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Linear-fractional Programming
In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP). Whereas the objective function in a linear program is a linear function, the objective function in a linear-fractional program is a ratio of two linear functions. A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function 1. Formally, a linear-fractional program is defined as the problem of maximizing (or minimizing) a ratio of affine functions over a polyhedron, : \begin \text \quad & \frac \\ \text \quad & A\mathbf \leq \mathbf, \end where \mathbf \in \mathbb^n represents the vector of variables to be determined, \mathbf, \mathbf \in \mathbb^n and \mathbf \in \mathbb^m are vectors of (known) coefficients, A \in \mathbb^ is a (known) matrix of coefficients and \alpha, \beta \in \mathbb are constants. The constraints have to restrict the feasible region to \, i.e. the region on which the denominator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
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 maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoconvex Function
In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function (mathematics), function that behaves like a convex function with respect to finding its local extrema, local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative. The property must hold in all of the function domain, and not only for nearby points. Formal definition Consider a differentiable function f:X \subseteq \mathbb^ \rightarrow \mathbb, defined on a (nonempty) convex set, convex open set X of the finite-dimensional Euclidean space \mathbb^n. This function is said to be pseudoconvex if the following property holds: Equivalently: Here \nabla f is the gradient of f, defined by: \nabla f = \left(\frac,\dots,\frac\right). Note that the definition may also be stated in terms of the directional derivative of f, in the direction given ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Algorithms And Methods
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 Argument of a function, input values from within an allowed set and computing the Value (mathematics), 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 categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior-point Method
Interior-point methods (also referred to as barrier methods or IPMs) are algorithms for solving linear and non-linear convex optimization problems. IPMs combine two advantages of previously-known algorithms: * Theoretically, their run-time is polynomial—in contrast to the simplex method, which has exponential run-time in the worst case. * Practically, they run as fast as the simplex method—in contrast to the ellipsoid method, which has polynomial run-time in theory but is very slow in practice. In contrast to the simplex method which traverses the ''boundary'' of the feasible region, and the ellipsoid method which bounds the feasible region from ''outside'', an IPM reaches a best solution by traversing the ''interior'' of the feasible region—hence the name. History An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Criss-cross Algorithm
In optimization (mathematics), mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear programming, linear inequality constraints and nonlinear programming, nonlinear optimization (mathematics), objective functions; there are criss-cross algorithms for linear-fractional programming problems, quadratic programming, quadratic-programming problems, and linear complementarity problems. Like the simplex algorithm of George Dantzig, George B. Dantzig, the criss-cross algorithm is not a time complexity, polynomial-time algorithm for linear programming. Both algorithms visit all 2''D'' corners of a (perturbed) unit cube, cube in dimension ''D'', the Klee–Minty cube (after Victor Klee and George J. Minty), in the worst-case complexity, worst case. However, when it is started at a random corner, the criss-cross algorithm Average-case com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Review
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics. SIAM is one of the four member organizations of the Joint Policy Board for Mathematics. Membership Membership is open to both individuals and organizations. By the end of its first full year of operation, SIAM had 130 membe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George B
George may refer to: Names * George (given name) * George (surname) People * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Papagheorghe, also known as Jorge / GEØRGE * George, stage name of Giorgio Moroder * George, son of Andrew I of Hungary Places South Africa * George, South Africa, a city ** George Airport United States * George, Iowa, a city * George, Missouri, a ghost town * George, Washington, a city * George County, Mississippi * George Air Force Base, a former U.S. Air Force base located in California Computing * George (algebraic compiler) also known as 'Laning and Zierler system', an algebraic compiler by Laning and Zierler in 1952 * GEORGE (computer), early computer built by Argonne National Laboratory in 1957 * GEORGE (operating system), a range of operating systems (George 1–4) for the ICT 1900 range of computers in the 1960s * GEORGE (programming language), an autocode system invented by Charles Le ... [...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 tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudolinear Function
In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative. The property must hold in all of the function domain, and not only for nearby points. Formal definition Consider a differentiable function f:X \subseteq \mathbb^ \rightarrow \mathbb, defined on a (nonempty) convex open set X of the finite-dimensional Euclidean space \mathbb^n. This function is said to be pseudoconvex if the following property holds: Equivalently: Here \nabla f is the gradient of f, defined by: \nabla f = \left(\frac,\dots,\frac\right). Note that the definition may also be stated in terms of the directional derivative of f, in the direction given by the vector v=y-x. This is because, as f is diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonicity
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), so i ... [...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] |
Quasiconvex Function
In mathematics, a quasiconvex function is a real number, real-valued function (mathematics), function defined on an interval (mathematics), interval or on a convex set, convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. Quasiconvexity is a more general property than convexity in that all convex functions are also quasiconvex, but not all quasiconvex functions are convex. ''Univariate'' Unimodality, unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple argument of a function, arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with Star_domain, star-convex sublevel sets can be unimodal without being quasiconvex. Def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
