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Decomposition Algorithms For Two Stage Stochastic Integer Programming
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Book Synopsis Decomposition Algorithms for Two-stage Stochastic Integer Programming by : John H. Penuel
Download or read book Decomposition Algorithms for Two-stage Stochastic Integer Programming written by John H. Penuel and published by . This book was released on 2009 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Decomposition Algorithms in Stochastic Integer Programming by : Babak Saleck Pay
Download or read book Decomposition Algorithms in Stochastic Integer Programming written by Babak Saleck Pay and published by . This book was released on 2017 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Time-staged Decomposition and Related Algorithms for Stochastic Mixed-integer Programming by : Yunwei Qi
Download or read book Time-staged Decomposition and Related Algorithms for Stochastic Mixed-integer Programming written by Yunwei Qi and published by . This book was released on 2012 with total page 103 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Computational Stochastic Programming by : Lewis Ntaimo
Download or read book Computational Stochastic Programming written by Lewis Ntaimo and published by Springer Nature. This book was released on with total page 518 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Stochastic Decomposition by : Julia L. Higle
Download or read book Stochastic Decomposition written by Julia L. Higle and published by Springer Science & Business Media. This book was released on 1996-02-29 with total page 254 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book summarizes developments related to a class of methods called Stochastic Decomposition (SD) algorithms, which represent an important shift in the design of optimization algorithms. Unlike traditional deterministic algorithms, SD combines sampling approaches from the statistical literature with traditional mathematical programming constructs (e.g. decomposition, cutting planes etc.). This marriage of two highly computationally oriented disciplines leads to a line of work that is most definitely driven by computational considerations. Furthermore, the use of sampled data in SD makes it extremely flexible in its ability to accommodate various representations of uncertainty, including situations in which outcomes/scenarios can only be generated by an algorithm/simulation. The authors report computational results with some of the largest stochastic programs arising in applications. These results (mathematical as well as computational) are the `tip of the iceberg'. Further research will uncover extensions of SD to a wider class of problems. Audience: Researchers in mathematical optimization, including those working in telecommunications, electric power generation, transportation planning, airlines and production systems. Also suitable as a text for an advanced course in stochastic optimization.
Book Synopsis Lectures on Stochastic Programming by : Alexander Shapiro
Download or read book Lectures on Stochastic Programming written by Alexander Shapiro and published by SIAM. This book was released on 2009-01-01 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: Optimization problems involving stochastic models occur in almost all areas of science and engineering, such as telecommunications, medicine, and finance. Their existence compels a need for rigorous ways of formulating, analyzing, and solving such problems. This book focuses on optimization problems involving uncertain parameters and covers the theoretical foundations and recent advances in areas where stochastic models are available. Readers will find coverage of the basic concepts of modeling these problems, including recourse actions and the nonanticipativity principle. The book also includes the theory of two-stage and multistage stochastic programming problems; the current state of the theory on chance (probabilistic) constraints, including the structure of the problems, optimality theory, and duality; and statistical inference in and risk-averse approaches to stochastic programming.
Book Synopsis Dual Decomposition in Stochastic Integer Programming by : Claus C. Carøe
Download or read book Dual Decomposition in Stochastic Integer Programming written by Claus C. Carøe and published by . This book was released on 1996 with total page 13 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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."
Book Synopsis A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems by : Hanif D. Sherali
Download or read book A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems written by Hanif D. Sherali and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the theory and applications of the Reformulation- Linearization/Convexification Technique (RL T) for solving nonconvex optimization problems. A unified treatment of discrete and continuous nonconvex programming problems is presented using this approach. In essence, the bridge between these two types of nonconvexities is made via a polynomial representation of discrete constraints. For example, the binariness on a 0-1 variable x . can be equivalently J expressed as the polynomial constraint x . (1-x . ) = 0. The motivation for this book is J J the role of tight linear/convex programming representations or relaxations in solving such discrete and continuous nonconvex programming problems. The principal thrust is to commence with a model that affords a useful representation and structure, and then to further strengthen this representation through automatic reformulation and constraint generation techniques. As mentioned above, the focal point of this book is the development and application of RL T for use as an automatic reformulation procedure, and also, to generate strong valid inequalities. The RLT operates in two phases. In the Reformulation Phase, certain types of additional implied polynomial constraints, that include the aforementioned constraints in the case of binary variables, are appended to the problem. The resulting problem is subsequently linearized, except that certain convex constraints are sometimes retained in XV particular special cases, in the Linearization/Convexijication Phase. This is done via the definition of suitable new variables to replace each distinct variable-product term. The higher dimensional representation yields a linear (or convex) programming relaxation.
Book Synopsis Transmission Expansion Planning: The Network Challenges of the Energy Transition by : Sara Lumbreras
Download or read book Transmission Expansion Planning: The Network Challenges of the Energy Transition written by Sara Lumbreras and published by Springer Nature. This book was released on 2020-11-19 with total page 311 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a panoramic look at the transformation of the transmission network in the context of the energy transition. It provides readers with basic definitions as well as details on current challenges and emerging technologies. In-depth chapters cover the integration of renewables, the particularities of planning large-scale systems, efficient reduction and solution methods, the possibilities of HVDC and super grids, distributed generation, smart grids, demand response, and new regulatory schemes. The content is complemented with case studies that highlight the importance of the power transmission network as the backbone of modern energy systems. This book will be a comprehensive reference that will be useful to both academics and practitioners.
Book Synopsis Online Optimization of Large Scale Systems by : Martin Grötschel
Download or read book Online Optimization of Large Scale Systems written by Martin Grötschel and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 789 pages. Available in PDF, EPUB and Kindle. Book excerpt: In its thousands of years of history, mathematics has made an extraordinary ca reer. It started from rules for bookkeeping and computation of areas to become the language of science. Its potential for decision support was fully recognized in the twentieth century only, vitally aided by the evolution of computing and communi cation technology. Mathematical optimization, in particular, has developed into a powerful machinery to help planners. Whether costs are to be reduced, profits to be maximized, or scarce resources to be used wisely, optimization methods are available to guide decision making. Opti mization is particularly strong if precise models of real phenomena and data of high quality are at hand - often yielding reliable automated control and decision proce dures. But what, if the models are soft and not all data are around? Can mathematics help as well? This book addresses such issues, e. g. , problems of the following type: - An elevator cannot know all transportation requests in advance. In which order should it serve the passengers? - Wing profiles of aircrafts influence the fuel consumption. Is it possible to con tinuously adapt the shape of a wing during the flight under rapidly changing conditions? - Robots are designed to accomplish specific tasks as efficiently as possible. But what if a robot navigates in an unknown environment? - Energy demand changes quickly and is not easily predictable over time. Some types of power plants can only react slowly.
Book Synopsis Algorithms and Reformulations for Large-scale Integer and Stochastic Integer Programs by : Dinakar Gade
Download or read book Algorithms and Reformulations for Large-scale Integer and Stochastic Integer Programs written by Dinakar Gade and published by . This book was released on 2012 with total page 137 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: In this dissertation, we develop methodologies to solve difficult classes of discrete optimization problems under uncertainty by using techniques from integer programming. First, we consider a class of two-stage stochastic integer programs with binary variables in the first stage, general integer variables in the second stage and random data with finitely many outcomes. We develop an L-shaped decomposition algorithm that iteratively tightens the linear relaxation of the scenario subproblems using Gomory cuts and maintains convex first stage approximations. We show that the algorithm is not only finitely convergent, but also computationally amenable allowing several alternative implementations. We develop a computer implementation of this algorithm and report computational results on the stochastic server location problem instances. With the goal of extending our methods to solve stochastic mixed integer programs, we develop extensions to the cutting plane tree algorithm by integrating Gomory cuts with disjunctive cuts. We report computational results with the Gomory-enhanced simple disjunctive cuts using benchmark test instances. Second, we introduce the concept of service levels into deterministic lot sizing problems and develop a polynomial time algorithm for a single item lot sizing problem with a ready rate service level constraint. Based on the algorithm, we develop compact extended reformulations for this problem and a relaxation. We show that although the extended reformulations are large, they outperform standard formulations of the problem while guaranteeing optimal solutions when they are used to solve capacitated multi-item instances.
Book Synopsis Decomposition Algorithms for Very Large Scale Stochastic Mixed-Integer Programs by :
Download or read book Decomposition Algorithms for Very Large Scale Stochastic Mixed-Integer Programs written by and published by . This book was released on 2007 with total page 8 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis 50 Years of Integer Programming 1958-2008 by : Michael Jünger
Download or read book 50 Years of Integer Programming 1958-2008 written by Michael Jünger and published by Springer Science & Business Media. This book was released on 2009-11-06 with total page 804 pages. Available in PDF, EPUB and Kindle. Book excerpt: In 1958, Ralph E. Gomory transformed the field of integer programming when he published a paper that described a cutting-plane algorithm for pure integer programs and announced that the method could be refined to give a finite algorithm for integer programming. In 2008, to commemorate the anniversary of this seminal paper, a special workshop celebrating fifty years of integer programming was held in Aussois, France, as part of the 12th Combinatorial Optimization Workshop. It contains reprints of key historical articles and written versions of survey lectures on six of the hottest topics in the field by distinguished members of the integer programming community. Useful for anyone in mathematics, computer science and operations research, this book exposes mathematical optimization, specifically integer programming and combinatorial optimization, to a broad audience.
Book Synopsis On Valid Inequalities for Polyhedra in Extended and Projected Spaces with Application to Two-stage Stochastic Integer Programming by :
Download or read book On Valid Inequalities for Polyhedra in Extended and Projected Spaces with Application to Two-stage Stochastic Integer Programming written by and published by . This book was released on 2015 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: The primary focus of this dissertation is on solution methods for stochastic integer programs (SIP). We investigate large-scale mixed-integer programs arising from two-stage SIPs and present several techniques to improve existing decomposition algorithms for solving them. Our main goal is to improve linear programming relaxations, which can lead to significant reductions in the time to solve these problems to optimality. We first study the problem of server scheduling in multiclass service systems under uncertainty in the customer arrival volumes. A common practice is to conduct the staffing and shift scheduling steps sequentially. We propose a two-stage SIP model that integrates these two decisions, which can yield similar quality-of-service with lower scheduling costs. As the standard Benders decomposition algorithm fails, we introduce a technique for using mixed-integer rounding to improve the Benders cuts, which is applicable to any SIP with integer first-stage variables. Numerical examples illustrate the computational efficiency of the proposed approach and the benefit of solving the integrated model. Next, we investigate two frameworks to use integrality constraints to obtain improved cuts within Benders decomposition: (i) cut-and-project, where integrality constraints are used to derive cuts in the extended variable space, and Benders cuts are used to project the resulting improved relaxation, and (ii) project-and-cut, where integrality constraints are used to derive cuts directly in the Benders reformulation. For the case of split cuts, we demonstrate that although these approaches yield equivalent relaxations when considering a single split disjunction, cut-and-project yields stronger relaxations when using multiple split disjunctions. Computational results demonstrate that the difference can be very large, and the cut-and-project can significantly improve the performance of Benders decomposition. Finally, we study extended formulations of linear relaxations of mixed-integer sets. We question whether there is a strongest possible extended formulation with respect to subsequent strengthening by lattice-free cuts. We give an explicit construction of one such extended formulation for general mixed-integer programs. For 0-1 mixed-integer programs with n integer and k continuous variables, we give an extended formulation with 2n + k -1 variables whose split closure is integral. Computational results show the strength of cutting planes derived from extended formulations.
Book Synopsis Applications of Stochastic Programming by : Stein W. Wallace
Download or read book Applications of Stochastic Programming written by Stein W. Wallace and published by SIAM. This book was released on 2005-01-01 with total page 724 pages. Available in PDF, EPUB and Kindle. Book excerpt: Consisting of two parts, this book presents papers describing publicly available stochastic programming systems that are operational. It presents a diverse collection of application papers in areas such as production, supply chain and scheduling, gaming, environmental and pollution control, financial modeling, telecommunications, and electricity.
Book Synopsis Global Optimization by : Reiner Horst
Download or read book Global Optimization written by Reiner Horst and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 705 pages. Available in PDF, EPUB and Kindle. Book excerpt: The enormous practical need for solving global optimization problems coupled with a rapidly advancing computer technology has allowed one to consider problems which a few years ago would have been considered computationally intractable. As a consequence, we are seeing the creation of a large and increasing number of diverse algorithms for solving a wide variety of multiextremal global optimization problems. The goal of this book is to systematically clarify and unify these diverse approaches in order to provide insight into the underlying concepts and their pro perties. Aside from a coherent view of the field much new material is presented. By definition, a multiextremal global optimization problem seeks at least one global minimizer of a real-valued objective function that possesses different local n minimizers. The feasible set of points in IR is usually determined by a system of inequalities. It is well known that in practically all disciplines where mathematical models are used there are many real-world problems which can be formulated as multi extremal global optimization problems.
Book Synopsis Electrical Power Unit Commitment by : Yuping Huang
Download or read book Electrical Power Unit Commitment written by Yuping Huang and published by Springer. This book was released on 2017-01-13 with total page 98 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume in the SpringerBriefs in Energy series offers a systematic review of unit commitment (UC) problems in electrical power generation. It updates texts written in the late 1990s and early 2000s by including the fundamentals of both UC and state-of-the-art modeling as well as solution algorithms and highlighting stochastic models and mixed-integer programming techniques. The UC problems are mostly formulated as mixed-integer linear programs, although there are many variants. A number of algorithms have been developed for, or applied to, UC problems, including dynamic programming, Lagrangian relaxation, general mixed-integer programming algorithms, and Benders decomposition. In addition the book discusses the recent trends in solving UC problems, especially stochastic programming models, and advanced techniques to handle large numbers of integer- decision variables due to scenario propagation
