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  • 1
    Electronic Resource
    Electronic Resource
    On structure and stability in stochastic programs with random technology matrix and complete integer recourse (1995)
    Springer
    Mathematical programming 70 (1995), S. 73-89 
    Details
    ISSN: 1436-4646
    Keywords: Stochastic integer programming ; Parametric integer programming ; Continuity ; Stability ; Weak convergence of probability measures
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract For two-stage stochastic programs with integrality constraints in the second stage, we study continuity properties of the expected recourse as a function both of the first-stage policy and the integrating probability measure. Sufficient conditions for lower semicontinuity, continuity and Lipschitz continuity with respect to the first-stage policy are presented. Furthermore, joint continuity in the policy and the probability measure is established. This leads to conclusions on the stability of optimal values and optimal solutions to the two-stage stochastic program when subjecting the underlying probability measure to perturbations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585929
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    A simple recourse model for power dispatch under uncertain demand (1995)
    Springer
    Annals of operations research 59 (1995), S. 135-164 
    Details
    ISSN: 1572-9338
    Keywords: Power dispatch under uncertainty ; stochastic programming ; asymptotic stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Optimal power dispatch under uncertainty of power demand is tackled via a stochastic programming model with simple recourse. The decision variables correspond to generation policies of a system comprising thermal units, pumped storage plants and energy contracts. The paper is a case study to test the kernel estimation method in the context of stochastic programming. Kernel estimates are used to approximate the unknown probability distribution of power demand. General stability results from stochastic programming yield the asymptotic stability of optimal solutions. Kernel estimates lead to favourable numerical properties of the recourse model (no numerical integration, the optimization problem is smooth convex and of moderate dimension). Test runs based on real-life data are reported. We compute the value of the stochastic solution for different problem instances and compare the stochastic programming solution with deterministic solutions involving adjusted demand portions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02031746
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On the Glivenko-Cantelli problem in stochastic programming: Mixed-integer linear recourse (1998)
    Springer
    Mathematical methods of operations research 47 (1998), S. 39-49 
    Details
    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
    URL: http://dx.doi.org/10.1007/BF01193835
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    Paper (German National Licenses)
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  • 4
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    Discontinuous Optimization Problems in Stochastic Integer Programming (1995)
    Details
    Publication Date: 2014-02-26
    Description: Integer stochastic linear programming is considered from the viewpoint of discontinuous optimization. After reviewing solution approaches via mollifier subgradients and decomposition we outline how to base a solution method on efficient pointwise calculation of the objective employing computer algebra.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 5
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    Dual Decomposition in Stochastic Integer Programming (1996)
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    Publication Date: 2014-02-26
    Description: We present an algorithm for solving stochastic integer programming problems with recourse, based on a dual decomposition scheme and Lagrangian relaxation. The approach can be applied to multi-stage problems with mixed-integer variables in each time stage. %We outline a branch-and-bound algorithm for obtaining primal feasible and %possibly optimal solutions. Numerical experience is presented for some two-stage test problems.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 6
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    On the Glivenko-Cantelli Problem in Stochastic Programming: Mixed-Integer Linear Recourse (1997)
    Details
    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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  • 7
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    On the Glivenko-Cantelli Problem in Stochastic Programming: Linear Recourse and Extensions (1996)
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    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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  • 8
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    A Two-Stage Stochastic Program for Unit Commitment Under Uncertainty in a Hydro-Thermal Power System (1998)
    Details
    Publication Date: 2014-02-26
    Description: We develop a two-stage stochastic programming model with integer first-stage and mixed-integer recourse for solving the unit commitment problem in power generation in the presence of uncertainty of load profiles. The solution methodology rests on a novel scenario decomposition method for stochastic integer programming. This method combines Lagrangian relaxation of non-anticipativity constraints with branch-and-bound. It can be seen as a decomposition algorithm for large-scale mixed-integer linear programs with block-angular structure. With realistic data from a German utility we validate our model and carry out test runs. Sizes of these problems go up to 20.000 integer and 150.000 continuous variables together with up to 180.000 constraints.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 9
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    Structure and Stability in Two-Stage Stochastic Programming (1995)
    Details
    Publication Date: 2020-08-05
    Description: {\begin{footnotesize} This thesis is concerned with structural properties and the stability behaviour of two-stage stochastic programs. Chapter~1 gives an introduction into stochastic programming and a summary of the main results of the thesis. In Chapter~2 we present easily verifiable sufficient conditions for the strong convexity of the expected-recourse function in a stochastic program with linear complete recourse. Different levels of randomness in the data are considered. We start with models where only the right-hand side of the constraints is random and extend these results to the situation where also the technology matrix contains random entries. The statements on strong convexity imply new stability estimates for sets of optimal solutions when perturbing the underlying probability measure. We work out Hölder estimates (in terms of the $\mbox{L}_1$-Wasserstein distance) for optimal solution sets to linear recourse models with random technology matrix. In Chapter~3 ({\it joint work with Werner Römisch, Berlin}) we are aiming at the Lipschitz stability of optimal solution sets to linear recourse models with random right-hand side. To this end , we first adapt the distance notion for the underlying probability measures to the structure of the model and derive a Lipschitz estimate for optimal solutions based on that distance. Here, the strong convexity established in Chapter~2 turns out as an essential assumption. For applications, however, a Lipschitz estimate with respect to a more accesssible probability distance is desirable. Structural properties of the expected-recourse function finally permit such an estimate in terms of the Kolmogorov-Smirnov distance of linear transforms of the underlying measures. The general analysis is specified to estimation via empirical measures. We obtain a law of iterated logarithm, a large deviation estimate and an estimate for the asymptotic distribution of optimal solution sets. Chapters~4 and~5 deal with two-stage linear stochastic programs where integrality constraints occur in the second stage. In Chapter~4 we study basic continuity properties of the expected-recourse function for models with random right-hand side and random technology matrix. The joint continuity with respect to the decision variable and the underlying probability measure leads to qualitative statements on the stability of local optimal values and local optimal solutions. In Chapter~5 we demonstrate that a variational distance of probability measures based on a suitable Vapnik-\v{C}ervonenkis class of Borel sets leads to convergence rates of the Hölder type for the expected recourse as a function of the underlying probability measure. The rates carry over to the convergence of local optimal values. As an application we again consider estimation via empirical measures. Beside qualitative asymptotic results for optimal values and optimal solutions we obtain a law of iterated logarithm for optimal values. \end{footnotesize}}
    Keywords: ddc:000
    Language: English
    Type: doctoralthesis , doc-type:doctoralThesis
    Format: application/postscript
    Format: application/pdf
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  • 10
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    Optimale Blockauswahl bei der Kraftwerkseinsatzplanung (1996)
    Details
    Publication Date: 2014-02-26
    Description: The paper addresses the unit commitment problem in power plant operation planning. For a real power system comprising coal and gas fired thermal as well as pumped storage hydro plants a large-scale mixed integer optimization model for unit commitment is developed. Then primal and dual approaches to solving the optimization problem are presented and results of test runs are reported.
    Keywords: ddc:000
    Language: German
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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