@book{Barlow75, Author = {R. Barlow and F. Proschan}, Title = {Statistical Theory of Reliability and Life Testing Probability Models}, Publisher ={Holt, Rinehart and Winston Inc.}, Year = {1975} } @book{Bazaraa90, Author = {M.S. Bazaraa and J.J Jarvis and H.D. Sherali}, Title = {Linear Programming and Network Flows, 2nd Edition}, Publisher = {Wiley}, Year = {1990}, note = { Chanpter: The Decomposition Principle } } @incollection{Bonferroni37, Author = { C.E. Bonferroni}, Title = { Teoria statistica delle classi e calcolo delle probabilit\'{a}}, Booktitle = {Volume in onore di Riccardo Dalla Volta}, Publisher = {Universit\'{a}\'{d}i Firenze}, Pages = {1-62}, Year = {1937} } @book{Boole54, Author = {G. Boole}, Title = { Laws of Thought}, Publisher = {American reprint of 1854 edition}, Address = {Dover, New York}, Year = {1854} } @article{Boole68, Author = {G. Boole}, Title ={Of Propositions Numerically Definite}, Journal = {Transactions of Cambridge Philosophical Society, Part II, XI}, Year = {1868} } @incollection{Boole52, Author = {G. Boole}, Title = {Collected Logical Works}, Booktitle = {Studies in Logic and Probability}, Publisher = {Open Court Publ. Co.}, Address = {LaSalle, Ill}, Year = {1952}, Volume = {I}, Editor = {R. Rhees} } @article{Boros89, Author = {E. Boros and A. Pr\'{e}kopa}, Title = { Closed Form Two-Sided Bounds for Probabilities That Exactly $r$ and At Least $r$ Out of $n$ Events Occur}, Journal = {Mathematics of Operations Research}, Year = {1989}, Volume = {14}, Pages = {317-342} } @book{Cottle92, Author = {R. Cottle and J. Pang and R. Stone}, Title = {The Linear Complementarity Problem}, Publisher = {Academic Press, Inc.}, Year = {1992}, Address = {Boston} } @inproceedings{Dawson67, Author = {D. Dawson and A. Sankoff}, Title = {An Inequality for Probabilities}, Booktitle = {Proceedings of the American Mathematical Society}, Volume = {18}, Year = {1967}, Pages = {504-507} } @article{Dentcheva, Author ={A. Pr\'{e}kopa and D. Dentcheva and A. Ruszczy\'{n}ski}, Title = {Concavity and efficient points of discrete distributions in probabilistic programming}, Journal = {Math. Program.}, Volume = {A 89}, Pages = {55-77}, Year = {2000} } @book{Dette97, Author = {H. Dette and W. Studden}, Title = {The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis}, Publisher = {John Wiley \& Sons, Inc}, Year = {1997}, Address = {New York} } @inbook{Dette97CanMon, Author = {H. Dette and W. Studden}, Title = {The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis}, Publisher = {John Wiley \& Sons, Inc}, Year = {1997}, Address = {New York}, Pages = {1-41} } @inbook{Dette97OrtPol, Author = {H. Dette and W. Studden}, Title = {The Theory of Canonical Moments with Applications in Statistics, Probability and Analysis}, Publisher = {John Wiley \& Sons, Inc}, Year = {1997}, Address = {New York}, Pages = {42-74} } @book{DeVore93, Author = {R. DeVore and G. Lorenz}, Title = {Constructive Approximation}, Publisher = {Springer-Verlag}, Year = {1993}, Address = {New York} } @inbook{DeVore93BesApp, Author = {R. DeVore and G. Lorenz}, Title = {Constructive Approximation}, Publisher = {Springer-Verlag}, Year = {1993}, Address = {New York}, Pages = {58-95} } @book{Devroye96, Author = {L. Devroye and L. Gy\"orfi and G. Lugosi}, Title ={A Probabilistic Theory of Pattern Recognation}, Publisher = {Springer}, Year = {1996}, Address = {New York} } @book{Feller50, Author = {W. Feller}, Title = {An Introduction to Probability Theory and Its Applications}, Year = {1950}, Publisher = { John Wiley}, Address = {New York}, note = {volume 1, third edition} } @book{Galambos96, Author = {J. Galambos and J. Simonelli}, Title = {Bonferroni-type Inequalities With Applications}, Publisher = {Springer Verlag}, Address ={New York}, Year = {1996} } @article{Galambos77, Author = {J. Galambos}, Title = {Bonferroni Inequalities}, Journal = {Annals of Probability}, Volume = {5}, Pages = {577-581}, Year ={1977} } @book{Gifi90, Author = {A. Gifi}, Title ={Nonlinear Multivariate Analysis}, Publisher = {John Wiley \& Sons}, Year = {1990}, Address = {New York} } @article{Hailperin65, Author = {Th. Hailperin}, Title = {Best Possible Inequalities for the Probability of a Logical Function of Events}, Journal = {The American Mathematical Monthly}, Year = {1965}, Volume = {72}, Pages = {343-359} } @article{Johnson74, Author = {D. S. Johnson}, Title = {Approximation Algorithms for Combinatorial Problems}, Journal = {Journal Comp. Sys. Sc.}, Volume = {9}, Pages = {256-278}, Year = {1974} } @book{Jordan65, Author = {C. Jordan}, Title = {Calculus of finite Differences, 3rd ed.}, Publisher = {Chelsea}, Year = {1965}, Address = {New York} } @book{Karlin66, Author = {S. Karlin and W. Studden}, Title = {Tchebycheff Systems: with Applications in Analysis and Statistics}, Publisher = {John Wiley \& Sons}, Address = {New York}, Year = {1966} } @inbook{Karlin66TchSys, Author = {S. Karlin and W. Studden}, Title = {Tchebycheff Systems: with Applications in Analysis and Statistics}, Publisher = {John Wiley \& Sons}, Address = {New York}, Year = {1966}, Pages = {1-36} } @inbook{Karlin66eg, Author = {S. Karlin and W. Studden}, Title = {Tchebycheff Systems: with Applications in Analysis and Statistics}, Publisher = {John Wiley \& Sons}, Address = {New York}, Year = {1966}, Pages = {37-78} } @book{KorteVygen2000), Author = {B. Korte and J. Vygen}, Title = {Combinatorial Optimization, Theory and Algorithms}, Year = {2000}, Publisher = {Springer-Verlag}, Address = {Berlin} } @article{Kuai00, Author = {H. Kuai and F. Alajaji and G. Takahara}, Title = {A Lower Bound on the Probability of a Finite Union of Events}, Journal = {Discrete Applied Mathematics}, Year = {2000}, Volume = {215}, Pages = {147-158} } @article{Kounias76, Author = {S. Kounias and J. Marin}, Title = {Best Linear Bonferroni Bounds}, Journal = {SIAM J. on Applied Mathematics}, Year = {1976}, Volume = {30}, Pages = {307-323} } @article{Kwerel75a, Author = {S.M. Kwerel}, Title ={Most Stingent Bounds on Aggregated Probabilities of Partially Specified Dependent Probability Systems}, Journal = {J. Am. Statist. Assoc.}, Volume = {70}, Pages = {472-479}, Year = {1975} } @article{Kwerel75b, Author ={S.M. Kwerel}, Title = {Bounds on Probability of a Union and Intersection of $m$ Events}, Journal = {Advances of Applied Probability}, Year = {1975}, Volume = {7}, Pages = {431-448} } @article{Lemke54, Author ={C.E. Lemke}, Title = {The Dual Method for Solving the Linear Programming Problem}, Journal = {Naval Research Logistic Quarterly}, Year = {1954}, Volume = {22}, Pages = {978-981} } @article{Nagy00, Author = {G. Nagy and A. Pr\'ekopa}, Year ={2000}, Title = {On Multivariate Discrete Moment Problems and Their Applications to Bounding Functions, Probabilities and Expectations}, Journal = {Rutcor Research Report 14-2000} } @article{Prekopa88, Author ={A. Pr\'ekopa}, Year = {1988}, Title ={Boole-Bonferroni Inequalities and Linear Programming}, Journal = {Operations Research}, Volume = {36}, Pages ={ 145-162} } @article{Prekopa90, Author ={A. Pr\'ekopa}, Title = {Sharp Bounds on Probabilities Using Linear Programming}, Journal = {Operations Research}, Year = {1990}, Volume = {38}, Pages ={227-239} } @article{Prekopa1990a, Author = {A. Pr\'ekopa}, Title = {The discrete moment problem and linear programming}, Journal ={Discrete Applied Mathematics}, Volume = {27}, Pages = {235-254}, Year = {1990} } @inproceedings{Prekopa92, Author = {A. Pr\'ekopa}, Title = {Inequalities on Expectations Based on the Knowledge of Multivariate Moments}, Year = {1992}, Booktitle = {Stochastic Inequalities}, Editor ={ M. Shaked and Y.L. Tong}, Organization= {Institute of Mathematical Statistics}, Series={Lecture Notes-Monograph Series}, Volume ={22}, Pages ={309-331} } @inproceedings{Prekopa2001, Author = {A. Pr\'ekopa}, Title = {Discrete higher order convex functions and their application}, Year = {2001}, Booktitle = {Generalized Convexity and Monotonicity}, Editor ={ N. Hadjisavvas and J.E. Martinez-Legaz and J. P. Penot}, Organization= {Springer}, Series={Lecture Notes in Economis and Mathematical Systems}, Pages ={21-47} } @book{Prekopa95, Author ={A. Pr\'ekopa}, Title = {Stochastic Programming}, Publisher ={Kluwer Scientific Publishers}, Year = {1995}, Address ={Dordrecht} } @article{Prekopa96, Author = {A. Pr\'ekopa}, Title ={A Brief Introduction to Linear Programming}, Journal ={ Math. Scientist}, Volume = {21}, Pages ={85-111}, Year = {1996} } @article{Prekopa98, Author = {A. Pr\'ekopa}, Title = {Bounds on Probabilities and Expectations Using Multivariate Moments of Discrete Distributions}, Journal ={ Studia Sci. Math. Hung.}, Volume = {34}, Pages = {349-378}, Year = {1998} } @article{Prekopa99, Author ={A. Pr\'ekopa}, Title = {The Use of Discrete Moment Bounds in Probabilistic Constrained Stochastic Programming Models}, Journal = {Annals of Operations Research}, Volume = {85}, Pages = {21-38}, Year = {1999} } @incollection{ Prekopa90b, Author ={A. Pr\'ekopa}, Title = {Totally Positive Linear Programming Problems}, Booktitle = {Functional Analysis, Optimization and Mathematical Economics (Collection of papers dedicated to the memory of L.V. Kantorovich, L.J. Leifman ed.)}, Publisher = {Oxford University Press}, Year = {1990}, Pages = {197-207} } @article{Prekopa00a, Author = {A. Pr\'ekopa and B. Vizv\'ari and G. Reg{\"{o}}s and L. Gao}, Title = {Bounding the Probability of the Union of Events by the Use of Aggregation and Disaggregation in Linear Programs}, Journal = {Rutcor Research Report}, Year = {2000}, note ={submitted to Operations Research} } @book{Rao99, Author = {R. Rao and H. Toutenburg}, Title ={Linear Models, Least Squares and Alternatives}, Publisher = {Springer}, Year = {1999}, Address = {New York} } @inbook{Rao99AppA, Author = {R. Rao and H. Toutenburg}, Title ={Linear Models, Least Squares and Alternatives}, Publisher = {Springer}, Year = {1999}, Address = {New York}, Pages = {353-390} } @proceedings{Roberts91, Editor ={F. Hwang and F. Roberts and C. Monma}, Year = {1991}, Title = {Reliability of Computer and CommunicationNetworks}, Series ={DIMACS Series in Discrete Mathematics and Theoretical Computer Science}, Number = {5} } @book{RobertsN81, Author = {N.H. Roberts and elt al}, Year = {1981}, Title ={Fault Free Handbook}, Publisher = {Nuclear Regulatory Commission}, Address = {Washington, D.C., Nureg-0492} } @book{Ross2000, Author = {S.M. Ross}, Year = {2000}, Title = {Introduction to Probability Models}, Pages = {137-141}, Publisher = {Academic Press}, Address = {New York}, note = {Seventh Edition} } @article{Szedmak2001, Author = {S. Szedmak}, Title = {An application of the Bose-Einstein distribution in binary optimization}, Journal = {Rutcor Research Report}, Year = {2001}, note ={submitted to The Journal of Heuristics} } @article{Yannakakis94, Author = {M. Yannakakis}, Title = {On the Approximation of Maximum Satisfiability}, Journal = {Journal of Algorithm}, Volume = {17}, Year = {1994}, Pages = {475-502} } @book{Press1986, author = {W.H. Press and B.P. Flannery and S.A. Teukolsky and W.T. Vetterling}, title = {Numerical Recipes: The Art of Scientific Computing}, publisher = {William H. Press}, address = {New York}, year = {1986} } @book{gnump, author = {GNU Open Source Community}, title = {GNU MP, The GNU Multiple Precision Arithmetic Library}, address = {http://www.swox.com/gmp/} } @book{mpfun, author = {D. Bailey and S. Chatterjee}, title = {MPFUN++, A Multiple Precison Floating Point Computation Package in C++}, address = {http://www.cs.unc.edu/Research/HARPOON/mpfun++} }