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Algorithmic Problems On Convex Sets
Many problems in mathematical programming can be formulated as problems on convex sets or convex bodies. Six kinds of problems are particularly important: optimization, violation, validity, separation, membership and emptiness. Each of these problems has a strong (exact) variant, and a weak (approximate) variant. In all problem descriptions, ''K'' denotes a compact and convex set in R''n''. Strong variants The strong variants of the problems are: * Strong optimization problem (SOPT): given a vector ''c'' in R''n'', find a vector ''y'' in ''K'' such that ''c''T''y'' ≥ ''c''T''x'' for all ''x'' in ''K'', or assert that ''K'' is empty. * Strong violation problem (SVIOL): given a vector ''c'' in R''n'' and a number ''t'', decide whether ''c''T''x'' ≤ ''t'' for all ''x'' in ''K'', or find ''y'' in ''K'' such that ''c''T''y'' > ''t''. * Strong validity problem (SVAL): given a vector ''c'' in R''n'' and a number ''t'', decide whether ''c''T''x'' ≤ ''t'' for all ''x'' in ''K''. * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Programming
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems 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 maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Optimization problems can be divided into two categories, depending on whether the variables ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Ellipsoid
In mathematics, the John ellipsoid or Löwner–John ellipsoid associated to a convex body in -dimensional Euclidean space can refer to the -dimensional ellipsoid of maximal volume contained within or the ellipsoid of minimal volume that contains . Often, the minimal volume ellipsoid is called the Löwner ellipsoid, and the maximal volume ellipsoid is called the John ellipsoid (although John worked with the minimal volume ellipsoid in his original paper). One can also refer to the minimal volume circumscribed ellipsoid as the outer Löwner–John ellipsoid, and the maximum volume inscribed ellipsoid as the inner Löwner–John ellipsoid. The German-American mathematician Fritz John proved in 1948 that each convex body in is circumscribed by a unique ellipsoid of minimal volume, and that the dilation of this ellipsoid by factor is contained inside the convex body.John, Fritz. "Extremum problems with inequalities as subsidiary conditions". ''Studies and Essays Presented to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographic Optimization
Lexicographic optimization is a kind of Multi-objective optimization. In general, multi-objective optimization deals with optimization problems with two or more objective functions to be optimized simultaneously. Often, the different objectives can be ranked in order of importance to the decision-maker, so that objective f_1 is the most important, objective f_2 is the next most important, and so on. Lexicographic optimization presumes that the decision-maker prefers even a very small increase in f_1, to even a very large increase in f_2, f_3, f_4, etc. Similarly, the decision-maker prefers even a very small increase in f_2, to even a very large increase in f_3, f_4, etc. In other words, the decision-maker has lexicographic preferences, ranking the possible solutions according to a lexicographic order of their objective function values. Lexicographic optimization is sometimes called preemptive optimization, since a small increase in one objective value preempts a much larger increas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leonid Khachiyan
Leonid Genrikhovich Khachiyan (; ; May 3, 1952April 29, 2005) was a Soviet and American mathematician and computer scientist. He was most famous for his ellipsoid algorithm (1979) for linear programming, which was the first such algorithm known to have a polynomial running time. Even though this algorithm was shown to be impractical, it has inspired other randomized algorithms for convex programming and is considered a significant theoretical breakthrough. Early life and education Khachiyan was born on May 3, 1952, in Leningrad to Armenian parents Genrikh Borisovich Khachiyan, a mathematician and professor of theoretical mechanics, and Zhanna Saakovna Khachiyan, a civil engineer. His grandparents were Karabakh Armenians. He had two brothers: Boris and Yevgeniy (Eugene). His family moved to Moscow in 1961, when he was nine. He received a master's degree from the Moscow Institute of Physics and Technology. In 1978 he earned his Ph.D. in computational mathematics/ theoretical mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Cone
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics. Dual cone In a vector space The dual cone ''C'' of a subset ''C'' in a linear space ''X'' over the reals, e.g. Euclidean space R''n'', with dual space ''X'' is the set :C^* = \left \, where \langle y, x \rangle is the duality pairing between ''X'' and ''X'', i.e. \langle y, x\rangle = y(x). ''C'' is always a convex cone, even if ''C'' is neither convex nor a cone. In a topological vector space If ''X'' is a topological vector space over the real or complex numbers, then the dual cone of a subset ''C'' ⊆ ''X'' is the following set of continuous linear functionals on ''X'': :C^ := \left\, which is the polar of the set -''C''. No matter what ''C'' is, C^ will be a convex cone. If ''C'' ⊆ then C^ = X^. In a Hilbert space (internal dual cone) Alternatively, many authors define the dual cone in the context of a real Hilbert space (such as R''n'' equipped with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Complexity
An ''n''-dimensional polyhedron is a geometric object that generalizes the 3-dimensional polyhedron to an ''n''-dimensional space. It is defined as a set of points in real affine (or Euclidean) space of any dimension ''n'', that has flat sides. It may alternatively be defined as the intersection of finitely many half-spaces. Unlike a 3-dimensional polyhedron, it may be bounded or unbounded. In this terminology, a bounded polyhedron is called a polytope... Analytically, a convex polyhedron is expressed as the solution set for a system of linear inequalities, ''ai''T''x'' ≤ ''bi'', where ''ai'' are vectors in R''n'' and ''bi'' are scalars. This definition of polyhedra is particularly important as it provides a geometric perspective for problems in linear programming. Examples Many traditional polyhedral forms are n-dimensional polyhedra. Other examples include: * A half-space is a polyhedron defined by a single linear inequality, ''a1''T''x'' ≤ ''b1''. * A hyperplane is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneous Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''p''/''q'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''p''/''q'' and ''α'' may not decrease if ''p''/''q'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is lar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex Complexity
An ''n''-dimensional polyhedron is a geometric object that generalizes the 3-dimensional polyhedron to an ''n''-dimensional space. It is defined as a set of points in real affine (or Euclidean) space of any dimension ''n'', that has flat sides. It may alternatively be defined as the intersection of finitely many half-spaces. Unlike a 3-dimensional polyhedron, it may be bounded or unbounded. In this terminology, a bounded polyhedron is called a polytope... Analytically, a convex polyhedron is expressed as the solution set for a system of linear inequalities, ''ai''T''x'' ≤ ''bi'', where ''ai'' are vectors in R''n'' and ''bi'' are scalars. This definition of polyhedra is particularly important as it provides a geometric perspective for problems in linear programming. Examples Many traditional polyhedral forms are n-dimensional polyhedra. Other examples include: * A half-space is a polyhedron defined by a single linear inequality, ''a1''T''x'' ≤ ''b1''. * A hyperplane is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Width
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system, the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in. Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, width, breadth, and depth. ''Height'' is used when there is a base from which vertical measurements can be taken. ''Width'' and ''breadth'' usually refer to a shorter dimension than ''length''. ''Depth'' is used for the measure of a third dimension. Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length squared) and volume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
