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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 89 (2000), S. 55-77 
    ISSN: 1436-4646
    Keywords: Key words: probabilistic programming – discrete distributions – generalized concavity – column generation Mathematics Subject Classification (1991): 90C15, 90C11, 65K05, 49M27
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract. We consider stochastic programming problems with probabilistic constraints involving integer-valued random variables. The concept of a p-efficient point of a probability distribution is used to derive various equivalent problem formulations. Next we introduce the concept of r-concave discrete probability distributions and analyse its relevance for problems under consideration. These notions are used to derive lower and upper bounds for the optimal value of probabilistically constrained stochastic programming problems with discrete random variables. The results are illustrated with numerical examples.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 58 (1993), S. 201-228 
    ISSN: 1436-4646
    Keywords: Stochastic programming ; dynamic programming ; decomposition ; parallel computing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A new decomposition method for multistage stochastic linear programming problems is proposed. A multistage stochastic problem is represented in a tree-like form and with each node of the decision tree a certain linear or quadratic subproblem is associated. The subproblems generate proposals for their successors and some backward information for their predecessors. The subproblems can be solved in parallel and exchange information in an asynchronous way through special buffers. After a finite time the method either finds an optimal solution to the problem or discovers its inconsistency. An analytical illustrative example shows that parallelization can speed up computation over every sequential method. Computational experiments indicate that for large problems we can obtain substantial gains in efficiency with moderate numbers of processors.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 19 (1980), S. 220-229 
    ISSN: 1436-4646
    Keywords: Stochastic Programming ; Feasible Direction Methods ; Point-to-Set Maps ; Convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A unified approach to stochastic feasible direction methods is developed. An abstract point-to-set map description of the algorithm is used and a general convergence theorem is proved. The theory is used to develop stochastic analogs of classical feasible direction algorithms.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical programming 35 (1986), S. 309-333 
    ISSN: 1436-4646
    Keywords: Large scale linear programming ; stochastic programming ; subgradient methods ; semidefinite quadratic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Annals of operations research 64 (1996), S. 289-309 
    ISSN: 1572-9338
    Keywords: Stochastic programming ; decomposition ; augmented Lagrangian ; Jacobi method ; parallel computation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract A general decomposition framework for large convex optimization problems based on augmented Lagrangians is described. The approach is then applied to multistage stochastic programming problems in two different ways: by decomposing the problem into scenarios and by decomposing it into nodes corresponding to stages. Theoretical convergence properties of the two approaches are derived and a computational illustration is presented.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Annals of operations research 85 (1999), S. 153-172 
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Stochastic programming problems have very large dimension and characteristic structureswhich are tractable by decomposition. We review some new developments in cutting planemethods, augmented Lagrangian and splitting methods for linear multi‐stage stochasticprogramming problems.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Mathematical methods of operations research 47 (1998), S. 39-49 
    ISSN: 1432-5217
    Keywords: Stochastic Programming ; Empirical Measures ; Uniform Convergence ; Value Functions of Mixed-Integer Linear Programs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.
    Type of Medium: Electronic Resource
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  • 8
    Book
    Book
    Philadelphia :SIAM, Soc. for Industrial and Applied Math.,
    Title: Lectures on stochastic programming : modeling and theory; 9
    Author: Shapiro, Alexander
    Contributer: Dentcheva, Darinka , Ruszczyński, Andrzej P.
    Edition: 2. ed.
    Publisher: Philadelphia :SIAM, Soc. for Industrial and Applied Math.,
    Year of publication: 2014
    Pages: XV, 436 S.
    Series Statement: MPS-SIAM series on optimization 9
    ISBN: 978-1-611973-42-6
    Type of Medium: Book
    Language: English
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  • 9
    Publication Date: 2014-02-26
    Description: Expected recourse functions in linear two-stage stochastic programs with mixed-integer second stage are approximated by estimating the underlying probability distribution via empirical measures. Under mild conditions, almost sure uniform convergence of the empirical means to the original expected recourse function is established.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 10
    Publication Date: 2014-02-26
    Description: Integrals of optimal values of random optimization problems depending on a finite dimensional parameter are approximated by using empirical distributions instead of the original measure. Under fairly broad conditions, it is proved that uniform convergence of empirical approximations of the right hand sides of the constraints implies uniform convergence of the optimal values in the linear and convex case.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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