zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Robust solutions of uncertain linear programs. (English) Zbl 0941.90053
Summary: We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time.

90C05Linear programming
90C25Convex programming
90C51Interior-point methods
Full Text: DOI
[1] Ben-Tal, A.: The entropic penalty approach to stochastic programming. Math. oper. Res 10, 263-279 (1985) · Zbl 0565.90052
[2] Ben-Tal, A.; Nemirovski, A.: Robust truss topology design via semidefinite programming. SIAM J. Optim 7, 991-1016 (1997) · Zbl 0899.90133
[3] Ben-Tal, A.; Nemirovski, A.: Robust convex optimization. Math. oper. Res 23, 769-805 (1998) · Zbl 0977.90052
[4] Ben-Tal, A.; Zibulevsky, M.: Penalty/barrier multiplier methods for convex programming problems. SIAM J. Optim 7, 347-366 (1997) · Zbl 0872.90068
[5] A. Ben-Tal, T. Margelit, A. Nemirovski, Robust modeling of multi-stage portfolio problems, in: Proc. Workshop on High-Performance Optimization, S. Zhang (Ed.), Rotterdam, August 1997, Kluwer Academic Press, Dordrecht, to appear, 1999.
[6] J.R. Birge, F. Louveaux, Introduction to Stochasatic Programming, Springer, Berlin, 1997. · Zbl 0892.90142
[7] J.E. Falk, Exact solutions to inexact linear programs, Oper. Res. 1976, 783--787. · Zbl 0335.90035
[8] M. Grötschel, L. Lovasz, A. Schrijver, The Ellipsoid Method and Combinatorial Optimization, Springer, Heidelberg, 1988.
[9] P. Kall, S.W. Wallace, Stochastic Programming, Wiley-Interscience, New York, 1994. · Zbl 0812.90122
[10] H.M. Markovitz, Portfolio Selection: Efficiency Diversification of Investment, Wiley, New York, 1959.
[11] Mulvey, J. M.; Vanderbei, R. J.; Zenios, S. A.: Robust optimization of large-scale systems. Oper. res 43, 264-281 (1995) · Zbl 0832.90084
[12] Yu. Nesterov, A. Nemirovski, Interior Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1994. · Zbl 0977.90081
[13] A. Prékopa, Stochastic Programming, Kluwer Academic Publishers, Dordrecht, 1995.
[14] Rockafellar, R. T.; Wets, R. J. -B.: Scenarios and policy aggregation in optimization under uncertainty. Math. oper. Res 16, 119-147 (1991) · Zbl 0729.90067
[15] Singh, C.: Convex programming with set-inclusive constraints and its applications to generalized linear and fractional programming. J. optim. Theory appl 38, No. 1, 33-42 (1982) · Zbl 0472.90047
[16] A.L. Soyster, Convex programming with set-inclusive constraints and applications to inexact linear programming, Oper. Res. 1973, 1154--1157. · Zbl 0266.90046
[17] K. Zhou, J.C. Doyle, K. Glover, Robust and Optimal Control, Prentice-Hall, Englewood Cliffs, NJ, 1996. · Zbl 0999.49500