Download or read book On Parallelizing Dual Decomposition in Stochastic Integer Programming written by MIles Lubin and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: For stochastic mixed-integer programs, we revisit the dual decomposition algorithm of Caroe and Schultz from a computational perspective with the aim of its parallelization. We address an important bottleneck of parallel execution by identifying a formulation that permits the parallel solution of the master program by using structure-exploiting interior-point solvers. Our results demonstrate the potential for parallel speedup and the importance of regularization (stabilization) in the dual optimization. Load imbalance is identified as a remaining barrier to parallel scalability.

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."

Download or read book An Integer Stochastic Dual Decomposition Method for Solving Two stage Stochastic Integer Programs written by Julius Rachmantio and published by . This book was released on 1998 with total page 306 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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.

Download or read book A Parallel Implementation of the Nested Decomposition Algorithm for Multistage Stochastic Linear Programs written by and published by . This book was released on with total page 29 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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.

Download or read book On High performance Benders decomposition based Exact Methods with Application to Mixed integer and Stochastic Problems written by Ragheb Rahmaniani and published by . This book was released on 2018 with total page 219 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic integer programming (SIP) combines the difficulty of uncertainty and non-convexity, and constitutes a class of extremely challenging problems to solve. Efficiently solving SIP problems is of high importance due to their vast applicability. Therefore, the primary focus of this dissertation is on solution methods for SIPs. We consider two-stage SIPs and present several enhanced decomposition algorithms for solving them. Our main goal is to develop new decomposition schemes and several acceleration techniques to enhance the classical decomposition methods, which can lead to efficiently solving various SIP problems to optimality. In the first essay of this dissertation, we present a state-of-the-art survey of the Benders decomposition algorithm. We provide a taxonomy of the algorithmic enhancements and the acceleration strategies of this algorithm to synthesize the literature, and to identify shortcomings, trends and potential research directions. In addition, we discuss the use of Benders decomposition to develop efficient (meta- )heuristics, describe the limitations of the classical algorithm, and present extensions enabling its application to a broader range of problems. Next, we develop various techniques to overcome some of the main shortfalls of the Benders decomposition algorithm. We propose the use of cutting planes, partial decomposition, heuristics, stronger cuts, and warm-start strategies to alleviate the numerical challenges arising from instabilities, primal inefficiencies, weak optimality/feasibility cuts, and weak linear relaxation. We test the proposed strategies with benchmark instances from stochastic network design problems. Numerical experiments illustrate the computational efficiency of the proposed techniques. In the third essay of this dissertation, we propose a new and high-performance decomposition approach, called Benders dual decomposition method. The development of this method is based on a specific reformulation of the Benders subproblems, where local copies of the master variables are introduced and then priced out into the objective function. We show that the proposed method significantly alleviates the primal and dual shortfalls of the Benders decomposition method and it is closely related to the Lagrangian dual decomposition method. Computational results on various SIP problems show the superiority of this method compared to the classical decomposition methods as well as CPLEX 12.7. Finally, we study parallelization of the Benders decomposition method. The available parallel variants of this method implement a rigid synchronization among the master and slave processors. Thus, it suffers from significant load imbalance when applied to the SIP problems. This is mainly due to having a hard mixed-integer master problem that can take hours to be optimized. We thus propose an asynchronous parallel Benders method in a branchand- cut framework. However, relaxing the synchronization requirements entails convergence and various efficiency problems which we address them by introducing several acceleration techniques and search strategies. In particular, we propose the use of artificial subproblems, cut generation, cut aggregation, cut management, and cut propagation. The results indicate that our algorithm reaches higher speedup rates compared to the conventional synchronized methods and it is several orders of magnitude faster than CPLEX 12.7.

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.

Download or read book Decomposition in Stochastic Integer Programming written by Claus C. Carøe and published by . This book was released on 1998 with total page 142 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Annotated Bibliographies in Combinatorial Optimization written by Mauro Dell'Amico and published by . This book was released on 1997-08-28 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wiley-Interscience Series in Discrete Mathematics and Optimization Advisory Editors Ronald L. Graham Jan Karel Lenstra Robert E. Tarjan Discrete Mathematics and Optimization involves the study of finite structures and is one of the fastest growing areas in mathematics today. The level and depth of recent advances in the area and the wide applicability of its evolving techniques point to the rapidity with which the field is moving and presage the ever-increasing interaction between it and computer science. The Series provides a broad coverage of discrete mathematics and optimization, ranging over such fields as combinatorics, graph theory, enumeration, mathematical programming and the analysis of algorithms, and including such topics as Ramsey theory, transversal theory, block designs, finite geometries, Polya theory, graph and matroid algorithms, network flows, polyhedral combinatorics and computational complexity. The Wiley-Interscience Series in Discrete Mathematics and Optimization will be a substantial part of the record in this extraordinary development. Recent titles in the Series: Local Search in Combinatorial Optimization Edited by Emile H. L. Aarts Philips Research Laboratories, Eindhoven and Eindhoven University of Technology, Eindhoven Jan Karel Lenstra Eindhoven University of Technology, Eindhoven and CWI Amsterdam In the past three decades local search has grown from a simple heuristic idea into a mature field of research in combinatorial optimization. Local search is still the method of choice for NP-hard problems as it provides a robust approach for obtaining high-quality solutions to problems of a realistic size in a reasonable time. This area of discrete mathematics is of great practical use and is attracting ever-increasing attention. The contributions to this book cover local search and its variants from both a theoretical and practical point of view, each with a chapter written by leading authorities on that particular aspect. Chapters 1 to 7 deal with the theory of local search and describe the principal search strategies such as simulated annealing, tabu search, genetic algorithms and neural networks. The remaining chapters present a wealth of results on applications of local search to problems in management science and engineering, including the traveling salesman problem, vehicle routing, machine scheduling, VLSI design and code design. This book is an important reference volume and an invaluable source of inspiration for advanced students and researchers in discrete mathematics, computer science, operations research, industrial engineering and management science.

Download or read book Encyclopedia of Optimization written by Christodoulos A. Floudas and published by Springer Science & Business Media. This book was released on 2008-09-04 with total page 4646 pages. Available in PDF, EPUB and Kindle. Book excerpt: The goal of the Encyclopedia of Optimization is to introduce the reader to a complete set of topics that show the spectrum of research, the richness of ideas, and the breadth of applications that has come from this field. The second edition builds on the success of the former edition with more than 150 completely new entries, designed to ensure that the reference addresses recent areas where optimization theories and techniques have advanced. Particularly heavy attention resulted in health science and transportation, with entries such as "Algorithms for Genomics", "Optimization and Radiotherapy Treatment Design", and "Crew Scheduling".

Download or read book A Primal dual Decomposition based Interior Point Approach to Two stage Stochastic Linear Programming written by Arjan Bastiaan Berkelaar and published by . This book was released on 1999 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Decomposition written by Julia L. Higle and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Large Scale Optimization in Supply Chains and Smart Manufacturing written by Jesús M. Velásquez-Bermúdez and published by Springer Nature. This book was released on 2019-09-06 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, theory of large scale optimization is introduced with case studies of real-world problems and applications of structured mathematical modeling. The large scale optimization methods are represented by various theories such as Benders’ decomposition, logic-based Benders’ decomposition, Lagrangian relaxation, Dantzig –Wolfe decomposition, multi-tree decomposition, Van Roy’ cross decomposition and parallel decomposition for mathematical programs such as mixed integer nonlinear programming and stochastic programming. Case studies of large scale optimization in supply chain management, smart manufacturing, and Industry 4.0 are investigated with efficient implementation for real-time solutions. The features of case studies cover a wide range of fields including the Internet of things, advanced transportation systems, energy management, supply chain networks, service systems, operations management, risk management, and financial and sales management. Instructors, graduate students, researchers, and practitioners, would benefit from this book finding the applicability of large scale optimization in asynchronous parallel optimization, real-time distributed network, and optimizing the knowledge-based expert system for convex and non-convex problems.

Download or read book Decomposition in Multistage Stochastic Programming and a Constraint Integer Programming Approach to Mixed integer Nonlinear Programming written by Stefan Vigerske and published by . This book was released on 2012 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming written by Ivo Nowak and published by Springer Science & Business Media. This book was released on 2006-03-28 with total page 213 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinearoptimizationproblemscontainingbothcontinuousanddiscretevariables are called mixed integer nonlinear programs (MINLP). Such problems arise in many ?elds, such as process industry, engineering design, communications, and ?nance. There is currently a huge gap between MINLP and mixed integer linear programming(MIP) solvertechnology.With a modernstate-of-the-artMIP solver itispossibletosolvemodelswithmillionsofvariablesandconstraints,whereasthe dimensionofsolvableMINLPsisoftenlimitedbyanumberthatissmallerbythree or four orders of magnitude. It is theoretically possible to approximate a general MINLP by a MIP with arbitrary precision. However, good MIP approximations are usually much larger than the original problem. Moreover, the approximation of nonlinear functions by piecewise linear functions can be di?cult and ti- consuming. In this book relaxation and decomposition methods for solving nonconvex structured MINLPs are proposed. In particular, a generic branch-cut-and-price (BCP) framework for MINLP is presented. BCP is the underlying concept in almost all modern MIP solvers. Providing a powerful decomposition framework for both sequential and parallel solvers, it made the success of the current MIP technology possible. So far generic BCP frameworks have been developed only for MIP, for example,COIN/BCP (IBM, 2003) andABACUS (OREAS GmbH, 1999). In order to generalize MIP-BCP to MINLP-BCP, the following points have to be taken into account: • A given (sparse) MINLP is reformulated as a block-separable program with linear coupling constraints.The block structure makes it possible to generate Lagrangian cuts and to apply Lagrangian heuristics. • In order to facilitate the generation of polyhedral relaxations, nonlinear c- vex relaxations are constructed. • The MINLP separation and pricing subproblems for generating cuts and columns are solved with specialized MINLP solvers.

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.