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Affine Scaling
In mathematical optimization, affine scaling is an algorithm for solving linear programming problems. Specifically, it is an interior point method, discovered by Soviet mathematician I. I. Dikin in 1967 and reinvented in the U.S. in the mid-1980s. History Affine scaling has a history of multiple discovery. It was first published by I. I. Dikin at Energy Systems Institute of Russian Academy of Sciences (Siberian Energy Institute, USSR Academy of Sc. at that time) in the 1967 ''Doklady Akademii Nauk SSSR'', followed by a proof of its convergence in 1974. Dikin's work went largely unnoticed until the 1984 discovery of Karmarkar's algorithm, the first practical polynomial time algorithm for linear programming. The importance and complexity of Karmarkar's method prompted mathematicians to search for a simpler version. Several groups then independently came up with a variant of Karmarkar's algorithm. E. R. Barnes at IBM, a team led by R. J. Vanderbei at AT&T, and several others rep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karmarkar
Narendra Krishna Karmarkar (born Circa 1956) is an Indian Mathematician. Karmarkar developed Karmarkar's algorithm. He is listed as an ISI highly cited researcher. He invented one of the first provably polynomial time algorithms for linear programming, which is generally referred to as an interior point method. The algorithm is a cornerstone in the field of Linear Programming. He published his famous result in 1984 while he was working for Bell Laboratories in New Jersey. Biography Karmarkar received his B.Tech in Electrical Engineering from IIT Bombay in 1978, Master of Science, MS from the California Institute of Technology in 1979, and PhD in Computer Science from the University of California, Berkeley in 1983 under the supervision of Richard M. Karp. Karmarkar was a post-doctoral research fellow at IBM research (1983), Member of Technical Staff and fellow at Mathematical Sciences Research Center, AT&T Bell Laboratories (1983-1998), professor of mathematics at M.I.T. (1991), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear 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 function#As a polynomial function, 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 mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rensselaer Polytechnic Institute
Rensselaer Polytechnic Institute () (RPI) is a private research university in Troy, New York, with an additional campus in Hartford, Connecticut. A third campus in Groton, Connecticut closed in 2018. RPI was established in 1824 by Stephen Van Rensselaer and Amos Eaton for the "application of science to the common purposes of life" and is the oldest technological university in the English-speaking world and the Western Hemisphere. Built on a hillside, RPI's campus overlooks the city of Troy and the Hudson River, and is a blend of traditional and modern architecture. The institute operates an on‑campus business incubator and the Rensselaer Technology Park. RPI is organized into six main schools which contain 37 departments, with emphasis on science and technology. It is classified among "R1: Doctoral Universities: Very High Research Activity" and many of its engineering programs are highly ranked. As of 2017, RPI's faculty and alumni included 6 members of the National Inve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MIT OpenCourseWare
MIT OpenCourseWare (MIT OCW) is an initiative of the Massachusetts Institute of Technology (MIT) to publish all of the educational materials from its Post-secondary education, undergraduate- and Quaternary education, graduate-level courses online, Free content, freely and Open access (publishing), openly available to anyone, anywhere. The project was announced on April 4, 2001, and uses Creative Commons License, Creative Commons Attribution-NonCommercial-ShareAlike license. The program was originally funded by the William and Flora Hewlett Foundation, the Andrew W. Mellon Foundation, and MIT. Currently, MIT OpenCourseWare is supported by MIT, corporate underwriting, major gifts, and donations from site visitors. The initiative inspired a number of other institutions to make their course materials available as open educational resources. , over 2,400 courses were available online. While a few of these were limited to chronological reading lists and discussion topics, a majority pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos (journal)
''Chaos: An Interdisciplinary Journal of Nonlinear Science'' is a monthly peer-reviewed scientific journal covering nonlinear systems and describing their manifestations in a manner comprehensible to researchers from a broad spectrum of disciplines. The editor-in-chief is Jürgen Kurths of the Potsdam Institute for Climate Impact Research. Abstracting and indexing The journal is abstracted and indexed in the Science Citation Index and Current Contents/Physical Chemical and Earth Sciences. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... of 3.267. References External links * {{Authority control American Institute of Physics academic journals Quarterly journals Publications established ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slack Variable
In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it into an equality. Introducing a slack variable replaces an inequality constraint with an equality constraint and a non-negativity constraint on the slack variable. Slack variables are used in particular in linear programming. As with the other variables in the augmented constraints, the slack variable cannot take on negative values, as the simplex algorithm requires them to be positive or zero. * If a slack variable associated with a constraint is ''zero'' at a particular candidate solution, the constraint is binding there, as the constraint restricts the possible changes from that point. * If a slack variable is ''positive'' at a particular candidate solution, the constraint is non-binding there, as the constraint does not restrict the possible changes from that point. * If a slack variable is ''negative'' at some point, the point is infeasible (not allowed), as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality. In general, the optimal values of the primal and dual problems need not be equal. Their difference is called the duality gap. For convex optimization problems, the duality gap is zero under a constraint qualification condition. This fact is called strong duality. Dual problem Usually the term "dual problem" refers to the ''Lagrangian dual problem'' but other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while an example of a 3×3 diagonal matrix is \left begin 6 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end\right/math>. An identity matrix of any size, or any multiple of it (a scalar matrix), is a diagonal matrix. A diagonal matrix is sometimes called a scaling matrix, since matrix multiplication with it results in changing scale (size). Its determinant is the product of its diagonal values. Definition As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix with ''n'' columns and ''n'' rows is diagonal if \forall i,j \in \, i \ne j \implies d_ = 0. However, the main diagonal entries are unrestricted. The term ''diagonal matrix'' may s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. Description Gradient descent is based on the observation that if the multi-variable function F(\mathbf) is def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projected Gradient
Projected is an American rock supergroup consisting of Sevendust members John Connolly and Vinnie Hornsby, Alter Bridge and Creed drummer Scott Phillips, and former Submersed and current Tremonti guitarist Eric Friedman. The band released their album, '' Human'', in June 2012, before falling into inactivity as members returned to their respective bands in late 2012. The band released their second studio album, the double album, ''Ignite My Insanity'', in July 2017. They reconvened in early 2020 and finished recording their third album in July 2020. History The band recorded as a four-piece, with each of the members coming from different bands. John Connolly and Vinnie Hornsby had both played together for many years in the alternative metal band Sevendust, as the guitarist and bassist respectively. Eric Friedman had previously been the guitarist for the now-defunct band Submersed, currently playing with Tremonti (formed with Mark Tremonti, Garrett Whitlock, Wolf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
