|
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] |
Mathematical Programming
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Optimization problems can be divided into two categories, depending on whether the variables ... [...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 and objective 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 polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Matrix
In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices. For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 block. : \left \begin a_ & a_ & a_ & b_ \\ a_ & a_ & a_ & b_ \\ \hline c_ & c_ & c_ & d \end \right Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned. This notion can be made more precise for an n by m matrix M by partitioning n into a collection \text, and then partitioning m into a collection \text. The original m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Programming
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, but follow known probability distributions. This framework contrasts with deterministic optimization, in which all problem parameters are assumed to be known exactly. The goal of stochastic programming is to find a decision which both optimizes some criteria chosen by the decision maker, and appropriately accounts for the uncertainty of the problem parameters. Because many real-world decisions involve uncertainty, stochastic programming has found applications in a broad range of areas ranging from finance to transportation to energy optimization. Methods Several stochastic programming methods have been developed: * Scenario-based methods including Sample Average Approximation * Stochastic integer programming for problems in which some var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques F
Jacques or Jacq are believed to originate from the Middle Ages in the historic northwest Brittany region in France, and have since spread around the world over the centuries. To date, there are over one hundred identified noble families related to the surname by the Nobility & Gentry of Great Britain & Ireland. Origins The origin of this surname comes from the Latin ' Iacobus', associated with the biblical patriarch Jacob. Ancient history A French knight returning from the Crusades in the Holy Lands probably adopted the surname from "Saint Jacques" (or "James the Greater"). James the Greater was one of Jesus' Twelve Apostles, and is believed to be the first martyred apostle. Being endowed with this surname was an honor at the time and it is likely that the Church allowed it because of acts during the Crusades. Indeed, at this time, the use of biblical, Christian, or Hebrew names and surnames became very popular, and entered the European lexicon. Robert J., a Knight Crusader ... [...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] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary (topology), boundary of a convex set in the plane is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval (mathematics), interval with the property that its epigraph (mathematics), epigraph (the set of points on or above the graph of a function, graph of the function) is a convex set. Convex minimization is a subfield of mathematical optimization, optimization that studies the problem of minimizing convex functions over convex sets. The branch of mathematics devoted to the study of properties of convex sets and convex f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recession Cone
In mathematics, especially convex analysis, the recession cone of a set A is a cone containing all vectors such that A ''recedes'' in that direction. That is, the set extends outward in all the directions given by the recession cone. Mathematical definition Given a nonempty set A \subset X for some vector space X, then the recession cone \operatorname(A) is given by :\operatorname(A) = \. If A is additionally a convex set then the recession cone can equivalently be defined by :\operatorname(A) = \. If A is a nonempty closed convex set then the recession cone can equivalently be defined as :\operatorname(A) = \bigcap_ t(A - a) for any choice of a \in A. Properties * If A is a nonempty set then 0 \in \operatorname(A). * If A is a nonempty convex set then \operatorname(A) is a convex cone. * If A is a nonempty closed convex subset of a finite-dimensional Hausdorff space (e.g. \mathbb^d), then \operatorname(A) = \ if and only if A is bounded. * If A is a nonempty set then A + \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duality Theory
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involution operation: if the dual of is , then the dual of is . In other cases the dual of the dual – the double dual or bidual – is not necessarily identical to the original (also called ''primal''). Such involutions sometimes have fixed point (mathematics), fixed points, so that the dual of is itself. For example, Desargues' theorem is self-dual in this sense under the ''standard duality (projective geometry), duality in projective geometry''. In mathematical contexts, ''duality'' has numerous meanings. It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics". Many mathematical dualities between objects of two type ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FortSP
FortSP is a software package for solving stochastic programming (SP) problems. It solves scenario-based SP problems with recourse as well as problems with chance constraints and integrated chance constraints. FortSP is available as a standalone executable that accepts input in SMPS format and as a library with an interface in the C programming language. The solution algorithms provided by FortSP include Benders' decomposition and a variant of level decomposition for two-stage problems, nested Benders' decomposition for multistage problems and reformulation of the problem as a deterministic equivalent. There is also an implementation of a cutting-plane algorithm for integrated chance constraints. FortSP supports external linear programming solvers such as CPLEX IBM ILOG CPLEX Optimization Studio (often informally referred to simply as CPLEX) is an optimization software package. History The CPLEX Optimizer was named after the simplex method implemented in the C programmin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerische Mathematik
''Numerische Mathematik'' is a peer review, peer-reviewed mathematics journal on numerical analysis. It was established in 1959 and is published by Springer Science+Business Media. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 Mathematical Citation Quotient, MCQ was 1.06, and its 2020 impact factor was 2.223. References External links * Numerical analysis journals Academic journals established in 1959 English-language journals Springer Science+Business Media academic journals Monthly journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
