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## 4. TITLE AND SUBTITLE Adaptive Selections of Sample Size and Solver Iterations in Stochastic Optimization with Application to Nonlinear Commodity Flow Problems (2009)

### Citations

342 | Stochastic Programming
- Kall, Wallace
- 1994
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Citation Context ...ization model for a stochastic lot-scheduling problem. One approach to approximately solve optimization problems defined in terms of expectations is to use Sample Average Approximations (SAA), e.g., (=-=Ruszczynski and Shapiro, 2007-=-). That is, one or more approximating problems (APs) are solved, problems that replace each expectation with a standard sample average approximation. The AP may be viewed as a deterministic mathematic... |

252 |
Optimization: Algorithms and Consistent Approximations,
- Polak
- 1997
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Citation Context ...for each stage, but all samples are independent between stages. For purposes of this research, we have chosen the Projected Gradient Method (PGM) as our nonlinear programming solver (for example, see =-=Polak 1997-=-, p. 66). However, it is worth mentioning that any linearly convergent nonlinear programming solver may be used. Our goal is to determine N k and n for each stage that approximately minimizes the tota... |

99 | Monte Carlo bounding techniques for determining solution quality in stochastic programs. - Mak, Morton, et al. - 1999 |

78 | A stochastic programming approach for supply chain network design under uncertainty, - Santoso, Ahmed, et al. - 2005 |

55 | A simulation-based approach to two-stage stochastic programming with recourse. - Shapiro, Mello - 1998 |

38 | Approximation and model management in aerodynamic optimization with variable- fidelity modeling
- Alexandrov, Lewis, et al.
- 2001
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Citation Context ... Numerous design and planning applications require the optimization of stochastic-programming problems. In aerodynamics, various problems arise in the optimization of a three-dimensional wing design (=-=Alexandrov et al., 2001-=-). In one civilengineering discipline, problems arise from structural optimization of bridges or support structures subject to failure probability constraints (Polak and Royset, 2007; Royset and Polak... |

37 | Congestion toll pricing of traffic networks. - Bergendorff, Hearn, et al. - 1997 |

29 | Stochastic programming by Monte Carlo simulation methods - Shapiro |

13 | Eficient sample sizes in stochastic nonlinear programing, - Polak, Royset - 2008 |

10 | Eds., Equilibrium and Advanced Transportation Modeling. - Marcotte, Nguyen - 1998 |

4 | Computational Methods for Congestion Toll Pricing Models - Hearn, Yildirim, et al. - 2001 |

4 | Adaptive and nonadaptive samples in solving stochastic linear programs Computational investigation, Submitted to Computational Optimization and Applications
- Higle, Zhao
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Citation Context ...ples are taken before the solution algorithm begins (or they could be taken before the algorithm begins), and no additional sampling is performed during the optimization process. In internal methods (=-=Higle and Zhao, 2004-=-), samples are intrinsic to the iterative solution process, performed whenever the algorithm requires the estimation of expectations. An alternative to an external sampling approach with a fixed sampl... |

2 |
LSSOL 1.0 User’s guide
- Gill, Hammarling, et al.
- 1986
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Citation Context ...onlinear-programming algorithm with Armijo step size rule; for example, see, Polak (1997, p. 67) and Bertsekas (1999, p. 31). The quadratic direction-finding problem in the PGM is solved using LSSOL (=-=Gill et al., 1986-=-) as implemented in TOMLAB 7.0 (Holmstrom, 2008). We use parameters α = 0.5 and β = 0.8 in the Armijo step-size rule and in Subroutine PE use an exponential smoothing parameter ψ = 1/ 3 and tolerance ... |

2 | Effective diagonalization strategies for the solution of a class of optimal design problems - He, Polak - 1990 |

2 | Implementable algorithm for stochastic optimization using sample average approximations - Royset, Polak |

1 | Network flows. Prentice-Hall, Upper Saddle River - Ahuja, Magnanti, et al. - 1993 |

1 | Robust solutions and risk measures for a supply chain planning problem under uncertainty - Poojari, Lucas, et al. - 2008 |

1 |
Adaptive control of sample size in stochastic optimization. Available from author
- Royset
- 2009
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Citation Context ... this research. Following his approach, we first identify the asymptotic distributions of the progress made by CA. We assume that the solver used in Step 3 of CA is uniformly linearly convergent (see =-=Royset, 2009-=-), i.e., there exists a θ ∈ ( 0,1) such that ∗ ∗ f ( P ( x)) − f ≤θ( f ( x) − f ) for all x ∈ X and N ∈ ¥ , where ¥ = {1,2,3,...} and PN N N N N N ( x ) is the iterate found after carrying out one ite... |

1 |
Extensions of stochastic optimization results toproblems with system failure probability functions
- Royset, Polak
- 2007
(Show Context)
Citation Context ...rov et al., 2001). In one civilengineering discipline, problems arise from structural optimization of bridges or support structures subject to failure probability constraints (Polak and Royset, 2007; =-=Royset and Polak, 2007-=-). Such problems may seek to optimize the cross-sectional dimensions of a support column consisting of a material of particular yield strength subjected to various bending moments. Structural loading ... |