Decomposition Algorithms In Stochastic Integer Programming
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Decomposition Algorithms In Stochastic Integer Programming
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Author : Babak Saleck Pay
language : en
Publisher:
Release Date : 2017
Decomposition Algorithms In Stochastic Integer Programming written by Babak Saleck Pay and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017 with Decomposition (Mathematics) categories.
In this dissertation we focus on two main topics. Under the first topic, we develop a new framework for stochastic network interdiction problem to address ambiguity in the defender risk preferences. The second topic is dedicated to computational studies of two-stage stochastic integer programs. More specifically, we consider two cases. First, we develop some solution methods for two-stage stochastic integer programs with continuous recourse; second, we study some computational strategies for two-stage stochastic integer programs with integer recourse. We study a class of stochastic network interdiction problems where the defender has incomplete (ambiguous) preferences. Specifically, we focus on the shortest path network interdiction modeled as a Stackelberg game, where the defender (leader) makes an interdiction decision first, then the attacker (follower) selects a shortest path after the observation of random arc costs and interdiction effects in the network. We take a decision-analytic perspective in addressing probabilistic risk over network parameters, assuming that the defender's risk preferences over exogenously given probabilities can be summarized by the expected utility theory. Although the exact form of the utility function is ambiguous to the defender, we assume that a set of historical data on some pairwise comparisons made by the defender is available, which can be used to restrict the shape of the utility function. We use two different approaches to tackle this problem. The first approach conducts utility estimation and optimization separately, by first finding the best fit for a piecewise linear concave utility function according to the available data, and then optimizing the expected utility. The second approach integrates utility estimation and optimization, by modeling the utility ambiguity under a robust optimization framework following \cite{armbruster2015decision} and \cite{Hu}. We conduct extensive computational experiments to evaluate the performances of these approaches on the stochastic shortest path network interdiction problem. In third chapter, we propose partition-based decomposition algorithms for solving two-stage stochastic integer program with continuous recourse. The partition-based decomposition method enhance the classical decomposition methods (such as Benders decomposition) by utilizing the inexact cuts (coarse cuts) induced by a scenario partition. Coarse cut generation can be much less expensive than the standard Benders cuts, when the partition size is relatively small compared to the total number of scenarios. We conduct an extensive computational study to illustrate the advantage of the proposed partition-based decomposition algorithms compared with the state-of-the-art approaches. In chapter four, we concentrate on computational methods for two-stage stochastic integer program with integer recourse. We consider the partition-based relaxation framework integrated with a scenario decomposition algorithm in order to develop strategies which provide a better lower bound on the optimal objective value, within a tight time limit.
Stochastic Decomposition
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Author : Julia L. Higle
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-27
Stochastic Decomposition written by Julia L. Higle and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-27 with Mathematics categories.
Motivation Stochastic Linear Programming with recourse represents one of the more widely applicable models for incorporating uncertainty within in which the SLP optimization models. There are several arenas model is appropriate, and such models have found applications in air line yield management, capacity planning, electric power generation planning, financial planning, logistics, telecommunications network planning, and many more. In some of these applications, modelers represent uncertainty in terms of only a few seenarios and formulate a large scale linear program which is then solved using LP software. However, there are many applications, such as the telecommunications planning problem discussed in this book, where a handful of seenarios do not capture variability well enough to provide a reasonable model of the actual decision-making problem. Problems of this type easily exceed the capabilities of LP software by several orders of magnitude. Their solution requires the use of algorithmic methods that exploit the structure of the SLP model in a manner that will accommodate large scale applications.
Time Staged Decomposition And Related Algorithms For Stochastic Mixed Integer Programming
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Author : Yunwei Qi
language : en
Publisher:
Release Date : 2012
Time Staged Decomposition And Related Algorithms For Stochastic Mixed Integer Programming written by Yunwei Qi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
Abstract: This dissertation focuses on solving two-stage stochastic mixed integer programs (SMIPs) with general mixed integer variables in both stages. Our setup allows randomness in all data elements influencing the recourse problem, and moreover, general integer variables are allowed in both stages. We develop a time-staged decomposition algorithm that uses multi-term disjunctive cuts to obtain convex approximation of the second-stage mixed-integer programs. We prove that the proposed method is finitely convergent. Among the main advantages of our decomposition scheme is that the subproblems are approximated by successive linear programming problems, and moreover these can be solved in parallel. Several variants of an SMIP example in the literature are included to illustrate our algorithms. To the best of our knowledge, the only previously known time-staged decomposition algorithm to address the two-stage SMIP in such generality used operations that are computationally impractical (e.g. requiring exact value functions of MIP subproblems). In contrast, our decomposition algorithm allows partially solving the subproblems. Following the studies of our decomposition algorithm, we proceed with computational studies related to some of the key ingredients of our decomposition algorithm. First, we investigate how well multi-term disjunctions can approximate feasible sets associated with stochastic mixed-integer programming problems. This part of our study is experimental in nature and we investigate both "wait-and-see" as well as "here-and-now" formulations of stochastic programming problems. In order to study the performance for the former class of problems, we use test problems from the integer programming literature (e.g. various versions of MIPLIB), whereas for the latter class of problems, we use the SSLP series of instances. Another important nugget of our decomposition algorithm is the use of multi-term disjunctions. Since the effectiveness of our scheme depends on this feature, we also investigate ways to improve the performance of cutting plane tree (CPT) algorithm for mixed integer programming problems. We compare different variable splitting rules in the computational experiment. A set of algorithms for solving multi-term CGLPs are also included and computational experiments with instances from MIPLIB are performed.
Computational Stochastic Programming
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Author : Lewis Ntaimo
language : en
Publisher: Springer Nature
Release Date :
Computational Stochastic Programming written by Lewis Ntaimo and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on with categories.
Decomposition Algorithms For Two Stage Stochastic Integer Programming
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Author : John H. Penuel
language : en
Publisher:
Release Date : 2009
Decomposition Algorithms For Two Stage Stochastic Integer Programming written by John H. Penuel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with categories.
ABSTRACT: Stochastic programming seeks to optimize decision making in uncertain conditions. This type of work is typically amenable to decomposition into first- and second-stage decisions. First-stage decisions must be made now, while second-stage decisions are made after realizing certain future conditions and are typically constrained by first-stage decisions. This work focuses on two stochastic integer programming applications. In Chapter 2, we investigate a two-stage facility location problem with integer recourse. In Chapter 3, we investigate the graph decontamination problem with mobile agents. In both problems, we develop cutting-plane algorithms that iteratively solve the first-stage problem, then solve the second-stage problem and glean information from the second-stage solution with which we refine first-stage decisions. This process is repeated until optimality is reached. If the second-stage problems are linear programs, then duality can be exploited in order to refine first-stage decisions. If the second-stage problems are mixed-integer programs, then we resort to other methods to extract information from the second-stage problem. The applications discussed in this work have mixed-integer second-stage problems, and accordingly we develop specialized cutting-plane algorithms and demonstrate the efficacy of our solution methods.
Stochastic Linear Programming Algorithms
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Author : Janos Mayer
language : en
Publisher: Taylor & Francis
Release Date : 2022-04-19
Stochastic Linear Programming Algorithms written by Janos Mayer and has been published by Taylor & Francis this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-19 with Computers categories.
A computationally oriented comparison of solution algorithms for two stage and jointly chance constrained stochastic linear programming problems, this is the first book to present comparative computational results with several major stochastic programming solution approaches. The following methods are considered: regularized decomposition, stochastic decomposition and successive discrete approximation methods for two stage problems; cutting plane methods, and a reduced gradient method for jointly chance constrained problems. The first part of the book introduces the algorithms, including a unified approach to decomposition methods and their regularized counterparts. The second part addresses computer implementation of the methods, describes a testing environment based on a model management system, and presents comparative computational results with the various algorithms. Emphasis is on the computational behavior of the algorithms.
Decomposition Algorithms For Very Large Scale Stochastic Mixed Integer Programs
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Author :
language : en
Publisher:
Release Date : 2007
Decomposition Algorithms For Very Large Scale Stochastic Mixed Integer Programs written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with categories.
The objectives of this project were to explore decomposition algorithms that solve optimization models under uncertainty. In order to accommodate a variety of future scenarios, our algorithms are designed to address large scale models. The main accomplishments of the project can be summarized as follows. 1) design and evaluate decomposition methods for stochastic mixed-integer programming (SMIP) problems (Yuan and Sen [2008]); 2) accelerate stochastic decomposition (SD) as a prelude to using SD for SMIP as well as a multi-stage version of SD (Sen et al [2007], Zhou and Sen [2008]); 3) develop a theory for parametric analysis of mixed-integer programs, and provide economically justifiable estimates of shadow prices from mixed-integer linear programming models (Sen and Genc [2008]). The first two relate to stochastic programming, whereas the last addresses one of the long-standing open questions in discrete optimization, namely, parametric analysis in MILP models. This paper (listed as [1]) is likely to have a long term impact on a variety of fields including discrete optimization, operations research, and computational economics.
Dual Decomposition In Stochastic Integer Programming
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Author : Claus C. Carøe
language : en
Publisher:
Release Date : 1996
Dual Decomposition In Stochastic Integer Programming written by Claus C. Carøe and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Stochastic programming categories.
Abstract: "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. Numerical experience is presented for some two-stage test problems."
Stochastic Linear Programming Algorithms
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Author : Janos Mayer
language : en
Publisher: CRC Press
Release Date : 1998-02-25
Stochastic Linear Programming Algorithms written by Janos Mayer and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-02-25 with Computers categories.
A computationally oriented comparison of solution algorithms for two stage and jointly chance constrained stochastic linear programming problems, this is the first book to present comparative computational results with several major stochastic programming solution approaches. The following methods are considered: regularized decomposition, stochastic decomposition and successive discrete approximation methods for two stage problems; cutting plane methods, and a reduced gradient method for jointly chance constrained problems. The first part of the book introduces the algorithms, including a unified approach to decomposition methods and their regularized counterparts. The second part addresses computer implementation of the methods, describes a testing environment based on a model management system, and presents comparative computational results with the various algorithms. Emphasis is on the computational behavior of the algorithms.
Decomposition Techniques In Mathematical Programming
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Author : Antonio J. Conejo
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-28
Decomposition Techniques In Mathematical Programming written by Antonio J. Conejo and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-28 with Technology & Engineering categories.
Optimization plainly dominates the design, planning, operation, and c- trol of engineering systems. This is a book on optimization that considers particular cases of optimization problems, those with a decomposable str- ture that can be advantageously exploited. Those decomposable optimization problems are ubiquitous in engineering and science applications. The book considers problems with both complicating constraints and complicating va- ables, and analyzes linear and nonlinear problems, with and without in- ger variables. The decomposition techniques analyzed include Dantzig-Wolfe, Benders, Lagrangian relaxation, Augmented Lagrangian decomposition, and others. Heuristic techniques are also considered. Additionally, a comprehensive sensitivity analysis for characterizing the solution of optimization problems is carried out. This material is particularly novel and of high practical interest. This book is built based on many clarifying, illustrative, and compu- tional examples, which facilitate the learning procedure. For the sake of cl- ity, theoretical concepts and computational algorithms are assembled based on these examples. The results are simplicity, clarity, and easy-learning. We feel that this book is needed by the engineering community that has to tackle complex optimization problems, particularly by practitioners and researchersinEngineering,OperationsResearch,andAppliedEconomics.The descriptions of most decomposition techniques are available only in complex and specialized mathematical journals, di?cult to understand by engineers. A book describing a wide range of decomposition techniques, emphasizing problem-solving, and appropriately blending theory and application, was not previously available.