## A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming (2000)

Venue: | Convex Programming, Mathematical Programming 104, 153 |

Citations: | 7 - 3 self |

### BibTeX

@TECHREPORT{Berkelaar00aprimal-dual,

author = {Arjan Berkelaar and Roy Kouwenberg and Shuzhong Zhang},

title = {A Primal-Dual Decomposition Algorithm for Multistage Stochastic Convex Programming},

institution = {Convex Programming, Mathematical Programming 104, 153},

year = {2000}

}

### OpenURL

### Abstract

This paper presents a new and high performance solution method for multistage stochastic convex programming. Stochastic programming is a quantitative tool developed in the field of optimization to cope with the problem of decision-making under uncertainty. Among others, stochastic programming has found many applications in finance, such as asset-liability and bond-portfolio management. However, many stochastic programming applications still remain computationally intractable because of their overwhelming dimensionality. In this paper we propose a new decomposition algorithm for multistage stochastic programming with a convex objective, based on the path-following interior point method combined with the homogeneous self-dual embedding technique. Our preliminary numerical experiments show that this approach is very promising in many ways for solving generic multistage stochastic programming, including its superiority in terms of numerical e#ciency, as well as the flexibility in testing and analyzing the model.

### Citations

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Citation Context ...age called MOSEK). The HSD method is also coded beyond linear programming. In fact, it is implemented for a much broader class of optimization problems known as semidefinite programming; see [20] and =-=[25]-=-. Furthermore, the HSD method is extended to a general convex programming framework by Andersen and Ye; see [2]. One of the advantages of the HSD method is that it requires no feasibility phase, allow... |

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Citation Context ...referred to the recent book on asset liability management edited by Mulvey and Ziemba [24]. However, e#ciently solving large-scale stochastic programming problems still remains a major challenge; see =-=[10]-=- and [19] for an introduction to stochastic programming. The main di#culty is caused by the nature of the stochastic programming model --- its overwhelming dimensionality. This happens when one models... |

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Citation Context ...ized decomposition; we refer to [10] and the references therein for a survey along this direction. A classical approach in that case is the so-called L-shaped method [27] and its multistage extension =-=[7]-=-, which are variants of the Benders decomposition. Recently, multistage linear stochastic programs with millions of variables and constraints have been solved with parallel implementations of Benders ... |

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Citation Context ...ct, our computational approach relies on the homogeneous self-dual method originally introduced by Xu, Hung and Ye [28] as a simplification of the self-dual embedding technique of Ye, Todd and Mizuno =-=[30]-=- for linear programming. This method was applied by Berkelaar, Dert, Oldenkamp, and Zhang [4] for solving two-stage stochastic linear programming problems. The crucial observation made by Berkelaar, D... |

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Citation Context ...edding technique of Ye, Todd and Mizuno [30]. This technique proves to be very e#cient in solving linear programs (a refined version of the HSD method is actually implemented by Andersen and Andersen =-=[1]-=- in an optimization package called MOSEK). The HSD method is also coded beyond linear programming. In fact, it is implemented for a much broader class of optimization problems known as semidefinite pr... |

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Citation Context ... of the size of the problem. This property is certainly of critical importance in solving large scale stochastic programming problems. Moreover, the IPMs are suitable to handle nonlinear problems. In =-=[11]-=- Birge and Qi showed how decomposition can be achieved based on Karmarkar's original interior point method for two-stage stochastic linear programming. Within the interior point method realm, in fact,... |

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Citation Context ...riants of the Benders decomposition. Recently, multistage linear stochastic programs with millions of variables and constraints have been solved with parallel implementations of Benders decomposition =-=[8, 14, 16]-=-. However, Benders decomposition is limited to problems with a linear objective function. There has been much less progress in solving large scale stochastic programming problems with a general convex... |

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Citation Context ...o types of decomposition methods have appeared. The first type, including [11], exploits the structure of two-stage stochastic linear programming which is viewed as large size linear programming; see =-=[21, 9, 12]-=- for serial algorithms and [18, 29, 13] for parallel implementations. The second type of interior point decomposition methods typically specializes some of the interior point methodology, such as cutt... |

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Citation Context ...tion package called MOSEK). The HSD method is also coded beyond linear programming. In fact, it is implemented for a much broader class of optimization problems known as semidefinite programming; see =-=[20]-=- and [25]. Furthermore, the HSD method is extended to a general convex programming framework by Andersen and Ye; see [2]. One of the advantages of the HSD method is that it requires no feasibility pha... |

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Citation Context ... broader class of optimization problems known as semidefinite programming; see [20] and [25]. Furthermore, the HSD method is extended to a general convex programming framework by Andersen and Ye; see =-=[2]-=-. One of the advantages of the HSD method is that it requires no feasibility phase, allowing one to freely select any interior starting point (possibly infeasible). Moreover, the method is capable of ... |

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Citation Context ...o types of decomposition methods have appeared. The first type, including [11], exploits the structure of two-stage stochastic linear programming which is viewed as large size linear programming; see =-=[21, 9, 12]-=- for serial algorithms and [18, 29, 13] for parallel implementations. The second type of interior point decomposition methods typically specializes some of the interior point methodology, such as cutt... |

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Citation Context ...riants of the Benders decomposition. Recently, multistage linear stochastic programs with millions of variables and constraints have been solved with parallel implementations of Benders decomposition =-=[8, 14, 16]-=-. However, Benders decomposition is limited to problems with a linear objective function. There has been much less progress in solving large scale stochastic programming problems with a general convex... |

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Citation Context ...d by Xu, Hung and Ye [28] as a simplification of the self-dual embedding technique of Ye, Todd and Mizuno [30] for linear programming. This method was applied by Berkelaar, Dert, Oldenkamp, and Zhang =-=[4]-=- for solving two-stage stochastic linear programming problems. The crucial observation made by Berkelaar, Dert, Oldenkamp, and Zhang in [4] is that it is possible to completely decompose the direction... |

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Citation Context ...e of interior point decomposition methods typically specializes some of the interior point methodology, such as cutting planes or barrier methods, to 2 solve stochastic programming problems; see e.g. =-=[5, 3, 31]-=-. However, the key di#culty for the first type of IPMs in this context is that in each iteration it usually involves solving a very large, perhaps even ill-conditioned, direction-finding linear system... |

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Citation Context ...o types of decomposition methods have appeared. The first type, including [11], exploits the structure of two-stage stochastic linear programming which is viewed as large size linear programming; see =-=[21, 9, 12]-=- for serial algorithms and [18, 29, 13] for parallel implementations. The second type of interior point decomposition methods typically specializes some of the interior point methodology, such as cutt... |

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Citation Context ...optimization models such as asset-liability and bond-portfolio management; in this particular application, one is referred to the recent book on asset liability management edited by Mulvey and Ziemba =-=[24]-=-. However, e#ciently solving large-scale stochastic programming problems still remains a major challenge; see [10] and [19] for an introduction to stochastic programming. The main di#culty is caused b... |

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Citation Context ...e of interior point decomposition methods typically specializes some of the interior point methodology, such as cutting planes or barrier methods, to 2 solve stochastic programming problems; see e.g. =-=[5, 3, 31]-=-. However, the key di#culty for the first type of IPMs in this context is that in each iteration it usually involves solving a very large, perhaps even ill-conditioned, direction-finding linear system... |