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Compact Quasi-Newton Representation
The compact representation for quasi-Newton methods is a matrix decomposition, which is typically used in gradient based optimization (mathematics), optimization algorithms or for solving nonlinear systems. The decomposition uses a low-rank representation for the direct and/or inverse Hessian matrix, Hessian or the Jacobian matrix and determinant, Jacobian of a nonlinear system. Because of this, the compact representation is often used for large problems and constrained optimization. Definition The compact representation of a quasi-Newton matrix for the inverse Hessian H_k or direct Hessian B_k of a nonlinear loss function, objective function f(x):\mathbb^n \to \mathbb expresses a sequence of recursive rank-1 or rank-2 matrix updates as one rank-k or rank-2k update of an initial matrix. Because it is derived from quasi-Newton updates, it uses differences of iterates and gradients \nabla f(x_k) = g_k in its definition \_^k . In particular, for r=k or r=2k the rectangular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Methods
In numerical analysis, a quasi-Newton method is an Iterative method, iterative numerical method used either to Root-finding algorithm, find zeroes or to Mathematical optimization, find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the Derivative, derivatives of the functions in place of exact derivatives. Newton's method requires the Jacobian matrix and determinant, Jacobian matrix of all Partial derivative, partial derivatives of a multivariate function when used to search for zeros or the Hessian matrix when used Newton's method in optimization, for finding extrema. Quasi-Newton methods, on the other hand, can be used when the Jacobian matrices or Hessian matrices are unavailable or are impractical to compute at every iteration. Some Iterative method, iterative methods that reduce to Newton's method, such as sequential quadratic programming, may also be considered quasi-Newton methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Householder Transformation
In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection (mathematics), reflection about a plane (mathematics), plane or hyperplane containing the origin. The Householder transformation was used in a 1958 paper by Alston Scott Householder. Definition Operator and transformation The Householder Operator (mathematics), operator may be defined over any finite-dimensional inner product space V with inner product \langle \cdot, \cdot \rangle and unit vector u\in V as : H_u(x) := x - 2\,\langle x,u \rangle\,u\,. It is also common to choose a non-unit vector q \in V, and normalize it directly in the Householder operator's expression: :H_q \left ( x \right ) = x - 2\, \frac\, q \,. Such an operator is Linear operator, linear and self-adjoint. If V=\mathbb^n, note that the reflection hyperplane can be defined by its ''normal vector'', a unit vector \vec v\in V (a vector wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IPOPT
IPOPT, short for "Interior Point OPTimizer, pronounced I-P-Opt", is a software library for large scale nonlinear optimization of continuous systems. It is written in C++ (after migrating from Fortran and C) and is released under the EPL (formerly CPL). IPOPT implements a primal-dual interior point method, and uses line searches based on Filter methods ( Fletcher and Leyffer). IPOPT can be called from various modeling environments: C, C++, Fortran, Java, R, Python, and others. IPOPT is part of the COIN-OR project. IPOPT is designed to exploit 1st derivative ( gradient) and 2nd derivative ( Hessian) information if provided (usually via automatic differentiation routines in modeling environments such as AMPL). If no Hessians are provided, IPOPT will approximate them using a quasi-Newton methods, specifically a BFGS update. IPOPT was originally developed by Ph.D. studenAndreas Wächterand ProfLorenz T. Bieglerof the Department of Chemical Engineering at Carneg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SciPy
SciPy (pronounced "sigh pie") is a free and open-source Python library used for scientific computing and technical computing. SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, fast Fourier transform, signal and image processing, ordinary differential equation solvers and other tasks common in science and engineering. SciPy is also a family of conferences for users and developers of these tools: SciPy (in the United States), EuroSciPy (in Europe) and SciPy.in (in India). Enthought originated the SciPy conference in the United States and continues to sponsor many of the international conferences as well as host the SciPy website. The SciPy library is currently distributed under the BSD license, and its development is sponsored and supported by an open community of developers. It is also supported by NumFOCUS, a community foundation for supporting reproducible and accessible science. Components The SciPy package is at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
R (programming Language)
R is a programming language for statistical computing and Data and information visualization, data visualization. It has been widely adopted in the fields of data mining, bioinformatics, data analysis, and data science. The core R language is extended by a large number of R package, software packages, which contain Reusability, reusable code, documentation, and sample data. Some of the most popular R packages are in the tidyverse collection, which enhances functionality for visualizing, transforming, and modelling data, as well as improves the ease of programming (according to the authors and users). R is free and open-source software distributed under the GNU General Public License. The language is implemented primarily in C (programming language), C, Fortran, and Self-hosting (compilers), R itself. Preprocessor, Precompiled executables are available for the major operating systems (including Linux, MacOS, and Microsoft Windows). Its core is an interpreted language with a na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
ACM Transactions On Mathematical Software
''ACM Transactions on Mathematical Software'' (''TOMS'') is a quarterly scientific journal that aims to disseminate the latest findings of note in the field of numeric, symbolic, algebraic, and geometric computing applications. The journal publishes two kinds of articles: Regular research papers that advance the development of algorithms and software for mathematical computing, and "algorithms papers" that describe a specific implementation of an algorithm and that are accompanied by the source code for this algorithm. Algorithms described in the transactions are generally published in the ''Collected Algorithms of the ACM (CALGO)''. Algorithms published since 1975 (and some earlier ones) are all still available. Software that accompanies algorithm papers is accessible by anyone via the CALGO website. History ACM Transactions on Mathematical Software is one of the oldest scientific journals specifically dedicated to mathematical algorithms and their implementation in software, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limited-memory BFGS
Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) using a limited amount of computer memory. It is a popular algorithm for parameter estimation in machine learning. The algorithm's target problem is to minimize f(\mathbf) over unconstrained values of the real-vector \mathbf where f is a differentiable scalar function. Like the original BFGS, L-BFGS uses an estimate of the inverse Hessian matrix to steer its search through variable space, but where BFGS stores a dense n\times n approximation to the inverse Hessian (''n'' being the number of variables in the problem), L-BFGS stores only a few vectors that represent the approximation implicitly. Due to its resulting linear memory requirement, the L-BFGS method is particularly well suited for optimization problems with many variables. Instead of the inverse Hessian H''k'', L-BFGS maintains a history of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karush–Kuhn–Tucker Conditions
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. Similar to the Lagrange approach, the constrained maximization (minimization) problem is rewritten as a Lagrange function whose optimal point is a global maximum or minimum over the domain of the choice variables and a global minimum (maximum) over the multipliers. The Karush–Kuhn–Tucker theorem is sometimes referred to as the saddle-point theorem. The KKT conditions were originally named after Harold W. Kuhn and Albert W. Tucker, who first published the conditions in 1951. Later scholars discovered that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Underdetermined System
In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting. Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom. Therefore, the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint removing a degree of freedom. An indeterminate system additional constraints that are not equations, such as restricting the solutions to integers. The underdetermined case, by contrast, occurs when the system has been underconstrained—that is, when the unknown ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woodbury Matrix Identity
In mathematics, specifically linear algebra, the Woodbury matrix identity – named after Max A. Woodbury – says that the inverse of a rank-''k'' correction of some matrix can be computed by doing a rank-''k'' correction to the inverse of the original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the identity appeared in several papers before the Woodbury report. The Woodbury matrix identity is \left(A + UCV \right)^ = A^ - A^U \left(C^ + VA^U \right)^ VA^, where ''A'', ''U'', ''C'' and ''V'' are conformable matrices: ''A'' is ''n''×''n'', ''C'' is ''k''×''k'', ''U'' is ''n''×''k'', and ''V'' is ''k''×''n''. This can be derived using blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broyden–Fletcher–Goldfarb–Shanno Algorithm
In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems. Like the related Davidon–Fletcher–Powell method, BFGS determines the descent direction by preconditioning the gradient with curvature information. It does so by gradually improving an approximation to the Hessian matrix of the loss function, obtained only from gradient evaluations (or approximate gradient evaluations) via a generalized secant method. Since the updates of the BFGS curvature matrix do not require matrix inversion, its computational complexity is only \mathcal(n^), compared to \mathcal(n^) in Newton's method. Also in common use is L-BFGS, which is a limited-memory version of BFGS that is particularly suited to problems with very large numbers of variables (e.g., >1000). The BFGS-B variant handles simple box constraints. The BFGS matrix also admits a compact representation, which makes it b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
