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Theory Of Two-level Planning
The theory of two-level planning (alternatively, Kornai–Liptak decomposition) is a method that Matrix decomposition , decomposes large problems of Linear programming, linear optimization into sub-problems. This decomposition simplifies the solution of the overall problem. The method also models a method of coordinating economic decisions so that decentralized firms behave so as to produce a global optimum. It was introduced by the Hungarian economist János Kornai and the mathematician Tamás Lipták in 1965. It is an alternative to Dantzig–Wolfe decomposition. Description The LP problem must have a special structure, known as a block angular structure. This is the same structure required for the Dantzig Wolfe decomposition: There are some constraints on overall resources (D) for which a central planning agency is assumed to be responsible, and n blocks of coefficients (F1 through Fn) that are the concern of individual firms. The central agency starts the process by prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Decomposition
In the mathematics, mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a Matrix (mathematics), matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. Example In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations A \mathbf = \mathbf, the matrix ''A'' can be decomposed via the LU decomposition. The LU decomposition factorizes a matrix into a lower triangular matrix ''L'' and an upper triangular matrix ''U''. The systems L(U \mathbf) = \mathbf and U \mathbf = L^ \mathbf require fewer additions and multiplications to solve, compared with the original system A \mathbf = \mathbf, though one might require significantly more digits in inexact arithmetic such as floating point. Similarly, the QR decomposition expresses ''A'' as ''QR'' with ''Q'' an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
János Kornai
János Kornai (21 January 1928 – 18 October 2021) was a Hungarian economist noted for his analysis and criticism of the command economies of Eastern European communist states. He also covered macroeconomic aspects in countries undergoing post-Soviet transition. He was emeritus professor at both Harvard University and Corvinus University of Budapest. Kornai was known to have coined the term shortage economy to reflect perpetual shortages of goods in the centrally-planned command economies of the Eastern Bloc. Biography Kornai was born János Kornhauser on 21 January 1928 in Budapest in a well-to-do Hungarian-Jewish family. The family spent time in an internment camp during the Nazi occupation of Hungary. His father was sent for slave labour and later killed at Auschwitz. Kornai studied philosophy for two years at Pázmány Péter University (now Eötvös Loránd University) in Budapest. Kornai gained his knowledge of economics on his own, and later held a candidate degree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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] |
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] |
