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## Dual decomposition in stochastic integer programming. (1999)

Venue: | Operations Research Letters, |

Citations: | 100 - 5 self |

### Citations

189 |
Scenarios and policy aggregation in optimization under uncertainty
- Rockafellar, Wets
- 1991
(Show Context)
Citation Context ... of constraints as an alternative to the well-known Lagrangian relaxation approach. The variable splitting method is equivalent to what is called Lagrangian decomposition of (2), by Guignard & Kim [5]. In stochastic programming, non-anticipativity conditions are “hard” since they couple constraints for the different scenarios. For linear problems without integer requirements there exist well developed theory and methodology for relaxing non-anticipativity constraints. Based on duality results involving augmented Lagrangians, algorithms like the progressive hedging method of Rockafellar & Wets [12] and the Jacobi method of Rosa & Ruszczynski [13] were developed and applied to a variety of problems. As elaborated above duality gaps occur in the presence of integer requirements such that the above methods are no longer formally justified. In the next section we will employ branch-and-bound to close the duality gap. This will also lead to optimality estimates for the feasible solutions that are generated in the course of the method. 3 A Branch-and-Bound algorithm Lagrangian duality provides upper bounds on the optimal value of problem (3) and corresponding optimal solutions (xj, yj), j = ... |

119 |
Proximity control in bundle methods for convex nondifferentiable optimization”,
- Kiwiel
- 1990
(Show Context)
Citation Context ... in Section 3. Since non-anticipativity constraints involve variables from all but the final time stage, branching in the multi-stage case has to concern the variables x1, . . . , xT−1 instead of only x1 in the two-stage case. Together with the increased dimension of the Lagrangian dual this more expensive branching is the main source of increased computational effort when extending the scheme from Section 3 to multi-stage models. 5 Numerical examples We have implemented the branch-and-bound algorithm of Section 3 using NOA 3.0 [7], which is an implementation of Kiwiels proximal bundle method [8] for non-differentiable optimization. The non-anticipativity condition was represented using the constraints x1 = x2, x1 = x2, . . . , x1 = xr. At each node of the branching tree we chose to branch on the component x(k) for which the dispersion maxj x j (k) −minj xj(k) was largest. For node selection we chose the node with the largest l∞-norm of dispersions. To obtain good lower bounds we used this rule intertwined with the Best-Bounds rule. The mixedinteger subproblems were solved using the CPLEX 4.0 Callable Library [4]. The experiments were carried out on a Sun SPARCstation 20 with 160 MB m... |

100 | A stochastic model for the unit commitment problem - Takriti, Birge, et al. - 1996 |

96 |
Lagrangean decomposition: a model yielding stronger Lagrangean bounds.
- Guignard, Kim
- 1987
(Show Context)
Citation Context ...ted to existing techniques in both combinatorial optimization and in stochastic programming. In combinatorial optimization, the idea of creating copies of variables and then relaxing the equality constraints for these 6 variables was introduced as variable splitting by Jornsten et al. [6]. The variable splitting approach was originally applied to optimization problems with a “hard” and an “easy” set of constraints as an alternative to the well-known Lagrangian relaxation approach. The variable splitting method is equivalent to what is called Lagrangian decomposition of (2), by Guignard & Kim [5]. In stochastic programming, non-anticipativity conditions are “hard” since they couple constraints for the different scenarios. For linear problems without integer requirements there exist well developed theory and methodology for relaxing non-anticipativity constraints. Based on duality results involving augmented Lagrangians, algorithms like the progressive hedging method of Rockafellar & Wets [12] and the Jacobi method of Rosa & Ruszczynski [13] were developed and applied to a variety of problems. As elaborated above duality gaps occur in the presence of integer requirements such that the... |

96 | The integer L-shaped method for stochastic integer programs with complete recourse. - Laporte, Louveaux - 1993 |

58 | On convergence of an augmented lagrangian decomposition method for sparse convex optimization”.
- Ruszczynski
- 1995
(Show Context)
Citation Context ...n Lagrangian relaxation approach. The variable splitting method is equivalent to what is called Lagrangian decomposition of (2), by Guignard & Kim [5]. In stochastic programming, non-anticipativity conditions are “hard” since they couple constraints for the different scenarios. For linear problems without integer requirements there exist well developed theory and methodology for relaxing non-anticipativity constraints. Based on duality results involving augmented Lagrangians, algorithms like the progressive hedging method of Rockafellar & Wets [12] and the Jacobi method of Rosa & Ruszczynski [13] were developed and applied to a variety of problems. As elaborated above duality gaps occur in the presence of integer requirements such that the above methods are no longer formally justified. In the next section we will employ branch-and-bound to close the duality gap. This will also lead to optimality estimates for the feasible solutions that are generated in the course of the method. 3 A Branch-and-Bound algorithm Lagrangian duality provides upper bounds on the optimal value of problem (3) and corresponding optimal solutions (xj, yj), j = 1, . . . , r, of the Lagrangian relaxation. In gen... |

43 | L-shaped decomposition of two-stage stochastic programs with integer recourse. - Carøe, Tind - 1998 |

31 |
The Value of the Stochastic Solution in Stochastic Linear Programs with Fixed Recourse.
- BIRGE
- 1982
(Show Context)
Citation Context ...m . The Lagrangian dual provides considerably better lower bounds than the LP-relaxation. For our test runs we used 10−3 as optimality tolerance in NOA which gave a duality gap at the root nodes of 0.2% – 0.3%. The LP-relaxation, however, gives a duality gap of 2.0% – 2.1%. Notice that a smaller optimality tolerance in NOA will produce better bounds but is also more time consuming. It should be noted that the size of the duality gap indicates that the scenario solutions have almost identical first-stage components. Hence we have calculated the value of the stochastic solution (VSS), see Birge [1], which measures the value of using a stochastic model instead of a deterministic model. For all three problems, the VSS was less than 0.8%, which means that the randomness has little influence on the optimal first-stage solution. 11 Problem Best solution CPU-time Lower bound SIZES3 224599.2 127 sec. 224360.0 SIZES5 224680.4 861 sec. 224369.0 SIZES10 224744.3 956 sec. 224311.4 Table 1: Table of results for SIZES-problems Acknowledgements We are grateful to Krzysztof C. Kiwiel for permission to use the NOA 3.0 package and to A. Løkketangen and D. L. Woodruff for providing us with data for the S... |

24 | Stochastic programming approaches to stochastic scheduling - Birge, Dempster - 1996 |

20 |
Progressive Hedging and Tabu Search Applied To Mixed Integer (0,1) Multi-Stage Stochastic Programming”,
- Løkketangen, Woodruff
- 1996
(Show Context)
Citation Context ... stopping enumeration after 500.000 nodes. Using our branch-and-bound algorithm the problem could be solved in 136 seconds CPU-time, yielding the optimal solution x = (0, 4) and corresponding value z = 61.32. Notice that the scenario subproblems are very small, so better runtimes may be achieved by grouping together scenarios. However, the problem is only meant as a benchmark for testing algorithms. Example 2: To compare the behavior of our algorithm with problems from the literature having larger second stages, we consider a family of two-stage mixed-integer minimization problems analyzed in [10]. The problems SIZES3, SIZES5 and SIZES10 have 3, 5 and 10 scenarios, respectively, and the scenario subproblems have 10 boolean variables, 65 bounded continuous variables and 31 constraints in each stage, with randomness occurring only in the right-hand side of the second-stage problem. The computational results are summarized in Table 1. The second column shows the time after which the best feasible solutions were found and the third column shows the lower bounds obtained after 1000 seconds of CPU-time, where the test runs were stopped. Contrary to the method in [10], we can estimate the fea... |

17 | Solving stochastic programs with integer recourse by enumeration: A framework using Grobner basis,
- Schultz, Stougie, et al.
- 1998
(Show Context)
Citation Context ...was represented using the constraints x1 = x2, x1 = x2, . . . , x1 = xr. At each node of the branching tree we chose to branch on the component x(k) for which the dispersion maxj x j (k) −minj xj(k) was largest. For node selection we chose the node with the largest l∞-norm of dispersions. To obtain good lower bounds we used this rule intertwined with the Best-Bounds rule. The mixedinteger subproblems were solved using the CPLEX 4.0 Callable Library [4]. The experiments were carried out on a Sun SPARCstation 20 with 160 MB memory. 10 Example 1: The following stochastic program was adapted from [14]: max{3 2 x1 + 4x2 +Q(x1, x2) : 0 ≤ x1, x2 ≤ 5 and integer} (16) where Q(x1, x2) is the expected value of the multiknapsack problem max 16y1 + 19y2 + 23y3 + 28y4 s.t. 2y1 + 3y2 + 4y3 + 5y4 ≤ ξ1 − x1, 6y1 + y2 + 3y3 + 2y4 ≤ ξ2 − x2, yi ∈ {0, 1}, i = 1, . . . , 4, and the random variable ξ = (ξ1, ξ2) is uniformly distributed on Ξ = {(5, 5), (5, 5.5), . . . (5, 15), (5.5, 5), . . . , (15, 15)}. The deterministic equivalent of (16) is an integer program with 1764 binary variables and 2 integer variables. Attempting to solve the problem with CPLEX yields an optimality gap of more than 25%, stopping... |

12 | Stochastic programming with integer recourse. - Vlerk - 1995 |

6 | User’s Guide for NOA 3.0: A Fortran Package for Convex Nondifferentiable Optimization - Kiwiel - 1993 |

6 | Progressive hedging and tabu search applied to mixed integer (0,1) multi-stage stochastic programming - Lkketangen, Woodru - 1996 |

3 | Wets, "Scenarios and policy aggregation in optimization under uncertainty - Rockafellar, J-B - 1991 |

2 |
Variable splitting–A New Lagrangean Relaxation Approach to
- Jornsten, Nasberg, et al.
- 1985
(Show Context)
Citation Context ...subgradient procedures in two ways. A small number of parameters give less controllability and the duality gap is increased, viz. Proposition 4. On the other hand, more control parameters mean that a larger space of parameter settings has to be searched and more iterations may be needed. Our approach can be related to existing techniques in both combinatorial optimization and in stochastic programming. In combinatorial optimization, the idea of creating copies of variables and then relaxing the equality constraints for these 6 variables was introduced as variable splitting by Jornsten et al. [6]. The variable splitting approach was originally applied to optimization problems with a “hard” and an “easy” set of constraints as an alternative to the well-known Lagrangian relaxation approach. The variable splitting method is equivalent to what is called Lagrangian decomposition of (2), by Guignard & Kim [5]. In stochastic programming, non-anticipativity conditions are “hard” since they couple constraints for the different scenarios. For linear problems without integer requirements there exist well developed theory and methodology for relaxing non-anticipativity constraints. Based on duali... |

2 | der Vlerk, "Solving stochastic programs with integer recourse by enumeration: a framework using Grobner basis reductions" (revised version), submitted to Mathematical Programming - Schultz, Stougie, et al. - 1996 |

2 | der Vlerk, "Stochastic Programming with Integer Recourse - van - 1995 |