|
Mixed Complementarity Problem
Mixed Complementarity Problem (MCP) is a problem formulation in mathematical programming. Many well-known problem types are special cases of, or may be reduced to MCP. It is a generalization of nonlinear complementarity problem (NCP). Definition The mixed complementarity problem is defined by a mapping F(x): \mathbb^n \to \mathbb^n, lower values \ell_i \in \mathbb \cup \ and upper values u_i \in \mathbb\cup\. The solution of the MCP is a vector x \in \mathbb^n such that for each index i \in \ one of the following alternatives holds: * x_i = \ell_i, \; F_i(x) \ge 0; * \ell_i < x_i < u_i, \; F_i(x) = 0; * . Another definition for MCP is: it is a on the |
Mathematical Programming
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems of sorts 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. More generally, optimization includes finding "best available" values of some objective function given a def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Complementarity Problem
In applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ..., a nonlinear complementarity problem (NCP) with respect to a mapping ''ƒ'' : R''n'' → R''n'', denoted by NCP''ƒ'', is to find a vector ''x'' ∈ R''n'' such that : x \geq 0,\ f(x) \geq 0 \text x^f(x)=0 where ''ƒ''(''x'') is a smooth mapping. The case of a discontinuous mapping was discussed by Habetler and Kostreva (1978). References * * Applied mathematics {{applied-math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variational Inequality
In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems, precisely the Signorini problem: in that model problem, the functional involved was obtained as the first variation of the involved potential energy. Therefore, it has a variational origin, recalled by the name of the general abstract problem. The applicability of the theory has since been expanded to include problems from economics, finance, optimization and game theory. History The first problem involving a variational inequality was the Signorini problem, posed by Antonio Signorini in 1959 and solved by Gaetano Fichera in 1963, according to the references and : the first papers of the theory were and , . Later on, Guido Stampacchia proved his generalization to the Lax–Milgram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term '' rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, the four concepts—''parallelepiped'' and ''cube'' in three dimensions, ''parallelogram'' and ''square'' in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only ''parallelograms'' and ''parallelepipeds'' exist. Three equivalent definitions of ''parallelepiped'' are *a polyhedron with six faces ( hexahedron), each of which is a parallelogram, *a hexahedron with three pairs of parallel faces, and *a prism of which the base is a parallelogram. The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all specific cases of parallelepiped. "Parallelepiped" is now usually pronounce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complementarity Theory
A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements (constraints) which include: that the inner product of the two vectors must equal zero, i.e. they are orthogonal. In particular for finite-dimensional real vector spaces this means that, if one has vectors ''X'' and ''Y'' with all ''nonnegative'' components (''x''''i'' ≥ 0 and ''y''''i'' ≥ 0 for all i: in the first quadrant if 2-dimensional, in the first octant if 3-dimensional), then for each pair of components ''x''''i'' and ''y''''i'' one of the pair must be zero, hence the name ''complementarity''. e.g. ''X'' = (1, 0) and ''Y'' = (0, 2) are complementary, but ''X'' = (1, 1) and ''Y'' = (2, 0) are not. A complementarity problem is a special case of a variational inequality. History Complementarity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adobe Photoshop
Adobe Photoshop is a raster graphics editor developed and published by Adobe Inc. for Windows and macOS. It was originally created in 1988 by Thomas and John Knoll. Since then, the software has become the industry standard not only in raster graphics editing, but in digital art as a whole. The software's name is often colloquially used as a verb (e.g. "to photoshop an image", " photoshopping", and "photoshop contest") although Adobe discourages such use. Photoshop can edit and compose raster images in multiple layers and supports masks, alpha compositing and several color models including RGB, CMYK, CIELAB, spot color, and duotone. Photoshop uses its own PSD and PSB file formats to support these features. In addition to raster graphics, Photoshop has limited abilities to edit or render text and vector graphics (especially through clipping path for the latter), as well as 3D graphics and video. Its feature set can be expanded by plug-ins; programs developed and di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
