zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Tractable approximations to robust conic optimization problems. (English) Zbl 1134.90026
Summary: In earlier proposals, the robust counterpart of conic optimization problems exhibits a lateral increase in complexity, i.e., robust linear programming problems (LPs) become second order cone problems (SOCPs), robust SOCPs become semidefinite programming problems (SDPs), and robust SDPs become NP-hard. We propose a relaxed robust counterpart for general conic optimization problems that (a) preserves the computational tractability of the nominal problem; specifically the robust conic optimization problem retains its original structure, i.e., robust LPs remain LPs, robust SOCPs remain SOCPs and robust SDPs remain SDPs, and (b) allows us to provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions.

MSC:
90C15Stochastic programming
90C31Sensitivity, stability, parametric optimization
WorldCat.org
Full Text: DOI
References:
[1] Ben-Tal, A., Nemirovski, A.: Robust convex optimization. Math. Oper. Res. 23, 769--805 (1998) · Zbl 0977.90052
[2] Ben-Tal, A., Nemirovski, A.: On the quality of SDP approximations of uncertain SDP programs, Research Report #4/98 Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Israel (1998)
[3] Ben-Tal, A., Nemirovski, A.: Robust solutions to uncertain programs. Oper. Res. Let. 25, 1--13 (1999) · Zbl 0941.90053
[4] Ben-Tal, A., Nemirovski, A.: Robust solutions of Linear Programming problems contaminated with uncertain data, Math. Progr. 88, 411--424 (2000) · Zbl 0964.90025
[5] Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPR-SIAM Series on Optimization, SIAM, Philadelphia 2001 · Zbl 0986.90032
[6] Ben-Tal, A., El-Ghaoui, L., Nemirovski, A.: Robust semidefinite programming. In: Saigal, R., Vandenberghe, L., Wolkowicz, H., (eds.), Semidefinite programming and applications, Kluwer Academic Publishers, (2000) · Zbl 0964.90025
[7] Bertsimas, D., Brown, D.: Constrainted stochastic LQC: A tractable approach. submitted for publication, 2004
[8] Bertsimas, D., Pachamanova, D., Sim, M.: Robust Linear Optimization under General Norms. Operations Research Letters, 32, 510--516 (2003) · Zbl 1054.90046
[9] Bertsimas, D., Sim, M.: Price of Robustness. Oper. Res. 52 (1), 35--53 (2004) · Zbl 1165.90565
[10] Bertsimas, D., Sim, M.: Robust Discrete Optimization and Network Flows. Math. Progr. 98, 49--71 (2003) · Zbl 1082.90067
[11] Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York, 1997 · Zbl 0892.90142
[12] El-Ghaoui, Lebret, H.: Robust solutions to least-square problems to uncertain data matrices. SIAM J. Matrix Anal. Appl. 18, 1035--1064 (1997) · Zbl 0891.65039
[13] El-Ghaoui, L., Oustry, F., Lebret, H.: Robust solutions to uncertain semidefinite programs. SIAM J. Optim. 9, 33--52 (1998) · Zbl 0960.93007
[14] Güler, 0, Tunçel, L.: Characterization of the barrier parameter of homogeneous convex cones. Math. Progr. 81, 55--76 (1998) · Zbl 0919.90124
[15] Nemirovski, A.: On tractable approximations of ramdomly perturbed convex constraints. In: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, December, pp. 2419--2422, 2003
[16] Nemirovski, A.: Regular Banach spaces and large deviations of random sums. http://iew3.technion.ac.il/Home/Users/Nemirovski.html#part4 2004 · Zbl 1106.90059
[17] Renegar, J.: A mathematical view of interior-point methods in convex optimization. MPS-SIAM Series on Optimization, SIAM, Philadelphia, 2001 · Zbl 0986.90075
[18] Soyster, A.L.: Convex programming with set-inclusive constraints and applications to inexact linear programming. Oper. Res. 21, 1154--1157 (1973) · Zbl 0266.90046