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Line Search
In optimization, line search is a basic iterative approach to find a local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. It first finds a descent direction along which the objective function f will be reduced, and then computes a step size that determines how far \mathbf should move along that direction. The descent direction can be computed by various methods, such as gradient descent or quasi-Newton method. The step size can be determined either exactly or inexactly. One-dimensional line search Suppose ''f'' is a one-dimensional function, f:\mathbb R\to\mathbb R, and assume that it is unimodal, that is, contains exactly one local minimum ''x''* in a given interval 'a'',''z'' This means that ''f'' is strictly decreasing in ,x*and strictly increasing in *,''z'' There are several ways to find an (approximate) minimum point in this case. Zero-order methods Zero-order methods use only function evaluations (i.e., a value oracle) - not derivatives: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Optimization (mathematics)
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] [Amazon] |
Superlinear Convergence
In mathematical analysis, particularly numerical analysis, the rate of convergence and order of convergence of a sequence that converges to a limit are any of several characterizations of how quickly that sequence approaches its limit. These are broadly divided into rates and orders of convergence that describe how quickly a sequence further approaches its limit once it is already close to it, called asymptotic rates and orders of convergence, and those that describe how quickly sequences approach their limits from starting points that are not necessarily close to their limits, called non-asymptotic rates and orders of convergence. Asymptotic behavior is particularly useful for deciding when to stop a sequence of numerical computations, for instance once a target precision has been reached with an iterative root-finding algorithm, but pre-asymptotic behavior is often crucial for determining whether to begin a sequence of computations at all, since it may be impossible or imprac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pattern Search (optimization)
Pattern search (also known as direct search, derivative-free search, or black-box search) is a family of numerical optimization methods that does not require a gradient. As a result, it can be used on functions that are not continuous or differentiable. One such pattern search method is "convergence" (see below), which is based on the theory of positive bases. Optimization attempts to find the best match (the solution that has the lowest error value) in a multidimensional analysis space of possibilities. History The name "pattern search" was coined by Hooke and Jeeves. An early and simple variant is attributed to Fermi and Metropolis when they worked at the Los Alamos National Laboratory. It is described by Davidon, as follows: Convergence Convergence is a pattern search method proposed by Yu, who proved that it converges using the theory of positive bases.*Yu, Wen Ci. 1979. âPositive basis and a class of direct search techniques€. ''Scientia Sinica'' 'Zhongguo Kexue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Learning Rate
In machine learning and statistics, the learning rate is a tuning parameter in an optimization algorithm that determines the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly acquired information overrides old information, it metaphorically represents the speed at which a machine learning model "learns". In the adaptive control literature, the learning rate is commonly referred to as gain. In setting a learning rate, there is a trade-off between the rate of convergence and overshooting. While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. A too high learning rate will make the learning jump over minima but a too low learning rate will either take too long to converge or get stuck in an undesirable local minimum. In order to achieve faster convergence, prevent oscillations and getting stuck in undesi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Grid Search
Grid, The Grid, or GRID may refer to: Space partitioning * Regular grid, a tessellation of space with translational symmetry, typically formed from parallelograms or higher-dimensional analogs ** Grid graph, a graph structure with nodes connected in a regular grid ** Square grid, a grid of squares ** Triangular grid, a grid of triangles ** Hexagonal grid, a grid of hexagons ** Unstructured grid, a tessellation of a space by simple shapes such as triangles or tetrahedra in an irregular pattern * Grid reference system, a coordinate system relative to a particular map projection * Grid (spatial index), a discretization of a geometric domain into a set of contiguous cells, used to organize information ** Discrete global grid (DGG), a grid that covers the entire Earth's surface * Grid (graphic design) (or typographic grid), organized lines for guiding graphic design * Grid plan, a city design with streets running at right angles * Grid paper, paper with a regular grid printe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
