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Robert J. Vanderbei
Robert J. Vanderbei (born 1955) is an American mathematician and Professor in the Department of Operations Research and Financial Engineering at Princeton University. Biography Robert J. Vanderbei was born in Grand Rapids, MI, in 1955. He received his BS in Chemistry in 1976 and an MS in Operations Research and Statistics in 1978 from Rensselaer Polytechnic Institute and his PhD in Applied Mathematics from Cornell University in 1981. In his thesis, he developed probabilistic potential theory for random fields consisting of tensor products of Wiener process, Brownian motions. He was postdoctoral research fellow at New York University's Courant Institute of Mathematical Sciences and then at the Mathematics Department at the University of Illinois Urbana-Champaign. In 1984, he left academia and joined Bell Labs, where he served as a team member of AT&T's Advanced Decision Support Systems venture. In 1990, Vanderbei returned to academia to teach at Princeton University. He is cur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karmarkar's Algorithm
Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems. It was the first reasonably efficient algorithm that solves these problems in polynomial time. The ellipsoid method is also polynomial time but proved to be inefficient in practice. Denoting n as the number of variables and L as the number of bits of input to the algorithm, Karmarkar's algorithm requires O(n^ L) operations on O(L) digit numbers, as compared to O(n^6 L) such operations for the ellipsoid algorithm. The runtime of Karmarkar's algorithm is thus :O(n^ L^2 \cdot \log L \cdot \log \log L) using FFT-based multiplication (see Big O notation). Karmarkar's algorithm falls within the class of interior point methods: the current guess for the solution does not follow the boundary of the feasible set as in the simplex method, but it moves through the interior of the feasible region, improving the approximation of the optimal solution by a definite fraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Programming
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard. Convex optimization has applications in a wide range of disciplines, such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, finance, statistics ( optimal experimental design), and structural optimization, where the approximation concept has proven to be efficient. With recent advancements in computing and optimization algorithms, convex programming is nearly as straightforward as linear programming. Definition A convex optimization problem is an optimization problem in which the objective function is a convex function and the feasible s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming. "Programming" in this context refers to a formal procedure for solving mathematical problems. This usage dates to the 1940s and is not specifically tied to the more recent notion of "computer programming." To avoid confusion, some practitioners prefer the term "optimization" — e.g., "quadratic optimization." Problem formulation The quadratic programming problem with variables and constraints can be formulated as follows. Given: * a real-valued, -dimensional vector , * an -dimensional real symmetric matrix , * an -dimensional real matrix , and * an -dimensional real vector , the objective of quadratic programming is to find an -dimensional vector , that wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semidefinite Programming
Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron. Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practical problems in operations research and combinatorial optimization can be modeled or approximated as semidefinite programming problems. In automatic control theory, SDPs are used in the context of linear matrix inequalities. SDPs are in fact a special case of cone programming and can be efficiently solved by interior point methods. All linear programs and (convex) quadratic programs can be expressed as SDPs, and via hierarchies of SDPs the solutions of polynomial optimization problems can be approximated. Semidefinite programming has been use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Point Method
Interior-point methods (also referred to as barrier methods or IPMs) are a certain class of algorithms that solve linear and nonlinear convex optimization problems. An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967 and reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, which runs in provably polynomial time and is also very efficient in practice. It enabled solutions of linear programming problems that were beyond the capabilities of the simplex method. Contrary to the simplex method, it reaches a best solution by traversing the interior of the feasible region. The method can be generalized to convex programming based on a self-concordant barrier function used to encode the convex set. Any convex optimization problem can be transformed into minimizing (or maximizing) a linear function over a convex set by converting to the epigraph form. The idea of encodi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Murray Hill, NJ
Murray Hill is an unincorporated community located within portions of both Berkeley Heights and New Providence, located in Union County in northern New Jersey, United States. It is the longtime central location of Bell Labs (part of Nokia since 2016), having moved there in 1941 from New York City when the division was still part of Western Electric. The first working transistor was demonstrated in Bell Labs' Murray Hill facility in 1947. The neighborhood shares its ZIP code 07974 with the neighboring borough of New Providence. Murray Hill was named and founded by Carl H. Schultz, founder of a mineral water business once located at First Avenue between 25th and 26th Streets in the Murray Hill district of Manhattan. Schultz purchased a large tract of land there during the 1880s where he built a residence for his family and donated land to be used for a train station with the condition that the area be known as "Murray Hill". Corporate residents * C. R. Bard, a manufacturer o ... [...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] |
Eugene Lawler
Eugene Leighton (Gene) Lawler (1933 – September 2, 1994) was an American computer scientist and a professor of computer science at the University of California, Berkeley... Reprinted in . Academic life Lawler came to Harvard as a graduate student in 1954, after a three-year undergraduate B.S. program in mathematics at Florida State University. He received a master's degree in 1957, and took a hiatus in his studies, during which he briefly went to law school and worked in the U.S. Army, at a grinding wheel company,. and as an electrical engineer at Sylvania from 1959 to 1961.Editorial staff (1995) ''In Memoriam: Eugene L. Lawler'', SIAM Journal on Computing 24(1), 1-2. He returned to Harvard in 1958, and completed his Ph.D. in applied mathematics in 1962 under the supervision of Anthony G. Oettinger with a dissertation entitled ''Some Aspects of Discrete Mathematical Programming''.. He then became a faculty member at the University of Michigan until 1971, when he moved to Be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, but pure mathematicians are not primarily motivated by such applications. Instead, the appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles. While pure mathematics has existed as an activity since at least Ancient Greece, the concept was elaborated upon around the year 1900, after the introduction of theories with counter-intuitive properties (such as non-Euclidean geometries and Cantor's theory of infinite sets), and the discovery of apparent paradoxes (such as continuous functions that are nowhere differentiable, and Russell's paradox). This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly, with a system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
US Patent
Under United States law, a patent is a right granted to the inventor of a (1) process, machine, article of manufacture, or composition of matter, (2) that is new, useful, and non-obvious. A patent is the right to exclude others, for a limited time (usually, 20 years) from profiting of a patented technology without the consent of the patent-holder. Specifically, it is the right to exclude others from: making, using, selling, offering for sale, importing, inducing others to infringe, applying for an FDA approval, and/or offering a product specially adapted for practice of the patent. United States patent law is codified in Title 35 of the United States Code, and authorized by the U.S. Constitution, in Article One, section 8, clause 8, which states: Patent law is designed to encourage inventors to disclose their new technology to the world by offering the incentive of a limited-time monopoly on the technology. For U.S. utility patents, this limited-time term of patent i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondegeneracy
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definition when ''V'' is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero ''x'' in ''V'' such that :f(x,y)=0\, for all \,y \in V. Nondegenerate forms A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if :f(x,y)=0 for all y \in V implies that x = 0. The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map V \to V^* be an isomorphism, not positivity. For example, a manifold with an inner product structure on its ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
