Duality (optimization)In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem.
Quadratic formIn mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form.
Cholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ʃəˈlɛski ) 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.
Conjugate gradient methodIn mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. 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.
Optimization problemIn mathematics, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete: An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
Isotropic quadratic formIn mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0. A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V.
Page (computer memory)A page, memory page, or virtual page is a fixed-length contiguous block of virtual memory, described by a single entry in the page table. It is the smallest unit of data for memory management in a virtual memory operating system. Similarly, a page frame is the smallest fixed-length contiguous block of physical memory into which memory pages are mapped by the operating system. A transfer of pages between main memory and an auxiliary store, such as a hard disk drive, is referred to as paging or swapping.
Ε-quadratic formIn mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called -quadratic forms, particularly in the context of surgery theory. There is the related notion of ε-symmetric forms, which generalizes symmetric forms, skew-symmetric forms (= symplectic forms), Hermitian forms, and skew-Hermitian forms.
LU decompositionIn numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.
File systemIn computing, a file system or filesystem (often abbreviated to fs) is a method and data structure that the operating system uses to control how data is stored and retrieved. Without a file system, data placed in a storage medium would be one large body of data with no way to tell where one piece of data stopped and the next began, or where any piece of data was located when it was time to retrieve it. By separating the data into pieces and giving each piece a name, the data are easily isolated and identified.
Linear programmingLinear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.
Matrix decompositionIn the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems. In numerical analysis, different decompositions are used to implement efficient matrix algorithms. For instance, when solving a system of linear equations , the matrix A can be decomposed via the LU decomposition.
