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Proximal Operator
In mathematical optimization, the proximal operator is an operator associated with a proper,An (extended) real-valued function ''f'' on a Hilbert space is said to be ''proper'' if it is not identically equal to +\infty, and -\infty is not in its image. lower semi-continuous convex function f from a Hilbert space \mathcal to \infty,+\infty/math>, and is defined by: ::\operatorname_f(v) = \arg \min_ \left(f(x) + \frac 1 2 \, x - v\, _\mathcal^2\right). For any function in this class, the minimizer of the right-hand side above is unique, hence making the proximal operator well-defined. The proximal operator is used in proximal gradient methods, which is frequently used in optimization algorithms associated with non-differentiable optimization problems such as total variation denoising. Properties The \text of a proper, lower semi-continuous convex function f enjoys several useful properties for optimization. * Fixed points of \text_f are minimizers of f: \ = \arg \min f. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
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 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. More generally, opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraction Mapping
In mathematics, a contraction mapping, or contraction or contractor, on a metric space (''M'', ''d'') is a function ''f'' from ''M'' to itself, with the property that there is some real number 0 \leq k < 1 such that for all ''x'' and ''y'' in ''M'', : The smallest such value of ''k'' is called the Lipschitz constant of ''f''. Contractive maps are sometimes called Lipschitzian maps. If the above condition is instead satisfied for ''k'' ≤ 1, then the mapping is said to be a . More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (''M'', ''d'') and (''N'', ''d) are two metric spaces, then is a contractive mapping if there is a constant such that : [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Python (programming Language)
Python is a high-level, general-purpose programming language. Its design philosophy emphasizes code readability with the use of significant indentation. Python is dynamically-typed and garbage-collected. It supports multiple programming paradigms, including structured (particularly procedural), object-oriented and functional programming. It is often described as a "batteries included" language due to its comprehensive standard library. Guido van Rossum began working on Python in the late 1980s as a successor to the ABC programming language and first released it in 1991 as Python 0.9.0. Python 2.0 was released in 2000 and introduced new features such as list comprehensions, cycle-detecting garbage collection, reference counting, and Unicode support. Python 3.0, released in 2008, was a major revision that is not completely backward-compatible with earlier versions. Python 2 was discontinued with version 2.7.18 in 2020. Python consistently ranks as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matlab
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico and starte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proximal Gradient Method
Proximal gradient methods are a generalized form of projection used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of the form \operatorname\limits_ \sum_^n f_i(x) where f_i: \mathbb^N \rightarrow \mathbb,\ i = 1, \dots, n are possibly non-differentiable convex functions. The lack of differentiability rules out conventional smooth optimization techniques like the steepest descent method and the conjugate gradient method, but proximal gradient methods can be used instead. Proximal gradient methods starts by a splitting step, in which the functions f_1, . . . , f_n are used individually so as to yield an easily implementable algorithm. They are called proximal because each non-differentiable function among f_1, . . . , f_n is involved via its proximity operator. Iterative shrinkage thresholding algorithm, projected Landweber, projected gradient, alternating projections, alternating-d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Optimization
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 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. More generally, opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subdifferential
In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connection to convex optimization. Let f:I \to \mathbb be a real-valued convex function defined on an open interval of the real line. Such a function need not be differentiable at all points: For example, the absolute value function ''f''(''x'')=, ''x'', is nondifferentiable when ''x''=0. However, as seen in the graph on the right (where ''f(x)'' in blue has non-differentiable kinks similar to the absolute value function), for any ''x''0 in the domain of the function one can draw a line which goes through the point (''x''0, ''f''(''x''0)) and which is everywhere either touching or below the graph of ''f''. The slope of such a line is called a ''subderivative'' (because the line is under the graph of ''f''). Definition Rigorously, a ''subderivat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moreau Envelope
The Moreau envelope (or the Moreau-Yosida regularization) M_f of a proper lower semi-continuous convex function f is a smoothed version of f. It was proposed by Jean-Jacques Moreau in 1965. The Moreau envelope has important applications in mathematical optimization: minimizing over M_f and minimizing over f are equivalent problems in the sense that set of minimizers of f and M_f are the same. However, first-order optimization algorithms can be directly applied to M_f, since f may be non-differentiable while M_f is always continuously differentiable. Indeed, many proximal gradient methods can be interpreted as a gradient descent method over M_f. Definition The Moreau envelope of a proper lower semi-continuous convex function f from a Hilbert space \mathcal to (-\infty,+\infty] is defined as M_(v) = \inf_ \left(f(x) + \frac \, x - v\, _2^2\right). Given a parameter \lambda \in \mathbb, the Moreau envelope of \lambda f is also called as the Moreau envelope of f with parameter \lam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Function (convex Analysis)
In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function, and one can freely convert between the two, but the characteristic function as defined below is better-suited to the methods of convex analysis. Definition Let X be a set, and let A be a subset of X. The characteristic function of A is the function :\chi_ : X \to \mathbb \cup \ taking values in the extended real number line defined by :\chi_ (x) := \begin 0, & x \in A; \\ + \infty, & x \not \in A. \end Relationship with the indicator function Let \mathbf_ : X \to \mathbb denote the usual indicator function: :\mathbf_ (x) := \begin 1, & x \in A; \\ 0, & x \not \in A. \end If one adopts the conventions that * for any a \in \mathbb \cup \, a + (+ \infty) = + \infty and a (+\infty) = + \infty, except 0(+\infty)=0; * \frac = + \infty; and * \f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator (mathematics)
In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an ''operator'', but the term is often used in place of ''function'' when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to be explicitly characterized (for example in the case of an integral operator), and may be extended to related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). See Operator (physics) for other examples. The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example \R^n to \R^n. Such operators often preserve properties, such as continuity. For example, differentiation and indef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (linear Algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P : V \to V such that P^2 = P. When V has an inner product and is complete (i.e. when V is a Hilbert space) the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection on a Hilbert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Variation Denoising
In signal processing, particularly image processing, total variation denoising, also known as total variation regularization or total variation filtering, is a noise removal process (filter). It is based on the principle that signals with excessive and possibly spurious detail have high ''total variation'', that is, the integral of the absolute image gradient is high. According to this principle, reducing the total variation of the signal—subject to it being a close match to the original signal—removes unwanted detail whilst preserving important details such as edges. The concept was pioneered by L. I. Rudin, S. Osher, and E. Fatemi in 1992 and so is today known as the ''ROF model''. This noise removal technique has advantages over simple techniques such as linear smoothing or median filtering which reduce noise but at the same time smooth away edges to a greater or lesser degree. By contrast, total variation denoising is remarkably effective edge-preserving filter, i.e., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
