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Gradient Method
In optimization (mathematics), optimization, a gradient method is an algorithm to solve problems of the form :\min_\; f(x) with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradient. See also * Gradient descent * Stochastic gradient descent * Coordinate descent * Frank–Wolfe algorithm * Landweber iteration * Random coordinate descent * Conjugate gradient method * Derivation of the conjugate gradient method * Nonlinear conjugate gradient method * Biconjugate gradient method * Biconjugate gradient stabilized method References * First order methods Optimization algorithms and methods Numerical linear algebra Gradient methods, {{linear-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization (mathematics)
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 defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Coordinate Descent
Randomized (Block) Coordinate Descent Method is an optimization algorithm popularized by Nesterov (2010) and Richtárik and Takáč (2011). The first analysis of this method, when applied to the problem of minimizing a smooth convex function, was performed by Nesterov (2010). In Nesterov's analysis the method needs to be applied to a quadratic perturbation of the original function with an unknown scaling factor. Richtárik and Takáč (2011) give iteration complexity bounds which do not require this, i.e., the method is applied to the objective function directly. Furthermore, they generalize the setting to the problem of minimizing a composite function, i.e., sum of a smooth convex and a (possibly nonsmooth) convex block-separable function: F(x) = f(x) + \Psi(x), where \Psi(x) = \sum_^n \Psi_i(x^), x\in R^N is decomposed into n blocks of variables/coordinates: x = (x^,\dots,x^) and \Psi_1,\dots, \Psi_n are (simple) convex functions. Example (block decomposition): If x = (x ... [...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 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 define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
First Order Methods
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Biconjugate Gradient Stabilized Method
In numerical linear algebra, the biconjugate gradient stabilized method, often abbreviated as BiCGSTAB, is an iterative method developed by H. A. van der Vorst for the numerical solution of nonsymmetric linear systems. It is a variant of the biconjugate gradient method (BiCG) and has faster and smoother convergence than the original BiCG as well as other variants such as the conjugate gradient squared method (CGS). It is a Krylov subspace method. Unlike the original BiCG method, it doesn't require multiplication by the transpose of the system matrix. Algorithmic steps Unpreconditioned BiCGSTAB To solve a linear system , BiCGSTAB starts with an initial guess and proceeds as follows: # # Choose an arbitrary vector such that , e.g., . denotes the dot product of vectors # # # For ## ## ## ## ## ## ## If is accurate enough, then set and quit ## ## ## ## ## If is accurate enough, then quit ## Preconditioned BiCGSTAB Preconditioners are usually used to acceler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biconjugate Gradient Method
In mathematics, more specifically in numerical linear algebra, the biconjugate gradient method is an algorithm to solve systems of linear equations :A x= b.\, Unlike the conjugate gradient method, this algorithm does not require the matrix A to be self-adjoint, but instead one needs to perform multiplications by the conjugate transpose . The algorithm # Choose initial guess x_0\,, two other vectors x_0^* and b^*\, and a preconditioner M\, # r_0 \leftarrow b-A\, x_0\, # r_0^* \leftarrow b^*-x_0^*\, A # p_0 \leftarrow M^ r_0\, # p_0^* \leftarrow r_0^*M^\, # for k=0, 1, \ldots do ## \alpha_k \leftarrow \, ## x_ \leftarrow x_k + \alpha_k \cdot p_k\, ## x_^* \leftarrow x_k^* + \overline\cdot p_k^*\, ## r_ \leftarrow r_k - \alpha_k \cdot A p_k\, ## r_^* \leftarrow r_k^*- \overline \cdot p_k^*\, A ## \beta_k \leftarrow \, ## p_ \leftarrow M^ r_ + \beta_k \cdot p_k\, ## p_^* \leftarrow r_^*M^ + \overline\cdot p_k^*\, In the above formulation, the computed r_k\, and r_k^* satisfy :r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Conjugate Gradient Method
In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function \displaystyle f(x) :: \displaystyle f(x)=\, Ax-b\, ^2, the minimum of f is obtained when the gradient is 0: :: \nabla_x f=2 A^T(Ax-b)=0. Whereas linear conjugate gradient seeks a solution to the linear equation \displaystyle A^T Ax=A^T b, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient \nabla_x f alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum and the second derivative is non-singular there. Given a function \displaystyle f(x) of N variables to minimize, its gradient \nabla_x f indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction: :: \Delta x_0=-\nabla_x f (x_0) with an adjustable s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivation Of The Conjugate Gradient Method
In numerical linear algebra, the conjugate gradient method is an iterative method for numerically solving the linear system :\boldsymbol=\boldsymbol where \boldsymbol is symmetric positive-definite. The conjugate gradient method can be derived from several different perspectives, including specialization of the conjugate direction methodConjugate Direction Methods http://user.it.uu.se/~matsh/opt/f8/node5.html for optimization, and variation of the Arnoldi/Lanczos iteration for eigenvalue problems. The intent of this article is to document the important steps in these derivations. Derivation from the conjugate direction method The conjugate gradient method can be seen as a special case of the conjugate direction method applied to minimization of the quadratic function :f(\boldsymbol)=\boldsymbol^\mathrm\boldsymbol\boldsymbol-2\boldsymbol^\mathrm\boldsymbol\text The conjugate direction method In the conjugate direction method for minimizing :f(\boldsymbol)=\boldsymbol^\mathrm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Gradient Method
In mathematics, the conjugate gradient method is an algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ... for the numerical solution of particular system of linear equations, systems of linear equations, namely those whose matrix is positive-definite matrix, positive-definite. The conjugate gradient method is often implemented as an iterative method, iterative algorithm, applicable to sparse matrix, sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems. The conjugate gradient method can also be used to solve unconstrained Mathematical optimization, optimization problems such as energ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landweber Iteration
The Landweber iteration or Landweber algorithm is an algorithm to solve ill-posed linear inverse problems, and it has been extended to solve non-linear problems that involve constraints. The method was first proposed in the 1950s by Louis Landweber, and it can be now viewed as a special case of many other more general methods. Basic algorithm The original Landweber algorithm attempts to recover a signal ''x'' from (noisy) measurements ''y''. The linear version assumes that y = Ax for a linear operator ''A''. When the problem is in finite dimensions, ''A'' is just a matrix. When ''A'' is Algebraic curve#Singularities, nonsingular, then an explicit solution is x = A^ y. However, if ''A'' is ill-conditioned, the explicit solution is a poor choice since it is sensitive to any noise in the data ''y''. If ''A'' is Mathematical singularity, singular, this explicit solution doesn't even exist. The Landweber algorithm is an attempt to Regularization (mathematics), regularize the problem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank–Wolfe Algorithm
The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient method, reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956. In each iteration, the Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer of this linear function (taken over the same domain). Problem statement Suppose \mathcal is a compact convex set in a vector space and f \colon \mathcal \to \mathbb is a convex, differentiable real-valued function. The Frank–Wolfe algorithm solves the optimization problem :Minimize f(\mathbf) :subject to \mathbf \in \mathcal. Algorithm :''Initialization:'' Let k \leftarrow 0, and let \mathbf_0 \! be any point in \mathcal. :Step 1. ''Direction-finding subproblem:'' Find \mathbf_k solving ::Minimize \mathbf^T \nabla f(\mathbf_k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
