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Ellipsoid Method
In mathematical optimization, the ellipsoid method is an iterative method for convex optimization, minimizing convex functions over convex sets. The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is polynomial in the input size. History The ellipsoid method has a long history. As an iterative method, a preliminary version was introduced by Naum Z. Shor. In 1972, an approximation algorithm for real convex optimization, convex minimization was studied by Arkadi Nemirovski and David B. Yudin (Judin). As an algorithm for solving linear programming problems with rational data, the ellipsoid algorithm was studied by Leonid Khachiyan; Khachiyan's achievement was to prove the Polynomial time, polynomial-time ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid 2
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a Surface (mathematics), surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar Cross section (geometry), cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is Bounded set, bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular Rotational symmetry, axes of symmetry which intersect at a Central symmetry, center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal ax ... [...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 by n the number of variables, ''m'' the number of inequality constraints, and L the number of bits of input to the algorithm, Karmarkar's algorithm requires O(m^ n^ L) operations on O(L)-digit numbers, as compared to O(n^3(n+m) L) such operations for the ellipsoid algorithm. In "square" problems, when ''m'' is in O(''n''), Karmarkar's algorithm requires O(n^ L) operations on O(L)-digit numbers, as compared to O(n^4 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: t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real RAM
In computing, especially computational geometry, a real RAM (random-access machine) is a mathematical model of a computer that can compute with exact real numbers instead of the binary fixed-point or floating-point numbers used by most actual computers. The real RAM was formulated by Michael Ian Shamos in his 1978 Ph.D. dissertation. Model The "RAM" part of the real RAM model name stands for "random-access machine". This is a model of computing that resembles a simplified version of a standard computer architecture. It consists of a stored program, a computer memory unit consisting of an array of cells, and a central processing unit with a bounded number of registers. Each memory cell or register can store a real number. Under the control of the program, the real RAM can transfer real numbers between memory and registers, and perform arithmetic operations on the values stored in the registers. The allowed operations typically include addition, subtraction, multiplication, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid 6
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal axes'', or simply axes of the ellipsoid. If the three axes have different lengths, the figure is a triaxial ellipsoid (r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid 1
An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a Surface (mathematics), surface that may be defined as the zero set of a polynomial of degree two in three variables. Among quadric surfaces, an ellipsoid is characterized by either of the two following properties. Every planar Cross section (geometry), cross section is either an ellipse, or is empty, or is reduced to a single point (this explains the name, meaning "ellipse-like"). It is Bounded set, bounded, which means that it may be enclosed in a sufficiently large sphere. An ellipsoid has three pairwise perpendicular Rotational symmetry, axes of symmetry which intersect at a Central symmetry, center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the ''principal ax ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Duality
In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0. This means that for any minimization problem, called the ''primal problem'', the solution to the primal problem is always greater than or equal to the solution to the dual maximization problem. Alternatively, the solution to a primal maximization problem is always less than or equal to the solution to the dual minimization problem. So, in short: weak duality states that any solution feasible for the dual problem is an upper bound to the solution of the primal problem. Weak duality is in contrast to strong duality, which states that the primal optimal objective and the dual optimal objective are ''equal''. Strong duality only holds in certain cases. Uses Many primal-dual approximation algorithms are based on the principle of weak duality.. Weak duality theorem Consider a linear programming problem, whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Oracle
A separation oracle (also called a cutting-plane oracle) is a concept in the mathematical theory of convex optimization. It is a method to describe a convex set that is given as an input to an optimization algorithm. Separation oracles are used as input to ellipsoid methods. Definition Let ''K'' be a convex and compact set in R''n''. A strong separation oracle for ''K'' is an oracle (black box) that, given a vector ''y'' in R''n'', returns one of the following: *Assert that ''y'' is in ''K''. * Find a hyperplane that separates ''y'' from ''K'': a vector a in R''n'', such that a\cdot y > a\cdot x for all ''x'' in ''K''. A strong separation oracle is completely accurate, and thus may be hard to construct. For practical reasons, a weaker version is considered, which allows for small errors in the boundary of ''K'' and the inequalities. Given a small error tolerance ''d''>0, we say that: * A vector ''y'' is ''d-near K'' if its Euclidean distance from ''K'' is at most ''d''; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Atallah
Mikhail Jibrayil (Mike) Atallah is a Lebanese American computer scientist, a distinguished professor of computer science at Purdue University. Biography Atallah received his bachelor's degree from the American University of Beirut in 1975. He then moved to Johns Hopkins University for his graduate studies, earning a master's degree in 1980 and a Ph.D. in 1982 under the supervision of S. Rao Kosaraju. Since that time he has been a member of the Purdue University faculty.Department faculty profile Purdue University, retrieved 2011-09-29. In 2001, Atallah co-founded Arxan Technologies, Inc., a provider of internet anti-piracy and anti-tampering software, and in 2007, he became its chief technology officer. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lovász
Lovász (): * Chris Lovasz (born 1980), Canadian-British content creator * Gyöngyi Lovász (born 1959), Hungarian retired footballer * Irén Lovász (born 1961), Hungarian folk singer and ethnographer * Lázár Lovász (1942–2023), Hungarian athlete who competed in hammer throw * László Lovász (born 1948), Hungarian mathematician best known for his work in combinatorics **Lovász conjecture (1970) ** Erdős–Faber–Lovász conjecture (1972) ** The Lovász local lemma (proved in 1975, by László Lovász & P. Erdős) ** The Lenstra–Lenstra–Lovász lattice basis reduction (algorithm) (LLL) ** Algorithmic Lovász local lemma (proved in 2009, by Robin Moser and Gábor Tardos) ** Lovász number (1979) * Zsuzsanna Lovász Zsuzsanna Lovász (formerly Zsuzsanna Pavlikné Lovász; born 17 December 1976) is a retired Hungarian handball player. She participated at the 2004 Summer Olympics, where she placed fifth with the Hungarian national team. Personal life Her ... (b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
