Dantzig–Wolfe Decomposition

	Dantzig–Wolfe Decomposition Dantzig–Wolfe decomposition is an algorithm for solving linear programming problems with special structure. It was originally developed by George Dantzig and Philip Wolfe and initially published in 1960. Many texts on linear programming have sections dedicated to discussing this decomposition algorithm. Dantzig–Wolfe decomposition relies on delayed column generation for improving the tractability of large-scale linear programs. For most linear programs solved via the revised simplex algorithm, at each step, most columns (variables) are not in the basis. In such a scheme, a master problem containing at least the currently active columns (the basis) uses a subproblem or subproblems to generate columns for entry into the basis such that their inclusion improves the objective function. Required form In order to use Dantzig–Wolfe decomposition, the constraint matrix of the linear program must have a specific form. A set of constraints must be identified as "connecting", ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Linear Programming Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Convex Combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the ''count'' of the weights as in a standard weighted average. More formally, given a finite number of points x_1, x_2, \dots, x_n in a real vector space, a convex combination of these points is a point of the form :\alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where the real numbers \alpha_i satisfy \alpha_i\ge 0 and \alpha_1+\alpha_2+\cdots+\alpha_n=1. As a particular example, every convex combination of two points lies on the line segment between the points. A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Benders' Decomposition Benders decomposition (or Benders' decomposition) is a technique in mathematical programming that allows the solution of very large linear programming problems that have a special block structure. This block structure often occurs in applications such as stochastic programming as the uncertainty is usually represented with scenarios. The technique is named after Jacques F. Benders. The strategy behind Benders decomposition can be summarized as ''divide-and-conquer''. That is, in Benders decomposition, the variables of the original problem are divided into two subsets so that a first-stage master problem is solved over the first set of variables, and the values for the second set of variables are determined in a second-stage subproblem for a given first-stage solution. If the subproblem determines that the fixed first-stage decisions are in fact infeasible, then so-called ''Benders cuts'' are generated and added to the master problem, which is then re-solved until no cuts can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Delayed Column Generation Column generation or delayed column generation is an efficient algorithm for solving large linear programs. The overarching idea is that many linear programs are too large to consider all the variables explicitly. The idea is thus to start by solving the considered program with only a subset of its variables. Then iteratively, variables that have the potential to improve the objective function are added to the program. Once it is possible to demonstrate that adding new variables would no longer improve the value of the objective function, the procedure stops. The hope when applying a column generation algorithm is that only a very small fraction of the variables will be generated. This hope is supported by the fact that in the optimal solution, most variables will be non-basic and assume a value of zero, so the optimal solution can be found without them. In many cases, this method allows to solve large linear programs that would otherwise be intractable. The classical example of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	GNU Linear Programming Kit The GNU Linear Programming Kit (GLPK) is a software package intended for solving large-scale linear programming (LP), mixed integer programming (MIP), and other related problems. It is a set of routines written in ANSI C and organized in the form of a callable library. The package is part of the GNU Project and is released under the GNU General Public License. Problems can be modeled in the language GNU MathProg (previously known as GMPL) which shares many parts of the syntax with AMPL and solved with standalone solver GLPSOL. GLPK can also be used as a C library. GLPK uses the revised simplex method and the primal-dual interior point method for non-integer problems and the branch-and-bound algorithm together with Gomory's mixed integer cuts for (mixed) integer problems. GLPK is supported in the free edition of the OptimJ modeling system An independent project provides a Java-based interface to GLPK (via JNI). This allows Java applications to call out to GLPK in a relativel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	AMPL AMPL (A Mathematical Programming Language) is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing (i.e., large-scale optimization and scheduling-type problems). It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories. AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files. AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions. One advantage of AMPL is the similarity of its syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization. Many modern solvers available on the NEOS Server (formerly hosted at the Argonne National Laboratory, currently hosted at the University of Wisconsin, Madison) accept AMPL input. Ac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Extreme Points In mathematics, an extreme point of a convex set S in a Real number, real or Complex number, complex vector space is a point in S which does not lie in any open line segment joining two points of S. In linear programming problems, an extreme point is also called vertex or corner point of S. Definition Throughout, it is assumed that X is a Real number, real or Complex number, complex vector space. For any p, x, y \in X, say that p x and y if x \neq y and there exists a 0 < t < 1 such that $p = t x + (1-t) y.$ If $K$ is a subset of $X$ and $p \in K,$ then $p$ is called an of $K$ if it does not lie between any two points of $K.$ That is, if there does exist $x, y \in K$ and $0 < t < 1$ such that $x \neq y$ and $p = t x + (1-t) y.$ The set of all extreme points of $K$ is denoted by $\operatorname(K).$ Generalizations If [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Convex Polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others''Mathematical Programming'', by Melvyn W. Jeter (1986) p. 68/ref> (including this article) allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In the influential textbooks of Grünbaum and Ziegler on the subject, as well as in many other texts in discrete geometry, convex polytopes are often simply called "polytopes". Grünbaum points out that this is solely to avoi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	George Dantzig George Bernard Dantzig (; November 8, 1914 – May 13, 2005) was an American mathematical scientist who made contributions to industrial engineering, operations research, computer science, economics, and statistics. Dantzig is known for his development of the simplex algorithm, an algorithm for solving linear programming problems, and for his other work with linear programming. In statistics, Dantzig solved two open problems in statistical theory, which he had mistaken for homework after arriving late to a lecture by Jerzy Neyman.Joe Holley (2005)"Obituaries of George Dantzig" In: ''Washington Post'', May 19, 2005; B06 At his death, Dantzig was the Professor Emeritus of Transportation Sciences and Professor of Operations Research and of Computer Science at Stanford University. Early life Born in Portland, Oregon, George Bernard Dantzig was named after George Bernard Shaw, the Irish writer.Richard W. Cottle, B. Curtis Eaves and Michael A. Saunders (2006)"Memorial Resolution: Ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	DW Block Angular Matrix DW may refer to: News media * Deutsche Welle, a Germany-based, international news publisher DW News DW-TV ** DW (Español) * Duowei News, or "DW News", an American Chinese-language news website * The Daily Wire, an American conservative news website Businesses and organizations * Daniel Wellington, a Swedish watch company * Development Workshop, a non-profit organization * Drum Workshop, or "DW Drums", an American drum kit and hardware manufacturer * DW Sports Fitness, a defunct British sports and fitness retailer * Dollywood, a theme park in Tennessee, United States Art and entertainment Film and television * Darkwing Duck, a cartoon character * ''Deadliest Warrior'', an American factual television program * ''Doctor Who'', a British science fiction television programme * D.W. Read, a character in the ''Arthur'' TV show and book series * DreamWorks SKG, an American movie studio Other media * ''Discworld'', a series of books by Terry Pratchett * ''Digimon World'', a vi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Simplex Algorithm In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming. The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are not actually used in the method, but one interpretation of it is that it operates on simplicial ''cones'', and these become proper simplices with an additional constraint. The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the constraints applied to the objective function. History George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator. During 1946 his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of Wassily Leontief, however, at that t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]