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Damped Newton Method
In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function f, which are solutions to the equation f(x)=0. However, to optimize a twice-differentiable f, our goal is to find the roots of f'. We can therefore use Newton's method on its derivative f' to find solutions to f'(x)=0, also known as the critical points of f. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function f. Newton's method The central problem of optimization is minimization of functions. Let us first consider the case of univariate functions, i.e., functions of a single real variable. We will later consider the more general and more practically useful multivariate case. Given a twice differentiable function f:\mathbb\to \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Optimization Vs Grad Descent
Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: People * Newton (surname), including a list of people with the surname * Newton (given name), including a list of people with the given name Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton (band), Spanish electronic music group * ''Newton'' (Blake), a print by William Blake * ''Newton'' (Paolozzi), a 1995 bronze sculpture by Eduardo Paolozzi * Cecil Newton (''Coronation Street''), a character in the British soap opera ''Coronation Street'' * Curtis Newton, "real" name of pulp magazine character Captain Future * George Newton, a character in the film series ''Beethoven Ludwig van Beethoven (baptised 17 December 177026 March 1827) was a German composer and pianist. He is one of the most revered figures in the history of Western music; his works rank among the most performed o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The gradient transforms like a vector under change of basis of the space of variables of f. If the gradient of a function is non-zero at a point p, the direction of the gradient is the direction in which the function increases most quickly from p, and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to minimize a function by gradient descent. In coordinate-free terms, the gradient of a function f(\mathbf) may be defined by: df=\nabla f \cdot d\mathbf where df is the total infinitesimal change in f for a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Residual Method
The conjugate residual method is an iterative numeric method used for solving systems of linear equations. It's a Krylov subspace method very similar to the much more popular conjugate gradient method, with similar construction and convergence properties. This method is used to solve linear equations of the form :\mathbf A \mathbf x = \mathbf b where A is an invertible and Hermitian matrix, and b is nonzero. The conjugate residual method differs from the closely related conjugate gradient method. It involves more numerical operations and requires more storage. Given an (arbitrary) initial estimate of the solution \mathbf x_0, the method is outlined below: : \begin & \mathbf_0 := \text \\ & \mathbf_0 := \mathbf - \mathbf_0 \\ & \mathbf_0 := \mathbf_0 \\ & \text k \text 0:\\ & \qquad \alpha_k := \frac \\ & \qquad \mathbf_ := \mathbf_k + \alpha_k \mathbf_k \\ & \qquad \mathbf_ := \mathbf_k - \alpha_k \mathbf_k \\ & \qquad \beta_k := \frac \\ & \qquad \mathbf_ := \mathbf_ + \beta_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Multipliers
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables). It is named after the mathematician Joseph-Louis Lagrange. Summary and rationale The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. In the general case, the Lagrangian is defined as \mathcal(x, \lambda) \equiv f(x) + \langle \lambda, g(x)\rangle for functions f, g; the notation \langle \cdot, \cdot \rangle denotes an inner product. The value \lambda is called the Lagrange multiplier. In simple ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constrained Optimization
In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized. Constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints, which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied. Relation to constraint-satisfaction problems The constrained-optimization problem (COP) is a significant generalization of the classic constraint-satisfaction problem (CSP) model. COP is a CSP that includes an ''objective function'' to be optimized. Many algorithms are used to hand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Gradient Method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to 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 optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it. The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear optimization problems. Description of the problem addres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cholesky Factorization
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices, and posthumously published in 1924. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations. Statement The Cholesky decomposition of a Hermitian positive-definite matrix , is a decomposition of the form \mathbf = \mathbf^, where is a lower triangular matrix with real and positive diagonal entries, and * denotes the conjugate transpose of . Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. The converse holds trivially: if can be wri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Methods
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with Algorithm#Termination, termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or Quasi-Newton method, quasi-Newton methods like Broyden–Fletcher–Goldfarb–Shanno algorithm, BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called ''Convergent series, convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
System Of Linear Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variable (math), variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A ''Solution (mathematics), solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the Tuple, ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A Nonlinear system, system of non-linear equations can often be Approximation, approximated by a linear system (see linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Backtracking Line Search
In (unconstrained) mathematical optimization, a backtracking line search is a line search method to determine the amount to move along a given search direction. Its use requires that the objective function is differentiable and that its gradient is known. The method involves starting with a relatively large estimate of the step size for movement along the line search direction, and iteratively shrinking the step size (i.e., "backtracking") until a decrease of the objective function is observed that adequately corresponds to the amount of decrease that is expected, based on the step size and the local gradient of the objective function. A common stopping criterion is the Armijo–Goldstein condition. Backtracking line search is typically used for gradient descent (GD), but it can also be used in other contexts. For example, it can be used with Newton's method if the Hessian matrix is positive definite. Motivation Given a starting position \mathbf and a search direction \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfe Conditions
In the unconstrained minimization problem, the Wolfe conditions are a set of inequalities for performing inexact line search, especially in quasi-Newton methods, first published by Philip Wolfe in 1969. In these methods the idea is to find \min_x f(\mathbf) for some smooth f\colon\mathbb R^n\to\mathbb R. Each step often involves approximately solving the subproblem \min_ f(\mathbf_k + \alpha \mathbf_k) where \mathbf_k is the current best guess, \mathbf_k \in \mathbb R^n is a search direction, and \alpha \in \mathbb R is the step length. The inexact line searches provide an efficient way of computing an acceptable step length \alpha that reduces the objective function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost ... 'sufficiently', rather than minimizing the objective function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
