luenberger: Introduction to linear and nonlinear programming. Addison-Wesley Publishing Comp., 1973. Zbl 0297.90044
[10] R. Hettich, P. Zehncke: Numerische Methoden der Approximation und semi-infiniten Optimierung. Teubner, Stuttgart, 1982. MR 0653476
[11] R. Hettich, K. O. Kortanek: Semi-infinite programming, theory, methods and applications. SIAM Review 35, 380–429. MR 1234637
[12] J. Nedoma: Linear independence and total separation of set families. Ekonomicko-matematický obzor 14 (1978). MR 0508972 | Zbl 0422.15015
[13] J. Nedoma: Vague matrices in linear programming. Ann. Oper. Res. 47 (1993), 483–496. DOI 10.1007/BF02023110 | MR 1260033 | Zbl 0793.90033
[14] J. Nedoma: Inaccurate linear equation systems with a restricted-rank error matrix. Linear and Multilinear Algebra 44 (1998), 29–44. DOI 10.1080/03081089808818545 | MR 1638938
[15] J. Nedoma: Positively regular vague matrices. (to appear). MR 1815951 | Zbl 1002.15020
[16] W. Oettli: On the solution set of a linear system with inaccurate coefficients. SIAM J. Numer. Anal. 2 (1965), 115–118. MR 0178567 | Zbl 0146.13404
[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 10.1007/BF01386090 | MR 0168106
[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 0274683 | Zbl 0193.18401
[20] J. Rohn: Systems of linear interval equations. Linear Algebra Appl. 126 (1989), 39–78. MR 1040771 | Zbl 0712.65029
[21] J. Stoer, C. Witzgall: Convexity and optimization in finite dimensions I. Springer, Berlin, 1970. MR 0286498
[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 1030790
[23] D. J. Thuente: Duality theory for generalized linear programs with computational methods. Operations Research 28 (1980), 1005–1011. DOI 10.1287/opre.28.4.1005 | MR 0584904 | Zbl 0441.90056
Search
Partner of
