Partitioning procedures for solving mixed-variables programming problems
TL;DR: This paper presented to the 8th International Meeting of the Institute of Management Sciences, Brussels, August 23-26, 1961 presents a meta-analyses of the determinants of infectious disease in eight operation rooms of the immune system and its consequences.
Abstract: Paper presented to the 8th International Meeting of the Institute of Management Sciences, Brussels, August 23-26, 1961.
Citations
More filters
Book•
23 May 2011TL;DR: It is argued that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas.
Abstract: Many problems of recent interest in statistics and machine learning can be posed in the framework of convex optimization. Due to the explosion in size and complexity of modern datasets, it is increasingly important to be able to solve problems with a very large number of features or training examples. As a result, both the decentralized collection or storage of these datasets as well as accompanying distributed solution methods are either necessary or at least highly desirable. In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to large-scale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn's method of partial inverses, Dykstra's alternating projections, Bregman iterative algorithms for l1 problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop MapReduce implementations.
17,433 citations
Additional excerpts
...…. . . . . . . . . . . . 31 6.3 General ℓ1 regularized loss minimization . . . . . . . . . . . . . . . . . . . . 32 6.4 Lasso . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.5 Sparse inverse covariance selection . . . . . . . . . . . . . . . . . . . . . . . . 33...
[...]
Book•
22 Jun 2009
TL;DR: This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling.
Abstract: A unified view of metaheuristics This book provides a complete background on metaheuristics and shows readers how to design and implement efficient algorithms to solve complex optimization problems across a diverse range of applications, from networking and bioinformatics to engineering design, routing, and scheduling. It presents the main design questions for all families of metaheuristics and clearly illustrates how to implement the algorithms under a software framework to reuse both the design and code. Throughout the book, the key search components of metaheuristics are considered as a toolbox for: Designing efficient metaheuristics (e.g. local search, tabu search, simulated annealing, evolutionary algorithms, particle swarm optimization, scatter search, ant colonies, bee colonies, artificial immune systems) for optimization problems Designing efficient metaheuristics for multi-objective optimization problems Designing hybrid, parallel, and distributed metaheuristics Implementing metaheuristics on sequential and parallel machines Using many case studies and treating design and implementation independently, this book gives readers the skills necessary to solve large-scale optimization problems quickly and efficiently. It is a valuable reference for practicing engineers and researchers from diverse areas dealing with optimization or machine learning; and graduate students in computer science, operations research, control, engineering, business and management, and applied mathematics.
2,735 citations
Cites background from "Partitioning procedures for solving..."
...It consists in fixing the values of complicated variables and solves the resulting reduced problem iteratively [69]....
[...]
Book•
01 Jan 2004TL;DR: Throughout, the authors focus on the traffic demands encountered in the real world of network design, and their generic approach allows problem formulations and solutions to be applied across the board to virtually any type of backbone communication or computer network.
Abstract: In network design, the gap between theory and practice is woefully broad. This book narrows it, comprehensively and critically examining current network design models and methods. You will learn where mathematical modeling and algorithmic optimization have been under-utilized. At the opposite extreme, you will learn where they tend to fail to contribute to the twin goals of network efficiency and cost-savings. Most of all, you will learn precisely how to tailor theoretical models to make them as useful as possible in practice. Throughout, the authors focus on the traffic demands encountered in the real world of network design. Their generic approach, however, allows problem formulations and solutions to be applied across the board to virtually any type of backbone communication or computer network. For beginners, this book is an excellent introduction. For seasoned professionals, it provides immediate solutions and a strong foundation for further advances in the use of mathematical modeling for network design. (Less)
1,093 citations
Cites methods from "Partitioning procedures for solving..."
...Benders decomposition (BD) is a general method applicable to LP and MIP problems that can be partitioned with respect to variables ([Ben62], [Las70], and [Min86])....
[...]
...Finally, we observe that although in the original work of Benders [Ben62] his decomposition method was developed for MIP problems and as such can be used in the BC techniques for solving, e....
[...]
...The BB technique can be made more effective using BD (note that BD was initially devised for MIP problems; see [Ben62])....
[...]
...Benders Decomposition, introduced in [Ben62], is also discussed in [Min89], [Min81], [Min84], and [PS01] for multicommodity network design applications....
[...]
01 Jan 2013
TL;DR: In today’s changing and competitive industrial environment, the difference between ad hoc planning methods and those that use sophisticated mathematical models to determine an optimal course of action can determine whether or not a company survives.
Abstract: Integer optimization problems are concerned with the efficient allocation of limited resources to meet a desired objective when some of the resources in question can only be divided into discrete parts. In such cases, the divisibility constraints on these resources, which may be people, machines, or other discrete inputs, may restrict the possible alternatives to a finite set. Nevertheless, there are usually too many alternatives to make complete enumeration a viable option for instances of realistic size. For example, an airline may need to determine crew schedules that minimize the total operating cost; an automotive manufacturer may want to determine the optimal mix of models to produce in order to maximize profit; or a flexible manufacturing facility may want to schedule production for a plant without knowing precisely what parts will be needed in future periods. In today’s changing and competitive industrial environment, the difference between ad hoc planning methods and those that use sophisticated mathematical models to determine an optimal course of action can determine whether or not a company survives.
1,066 citations
TL;DR: This paper proposes a stochastic programming model and solution algorithm for solving supply chain network design problems of a realistic scale and integrates a recently proposed sampling strategy, the sample average approximation scheme, with an accelerated Benders decomposition algorithm to quickly compute high quality solutions.
1,044 citations
Cites methods from "Partitioning procedures for solving..."
...Cutting plane algorithms such as Benders decomposition [3] (also known as the L-shaped decomposition method in the stochastic programming literature [30]) is quite suitable for this class of problems....
[...]
...The authors proposed a modification of the Benders decomposition algorithm [3], commonly used for deterministic network design problems [8], to generate robust designs....
[...]