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  • 1
    Electronic Resource
    Electronic Resource
    Finite master programs in regularized stochastic decomposition (1994)
    Springer
    Mathematical programming 67 (1994), S. 143-168 
    Details
    ISSN: 1436-4646
    Keywords: Stochastic programming ; Decomposition ; Cutting plane methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Stochastic decomposition is a stochastic analog of Benders' decomposition in which randomly generated observations of random variables are used to construct statistical estimates of supports of the objective function. In contrast to deterministic Benders' decomposition for two stage stochastic programs, the stochastic version requires infinitely many inequalities to ensure convergence. We show that asymptotic optimality can be achieved with a finite master program provided that a quadratic regularizing term is included. Our computational results suggest that the elimination of the cutting planes impacts neither the number of iterations required nor the statistical properties of the terminal solution.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582219
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Inexact subgradient methods with applications in stochastic programming (1994)
    Springer
    Mathematical programming 63 (1994), S. 65-82 
    Details
    ISSN: 1436-4646
    Keywords: Subgradient methods ; Approximations ; stochastic optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In many instances, the exact evaluation of an objective function and its subgradients can be computationally demanding. By way of example, we cite problems that arise within the context of stochastic optimization, where the objective function is typically defined via multi-dimensional integration. In this paper, we address the solution of such optimization problems by exploring the use of successive approximation schemes within subgradient optimization methods. We refer to this new class of methods as inexact subgradient algorithms. With relatively mild conditions imposed on the approximations, we show that the inexact subgradient algorithms inherit properties associated with their traditional (i.e., exact) counterparts. Within the context of stochastic optimization, the conditions that we impose allow a relaxation of requirements traditionally imposed on steplengths in stochastic quasi-gradient methods. Additionally, we study methods in which steplengths may be defined adaptively, in a manner that reflects the improvement in the objective function approximations as the iterations proceed. We illustrate the applicability of our approach by proposing an inexact subgradient optimization method for the solution of stochastic linear programs.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582059
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Network planning with random demand (1994)
    Springer
    Telecommunication systems 3 (1994), S. 11-30 
    Details
    ISSN: 1572-9451
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract We study a planning problem associated with networks for private line services. In these networks, demands are known to exhibit considerable variability, and as such, they should be treated as random variables. The proposed planning model is a two-stage stochastic linear program (SLP) with recourse. Due to the enormous size of the deterministic equivalent, we choose a sampling based algorithm calledstochastic decomposition (SD). For very large-scale SLPs, such as the ones solved in this application, SD provides an effective methodology. The model presented in this paper is validated by using a detailed simulation of the network. We report results with a network that has 86 demand pairs, 89 links and 706 potential routes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02110042
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    A Stochastic Programming Model for Currency Option Hedging (2000)
    Springer
    Annals of operations research 100 (2000), S. 227-249 
    Details
    ISSN: 1572-9338
    Keywords: stochastic programming ; currency options ; hedging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper we use a stochastic programming approach to develop currency option hedging models which can address problems with multiple random factors in an imperfect market. The portfolios considered in our model are rebalanced at the end of each time period, and reinvestments are allowed during the hedging process. These sequential decisions (reinvestments) are based on the evolution of random parameters such as exchange rates, interest rates, etc. We also allow the inclusion of a variety of instruments in the hedging portfolio, including short term derivative securities, short term options, and futures. These instruments help generate strategies that provide good liquidity and low trade intensity. One of the important features of the model is that it incorporates constraints on sensitivity measures such as Delta and Gamma. By ensuring that these hedge parameters track a desired trajectory (e.g., the parameters of a target option), the new model provides investment strategies that are robust with respect to the perturbations measured by Delta and Gamma. In order to manage the explosion of scenarios due to multiple random factors, we incorporate sampling within a scenario aggregation algorithm. We illustrate that when compared with other myopic hedging methods in imperfect markets, the new stochastic programming model can provide better performance. Our examples also illustrate stochastic programming as a practical computational tool for realistic hedging problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019296422231
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    Paper (German National Licenses)
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  • 5
    Electronic Resource
    Electronic Resource
    Statistical verification of optimality conditions for stochastic programs with recourse (1991)
    Springer
    Annals of operations research 30 (1991), S. 215-239 
    Details
    ISSN: 1572-9338
    Keywords: Stochastic programming ; optimality criteria ; large scale optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Statistically motivated algorithms for the solution of stochastic programming problems typically suffer from their inability to recognize optimality of a given solution algorithmically. Thus, the quality of solutions provided by such methods is difficult to ascertain. In this paper, we develop methods for verification of optimality conditions within the framework of Stochastic Decomposition (SD) algorithms for two stage linear programs with recourse. Consistent with the stochastic nature of an SD algorithm, we provide termination criteria that are based on statistical verification of traditional (deterministic) optimality conditions. We propose the use of “bootstrap methods” to confirm the satisfaction of generalized Kuhn-Tucker conditions and conditions based on Lagrange duality. These methods are illustrated in the context of a power generation planning model, and the results are encouraging.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02204818
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Statistical approximations for recourse constrained stochastic programs (1995)
    Springer
    Annals of operations research 56 (1995), S. 157-175 
    Details
    ISSN: 1572-9338
    Keywords: Stochastic programming ; approximations ; risk models
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract The two stage stochastic program with recourse is known to have numerous applications in financial planning, energy modeling, telecommunications systems etc. Notwithstanding its applicability, the two stage stochastic program is limited in its ability to incorporate a decision maker's attitudes towards risk. In this paper we present an extension via the inclusion of a recourse constraint. This results in a convex integrated chance constraint (ICC), which inherits the convexity properties of two stage programs. However, it also inherits some of the difficulties associated with the evaluation of recourse functions. This motivates our study of conditions that may be applicable to algorithms using statistical approximations of such ICC. We present a set of sufficient conditions that these approximations may satisfy in order to assure convergence. Our conditions are satisfied by a wide range of statistical approximations, and we demonstrate that these approximations can be generated within standard algorithmic procedures.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02031705
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Statistical approximations forstochastic linear programming problems (1999)
    Springer
    Annals of operations research 85 (1999), S. 173-193 
    Details
    ISSN: 1572-9338
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Sampling and decomposition constitute two of the most successful approaches foraddressing large‐scale problems arising in statistics and optimization, respectively. In recentyears, these two approaches have been combined for the solution of large‐scale stochasticlinear programming problems. This paper presents the algorithmic motivation for suchmethods, as well as a broad overview of issues in algorithm design. We discuss both basicschemes as well as computational enhancements and stopping rules. We also introduce ageneralization of current algorithms to handle problems with random recourse.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018917710373
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    Duality Gaps in Stochastic Integer Programming (2000)
    Springer
    Journal of global optimization 18 (2000), S. 189-194 
    Details
    ISSN: 1573-2916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this note, we explore the implications of a result that suggests that the duality gap caused by a Lagrangian relaxation of the nonanticipativity constraints in a stochastic mixed integer (binary) program diminishes as the number of scenarios increases. By way of an example, we illustrate that this is not the case. In general, the duality gap remains bounded away from zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008314824754
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