In economics , convexpreferences refer to a property of an individual s ordering of various outcomes which roughly corresponds to the idea that averages are better than the extremes . The concept roughly corresponds to the concept of marginal utility Diminishing marginal utility diminishing marginal utility but uses modern theory to represent the concept without requiring the use of utility function s. Comparable to the greater than or equal to Order theory Partially ordered sets ordering relation math geq math for real numbers, the notation math succeq math below can be translated as is at least as good as in Preference economics preference satisfaction . Use x , y , and z to denote three consumption bundles combinations of various quantities of various goods . Formally, a preference relation P on the consumption set X is Convex set convex if for any math x, y, z in X math where math y succeq x math and math z succeq x math , it is the case that math theta y 1 theta z succeq x math for any math theta in 0,1 math . That is, the preference ordering P is convex if for any two goods bundles that are each viewed as being at least as good as a third bundle, a weighted average of the two ... amount of each bundle is viewed as being better than the third bundle. A set of Convex function convex shaped indifference curve s displays convexpreferences Given a convex indifference curve ... is a convex set . Convexpreferences with their associated convex indifference mapping arise from Quasi convex function quasi concave utility functions, although these are not necessary for the analysis of preferences. References Hal R. Varian Intermediate Microeconomics A Modern Approach . New ... convex if for any math x, y, z in X math where math y succeq x math , math z succeq x math , and math ... ordering P is strictly convex if for any two distinct goods bundles that are each viewed ... 9 See also Convex function Level set Quasi convex function Semi continuous function Shapley Folkman ... more details
Image Convex polygon illustration1.png right thumb A convex set. wiktionary convex convexity The word convex means curving out or bulging outward , as opposed to Concave disambiguation concave . Convex or convexity may refer to tocright Mathematics Convex set , a set of points containing all line segments between each pair of its points Convex function , a function with the Epigraph forming a convex set Convex polytope , a polytope which forms a convex set. These include convex polygon s. Convex hull , the minimal convex set containing a set of points X Convex combination , a linear combination of points with non negative coefficients that sum up to 1 Convex conjugate , a generalization of the Legendre transformation Convex bipartite graph , a special kind of bipartite graph Convex polygon , a polygon which is not concave Convex plane graph , a plane graph with convex faces Convex optimization , a subfield of optimization, studies the problem of minimizing convex functions Convex geometry , the branch of geometry studying convex sets, mainly in Euclidean space Convex curve , the boundary of a convex set Convex body , a compact convex set with non empty interior Economics Convexity in economics Convexpreferences , a preference relation with convex upper contour sets Non convexity economics , refers to violations of the convexity assumptions of elementary economics Finance Bond convexity , a measure of the sensitivity of the price of a bond to changes in interest rates Convexity finance , second derivatives in financial modeling generally Optics Convex lens , a lens with surfaces that curve outward Art Convex and Concave , a 1955 lithograph print by the artist M. C. Escher Proper names Convex Computer , a company that produced a number of vector supercomputers, bought by HP in 1995 Convex Software Library , a client side open source solution for Internet Explorer which uses a hidden Java applet to process XForms Convex application, an iPhone iPad software created by Ergonotics ... more details
In economics, an agent s preferences are said to be weakly monotonic if, given a consumption bundle math x math , the agent prefers all consumption bundles math y math that have more of every good. That is, math y gg x math implies math y succ x math . An agent s preferences are said to be strongly monotonic if, given a consumption bundle math x math , the agent prefers all consumption bundles math y math that have more of at least one good. That is, math y geq x math and math y neq x math imply math y succ x math . This definition defines monotonic increasing preferences. Monotonic decreasing preferences can often be defined to be compatible with this definition. For instance, an agent s preferences for pollution may be monotonic decreasing less pollution is better . In this case, the agent s preferences for lack of pollution are monotonic increasing. Much of consumer theory relies on a weaker assumption, local nonsatiation . An example of preferences which are weakly monotonic but not strongly monotonic are those represented by a Leontief Utilities Leontief utility function . References Andreu Mas Colell Mas Colell, Andreu , Whinston, Michael D., Green, Jerry R. Microeconomic Theory. Oxford University Press. 1995. See also Monotonic function Monotonicity in calculus and analysis Strict Category Microeconomics Category Consumer theory microeconomics stub ru uk ... more details
refimprove date December 2011 In economics, a consumer is said to have homothetic preferences when its preferences can be represented by a homothetic utility function. ref cite book last Varian first Hal title Microeconomic Analysis year 1992 pages 147 ref Both homogenous function homogenous and homothetic preferences are common functional forms used to represent consumer preferences when analyzing the demand for Consumption economics consumption of various goods and services . A homothetic function is a monotonic transformation of a function which is homogenous function homogenous of degree 1. However, since ordinal utility functions are only defined up to a monotonic transformation , there is little distinction between the two concepts in practice. In a model where competitive consumers optimize homothetic utility functions subject to a budget constraint, the ratios of goods demanded by consumers will depend only on relative prices, not on income or scale. Intratemporally vs. Intertemporally homothetic preferences Assuming homothetic individual preferences over goods consumed in the same time period means that consumers with different incomes but facing the same prices will demand goods in the same proportions. Models of modern macroeconomics and public finance often assume the constant relative risk aversion form for within period utility also called the power utility power or isoelastic form . The reason is that, in combination with additivity over time, this gives homothetic intertemporal preferences and this homotheticity is of considerable analytic convenience for example, it allows for the analysis of steady states in growth models . These assumptions imply that the elasticity of intertemporal substitution , and its inverse, risk aversion fluctuation risk aversion , are constant rich and poor decision makers are equally averse to proportional fluctuations in consumption ... model with heterogeneous agents. reflist DEFAULTSORT Homothetic Preferences Category Utility Category ... more details
Social preferences are a type of preference studied in behavioral economics behavioral and experimental economics experimental economics and social psychology , including interpersonal altruism , Fair division fairness , reciprocity , and inequity aversion . The term social preferences incorporates obstreperous esp. the Fehr Schmidt inequity aversion model and non obstreperous e.g., vulnerability based theories. Much of the recent evidence used to test society ideas and models has come from economics experiments. However, social preferences also matter outside the laboratory. ref Gary Becker, The Economics of Discrimination ref ref Erzo F.P. Luttmer, Neighbors as Negatives Relative Earnings and Well Being, Quarterly Journal of Economics ref References references Category Social psychology Category Social sciences socio stub econ stub ... more details
Lexicographic preferences lexicographical order based on the order of amount of each good describe comparative preferences where an agent economics economic agent infinitely prefers one good X to another Y . Thus if offered several bundles of goods, the agent will choose the bundle that offers the most X, no matter how much Y there is. Only when there is a tie of Xs between bundles will the agent start comparing Ys. For example, if for a given bundle X Y Z an agent orders his preferences according to the rule X Y Z, then the bundles 5 3 3 , 5 1 6 , 3,5,3 would be ordered, from most to least preferred 5 3 3 5 1 6 3 5 3 Even though the first option contains fewer total goods than the second option, it is preferred because it has more Y. Note that the number of X s is the same, and so the agent is comparing Y s. Even though the third option has the same total goods as the first option, the first option is still preferred. Even though the third option has far more Y than the second option, the second option is still preferred because it has slightly more X. Implications If all agents have the same lexicographic preferences, then general equilibrium cannot exist because agents won t sell to each other as long as price of the less preferred is more than 0 number zero . But if the price of the less wanted is zero, then all agents want an infinite amount of the good. Equilibrium cannot be attained. Lexicographic preferences can still exist with general equilibrium. For example, Different people have different bundles of lexicographic preferences such that different individuals value items in different orders. Some, but not all people have lexicographic preferences. Lexicographic preferences extend only to a certain quantity of the good. Lexicographic preferences are the classical example of rational preferences that are not representable by a Utility Utility functions utility function , if amounts can be any non negative real value. If there were such a function U then, e.g. ... more details
Unreferenced date June 2008 Endogenous preferences are preference s that cannot be taken as given, but are affected by individual internal responses to the external state of affairs. They are interdependent, in part determined by social institutions, marketed advertisement, and subject to learning experience and observation and habit formation past experience . See also Acquired taste economics stub Category Economics ... more details
noreferences date November 2007 Infobox Software name System Preferences logo Image System Preferences icon.png 64px screenshot File SystemPreferences.png 250px caption System Preferences application in Mac OS X Lion . developer Apple Inc. latest release version 11.0 latest release date July 20, 2011 operating system Mac OS X genre Computer configuration Settings license Proprietary software Proprietary Deleted image removed Image Preference Pane 10.4.png 90px thumb right Preference Pane from Mac OS X 10.4. Commented out because image was deleted Image Screen Saver 10.4.png 90px thumb right Screen Saver from Mac OS X 10.4. System Preferences is an Application software application included with the Mac OS X operating system that allows users to modify various system settings which are divided into separate preference pane s. The System Preferences application was introduced in the first version of Mac OS X to replace the control panel Mac OS control panel that was included in previous versions ... in System Preferences, were separate applications that were accessed through the Apple menu . Mac ... but subsections of the System Preferences application. By default, System Preferences organizes preference panes into several categories. In the latest version of System Preferences, included with Mac ... to sort preference panes alphabetically. Originally, System Preferences included a customizable toolbar ... OS X v10.3 , a corresponding preference pane was added to System Preferences. This was replaced ... and processor usage. Mission Control Mac OS X Mission Control changes the preferences for the Mission ... settings MobileMe used to set preferences for the user s MobileMe account and iDisk. Mouse computing Mouse set mouse preferences. If using a Magic Mouse , provides preferences for the multitouch gestures ... speech settings. Spotlight software Spotlight set the preferences for the Spotlight system wide search ... Preferences ... more details
In the Color psychology psychology of color , color preferences are the tendency for an individual or a group to prefer some color s over others, including a favorite color . Introduction In general, people have a connection with certain colors due to their experiences with objects of those colors. Children with favorite purple stuffed animals will generally prefer the color purple into adulthood. This works in a negative manner as well. In a study with Berkley students, they found that students with school spirit s favorite colors were blue and gold their school s colors . They also found that they did not like the colors red and white, which are the colors of their Stanford rivals. ref cite web last Sohn first Emily title Color Preferences Determined by Experience url http news.discovery.com human colors preferences evolution style.html publisher Discovery News accessdate 2 October 2011 ref Children s color preferences The age when infants begin showing a preference for color is at about 12 weeks old. Generally, children prefer the colors red and blue, and cool colors are preferred ... 3 5 years of age is an indicator of their developmental stage. Color preferences tend to change as people ..., D. 2009 . Young Children s Color Preferences in the Interior Environment. Early Childhood Education Journal, 36 6 , 491 496. doi 10.1007 s10643 009 0311 6 ref Color preferences in different societies ... learning is the process where an individual develops color preferences. In different countries, color ... year 1999 first W. Ray title The meanings of colour preferences among hues journal Pigment & Resin ... date December 2001 first1 Lee first2 Christopher title Color preferences according to gender and sexual ... Managing Images in Different Cultures A Cross National Study of Color Meanings and Preferences journal ... 2004 first1 Davida first2 Andrea first3 Kevin title Infants spontaneous color preferences are not due ... preferences journal Vision Research volume 47 issue 10 pages 1368 1381 doi 10.1016 j.visres.2006.09.024 ... more details
Legacy preferences or legacy admission is a type of preference given by educational institutions to certain applicants on the basis of their familial relationship to alumni of that institution. Students so admitted are referred to as legacies or legacy students . This preference is most common in American universities and colleges ref citation author Daniel Golden chapterurl http www.tcf.org publications education Legacy ch4.pdf title Affirmative Action for the Rich chapter Chapter 4 An Analytic Survey of Legacy Preference year 2010 ISBN 978 0870785184 ref and emerged after World War I, primarily in response to the resulting immigrant influx ref cite book title Color and Money How Rich White Kids Are Winning the War over College Affirmative Action author Peter G. Schmidt isbn 978 1403976017 year 2007 ref . The Ivy League institutions are estimated to admit 10 to 30 of each entering class using this factor. ref cite news url http www.economist.com displaystory.cfm?story id 2333345 work The Economist title The curse of nepotism date January 8, 2004 ref ref name test http www.thecrimson.com ... preferences in comparison to other programmes At some schools, legacy preferences have an effect ... title The Opportunity Cost of Admission Preferences at Elite Universities author Thomas J. Espenshade ... that legacy preferences are a way to indirectly sell university placement. Opponents accuse these programs ... Analysis of the Impact of Legacy Preferences on Alumni Giving at Top Universities year 2010 ISBN 978 ... or opposing both affirmative action and legacy preferences simultaneously. For example, the conservative ... of all non academic preferences also point out that many European universities, including highly ... , do not use any racial, legacy, or athletic preferences in admissions decisions. ref cite news title ... legacy preferences in government schools, which argues that they violate the Nobility Clause of the constitution ... Preferences in Public School Admissions , Washington University Law Review , Volume 84, page 1375 ... more details
EDIT BELOW THIS LINE Mate preferences in humans refers to why one human chooses or chooses not to mate with another human and their reasoning why Evolutionary Psychology Mating see Evolutionary Psychology, mating . Men and women have been observed having different criteria as what makes a good or ideal mate gender differences . A potential mate s socioeconomic status has also been seen as having a noticeable effect, especially in developing areas where social status is more emphasized. ref Stone, Shackelford, & Buss 2008 Socioeconomic Development and Shifts in Mate Preferences. Evolutionary Psychology Vol. 6 ref Gender Differences Several studies have shown that females are more selective of their mates than males are, with men requiring less time to consent to sexual behavior than women, as well as having a greater desire for short term relationships and wanting a higher number of sexual partners in their lifetime ref Schmitt, Shackelford, & Buss 2001 Are men really more oriented toward short term mating than women? A critical review of theory and research. Psychology, Evolution and Gender, Vol 3 . ref The difference between short term and long term relationships can change how an individual pursues a mate. For instance, when searching for a long term mate, women often tend to place a heavier emphasis on resources and if their mate can provide well enough for her and potential family. However in short term situations, a potential mate s physical condition is weighed more, because it is a good indicator if they have desirable heritable genes ref Li, Valentine, & Patel 2010 Mate preferences in the US and Singapore A cross cultural test of the mate preference priority model. Personality and Individual Differences. Vol 50 ref ref Kenrick, Groth, Trost, & Sadalla 1993 Integrating evolutionary and social exchange perspectives on relationships Effects of gender, self appraisal ... mate preferences Personality and Individual Differences 39 ref four dimensions were found that seem ... more details
In mathematics , a convex graph may be a convex bipartite graph a convex plane graph the graph of a function graph of a convex function disambig ... more details
wiktionarypar plano convex Plano convex may refer to Plano convex lenses, in optics see Lens optics Types of simple lenses The plano convex type of mudbrick , used by the ancient Sumerians disambig ... more details
MergeTo Convex set date December 2011 In mathematics , a convex curve is the boundary topology boundary of a convex set . See also Secant line Category Convex geometry fr Courbe convexe ... more details
Technical date June 2011 Image Extreme points illustration.png thumb right The convex hull of the red set is the blue convex set . See also Convex set Convex combination In mathematics , the convex hull or convex envelope for a set X of points in the Euclidean plane or Euclidean space is the minimal convex set containing X . For instance, when X is a bounded subset of the plane, the convex hull may ... de Berg van Kreveld Overmars Schwarzkopf 2000 , p. 3. ref Formally, the convex hull may be defined as the intersection of all convex sets containing X , the intersection of all halfspace s containing X , or the set of all convex combination s of points in X . With the latter definition, convex ... further, to oriented matroid s. sfnp Knuth 1992 The algorithm ic problem of finding the convex ... problems of computational geometry . Definitions The convex hull of a given set X may be defined as The unique minimal convex set containing X The intersection of all convex sets containing X The intersection of all halfspaces containing X The set of all convex combination s of points in X It is not obvious from the first definition that there is a unique minimal convex set containing X , for every X however, the intersection of all convex sets containing X is well defined, and it is easy to see that it is a subset of every other convex set containing X , so it is exactly the unique minimal convex set containing X . Each convex set containing X must by the assumption that it is convex contain all convex combinations of points in X , so the set of all convex combinations is contained in the intersection of all convex sets containing X . Conversely, the set of all convex combinations is itself a convex set containing X , so it also contains the intersection of all convex sets ... convex hull Carath odory s theorem , if X is a subset of an N dimensional vector space, convex combinations ... to saying that the convex hull of X is the union of all simplex simplices with at most N 1 vertices ... more details
Unreferenced date December 2009 Image Convex combination illustration.svg right thumb Given three points math x 1, x 2, x 3 math in a plane as shown in the figure, the point math P math is a convex combination of the three points, while math Q math is not. br math Q math is however an affine combination of the three points, as their affine hull is the entire plane. In convex geometry , a convex combination is a linear combination of point geometry points which can be vector geometric vector s, scalar ... , math in a real vector space , a convex combination of these points is a point of the form math alpha ... i ge 0 math and math alpha 1 alpha 2 cdots alpha n 1. math As a particular example, every convex combination of two points lies on the line segment between the points. All convex combinations are within the convex hull of the given points. In fact, the collection of all such convex combinations of points in the set constitutes the convex hull of the set. There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval math 0,1 math is convex but generates the real number line under linear combinations. Another example is the convex set of probability distribution s, as linear combinations preserve neither nonnegativity nor affinity i.e., having total integral one . Other objects Similarly, a convex combination ... combination Affine, conical, and convex combinations A conical combination is a linear combination with nonnegative coefficients Weighted mean s are functionally the same as convex combinations, but they use ... s are like convex combinations, but the coefficients are not required to be non negative. Hence ... hull Carath odory s theorem convex hull Convex hull Simplex Barycentric coordinate system mathematics Barycentric coordinate system DEFAULTSORT Convex Combination Category Convex geometry Category Mathematical analysis Category Convex hulls de Linearkombination Spezialf lle es Combinaci n convexa ... more details
Convex geometry is the branch of geometry studying convex set s, mainly in Euclidean space . Convex sets occur naturally in many areas of mathematics computational geometry , convex analysis , discrete ... branches of the mathematical discipline Convex and Discrete Geometry are General Convexity , Polytopes ... list axiomatic and generalized convexity convex sets without dimension restrictions convex sets in topological vector spaces convex sets in 2 dimensions including convex curves convex sets in 3 dimensions including convex surfaces convex sets in n dimensions including convex hypersurfaces finite dimensional Banach spaces random convex sets and integral geometry asymptotic theory of convex bodies approximation by convex sets variants of convex sets star shaped, m, n convex, etc. Helly ..., volume mixed volume s and related topics inequalities and extremum problems convex functions and convex programs spherical and hyperbolic convexity The phrase convex geometry is also used in combinatorics as the name for an abstract model of convex sets based on antimatroid s. Historical note Convex ... to convex geometry date back to antiquity and can be traced in the works of Euclid and Archimedes , it became ... Fenchel W. Fenchel gave a comprehensive survey of convex geometry in Euclidean space R sup n sup . Further development of convex geometry in the 20th century and its relations to numerous mathematical disciplines are summarized in the Handbook of convex geometry edited by P. M. Gruber and J. M. Wills. See also List of convexity topics References Expository articles on convex geometry K. Ball, An elementary introduction to modern convex geometry, in Flavors of Geometry, pp. 1 58, Math. Sci ... , 47 83. V. Klee, What is a convex set? Amer. Math. Monthly, Vol. 78 1971 , 616 631. Some books on convex .... English translation Theory of convex bodies, BCS Associates, Moscow, ID, 1987. R. J. Gardner, Geometric .... M. Gruber , Convex and discrete geometry, Springer Verlag, New York, 2007. P. M. Gruber, J. M. Wills ... more details
Image ConvexFunction.svg thumb 300px right Convex function on an interval. Image Epigraph convex.svg right thumb 300px A function in black is convex if and only if the region above its Graph of a function graph in green is a convex set . Merge from Proper convex function discuss Talk Convex function ... on an interval mathematics interval is called convex or convex downward or concave upward if the graph ..., a function is convex if its epigraph mathematics epigraph the set of points on or above the graph of the function is a convex set . More generally, this definition of convex functions makes sense for functions defined on a convex subset of any vector space . Convex functions play an important role ... where they are distinguished by a number of convenient properties. For instance, a strictly convex ... additional hypotheses, convex functions continue to satisfy such properties and, as a result, they are the most ... , a convex function applied to the expected value of a random variable is always less or equal to the expected value of the convex function of the random variable. This result, known as Jensen s inequality ... defined on a convex set X in a vector space is called convex if, for any two points math x 1 math and math ... The function is called strictly convex if math f tx 1 1 t x 2 t f x 1 1 t f x 2 , math for every math ... function concave if &minus f is strictly convex. Properties Suppose f is a function of one real ... R is symmetric in math x 1,x 2 math . f is convex if and only if math R x 1,x 2 math is monotonically ... of convexity is quite useful to prove the following results. A convex function f defined on some ... in the examples section . A function is midpoint convex on an interval C if math f left frac x 1 ... that is midpoint convex will be convex. ref Sierpinski Theorem, Donoghue 1969 , http books.google.com ... convex will be convex. A differentiable function of one variable is convex on an interval if and only ... and convex then it is also continuously differentiable . A continuously differentiable function ... more details
Image Convex polygon illustration1.png right thumb alt Illustration of a convex set, which looks somewhat like a disk A green convex set contains the black line segment joining the points x and y. The entire line segment lies in the interior of the convex set A convex set. Image Convex polygon illustration2.png right thumb alt Illustration of a non convex set, which looks somewhat like a boomerang or wedge. A green non convexconvex set contains the black line segment joining the points x and y. Part of the line segment lies outside of the green non convex set. A non convex set, with a line segment outside the set. In Euclidean space , an object is convex if for every pair of points within the object ..., a solid cube geometry cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex. The notion can be generalized to other spaces as described below. In vector spaces Image Convex supergraph.png right thumb A convex function function is convex ... graph in blue , is a convex set. Let S be a vector space over the real number s, or, more generally, some ordered field . This includes Euclidean spaces. A set mathematics set C in S is said to be convex ... that a convex set in a real number real or complex number complex topological vector space is path connected , thus connected space connected . A set C is called absolutely convex if it is convex and balanced set balanced . The convex subset s of R the set of real numbers are simply the intervals of R . Some examples of convex subsets of the Euclidean space Euclidean plane are solid regular polygon s, solid triangles, and intersections of solid triangles. Some examples of convex subsets ... s. The Kepler Poinsot polyhedra are examples of non convex sets. Properties If math S math is a convex ... is known as a convex combination of math u 1,u 2, ldots,u r math . Intersections and unions The collection of convex subsets of a vector space has the following properties ref name Soltan Soltan ... more details
Unreferenced date December 2009 In linear algebra , a convex cone is a subset of a vector space over .... Image Convex cone illust.svg right thumb A convex cone light blue . Inside of it, the light red convex cone consists of all points x y with > 0 and > 0, for the depicted x and y . The curves ... of a vector space V is a convex cone if x y belongs to C , for any positive scalars , , and any ... subspace null vector vector space 0 are convex cones by this definition. Other examples are the set ... vectors x such that is a positive scalar and x is an element of some convex set convex subset X ... not contain 0 , this construction gives an open resp. closed convex circular cone . The intersection of two convex cones in the same vector space is again a convex cone, but their union may fail to be one. The class of convex cones is also closed under arbitrary linear map s. In particular, if C is a convex cone, so is its opposite C and C math cap math C is the largest linear subspace contained in C . Convex cones are linear cones If C is a convex cone, then for any positive scalar and any x in C the vector x 2 x 2 x is in C . It follows that a convex cone C is a special case of a cone linear algebra linear cone . Alternative definitions It follows from the above property that a convex cone can also be defined as a linear cone that is closed under convex combination s, or just under addition s. More succinctly, a set C is a convex cone if and only if C C and C C C , for any ... of convex cone by non negative scalars , , not both zero . Blunt and pointed cones According to the above definition, if C is a convex cone, then C math cup math 0 is a convex cone, too. A convex cone is said to be pointed or blunt depending on whether it includes the null vector 0 or not. Blunt cones can be excluded from the definition of convex cone by substituting non negative ... are convex cones. Moreover, any convex cone C that is not the whole space V must be contained in some ... more details
Convex analysis is the branch of mathematics devoted to the study of properties of convex function s and convex set s, often with applications in convex optimization convex minimization , a subdomain of optimization mathematics optimization theory . Convex sets main Convex set A convex set is a set math ... Rockafellar, R. Tyrrell title Convex Analysis publisher Princeton University Press location Princeton, NJ year 1997 origyear 1970 isbn 9780691015866 ref Convex functions main Convex function A convex ..., a convex function is any extended real valued function such that its epigraph mathematics epigraph math left x,r in X times mathbb R f x leq r right math is a convex set. ref name Rockafellar Convex conjugate main Convex conjugate The convex conjugate of an extended real valued not necessarily convex function math f X to mathbb R cup pm infty math is math f X to mathbb R cup pm infty math ... from the Fenchel Young inequality . For Proper convex function proper functions , math f f math if and only if math f math is convex and lower semi continuous by Fenchel Moreau theorem . ref name ... Adrian title Convex Analysis and Nonlinear Optimization Theory and Examples edition 2 year 2006 publisher Springer isbn 9780387295701 pages 76 77 ref Convex minimization main Convex optimization A convex ... R cup pm infty math is a convex function and math M subseteq X math is a convex set. Dual problem ... separated locally convex space s math left X,X right math and math left Y,Y right math . Then given ... function convex analysis indicator function . Then let math F X times Y to mathbb R ... function is given by math sup y in Y F 0,y math where math F math is the convex conjugate ... Overcoming the failure of the classical generalized interior point regularity conditions in convex ... Zalinescu cite book last Z linescu first Constantin title Convex analysis in general vector spaces ... function relating the primal and dual problems and math F math is the convex conjugate biconjugate ... more details
In mathematics , a convex body in n dimension al Euclidean space R sup n sup is a compact space compact convex set with non empty set empty interior topology interior . A convex body K is called symmetric if it is centrally symmetric with respect to the origin, i.e. a point x lies in K if and only if its Antipodal point antipode , &minus x , also lies in K . Symmetric convex bodies are in a bijection one to one correspondence with the unit ball s of norm mathematics norms on R sup n sup . Important examples of convex bodies are the Euclidean ball , the hypercube and the cross polytope . References cite journal last Gardner first Richard J. title The Brunn Minkowski inequality journal Bulletin of the American Mathematical Society Bull. Amer. Math. Soc. N.S. volume 39 issue 3 year 2002 pages 355&ndash 405 electronic doi 10.1090 S0273 0979 02 00941 2 Category Multi dimensional geometry es Cuerpo convexo zh ... more details
Image 3dpoly.svg thumb right A 3 dimensional convex polytope A convex polytope is a special case of a polytope , having the additional property that it is also a convex set of points in the n dimensional space R sup n sup . ref name grun Some authors use the terms convex polytope and convex polyhedron ...?id ofrBsl61lq8C&pg PA67&dq 22unbounded convex polyhedron 22&sig ACfU3U1Yv3iG XIn3hiuh84nK2e8UIcdAA ... convex polytope will be used below whenever the boundedness is critical to the discussed issue. Yet other texts treat a convex n polytope as a surface or n 1 manifold. Convex polytopes play an important ... . A comprehensive and influential book in the subject, called Convex Polytopes , was published ... texts in discrete geometry , convex polytopes are often simply called polytopes . Gr nbaum points out that this is solely to avoid the endless repetition of the word convex , and that the discussion should throughout be understood as applying only to the convex variety. A polytope is called full dimensional ... Many examples of bounded convex polytopes can be found in the article polyhedron . In the 2 dimensional ... shape the intersection of two non parallel half planes , a shape defined by a convex polygonal chain with two ray geometry ray s attached to its ends, and a convex polygon . Special cases of an unbounded convex polytope are a slab between two parallel hyperplanes, a wedge defined by two non parallel ... A convex polytope may be defined in a number of ways, depending on what is more suitable for the problem at hand. Gr nbaum s definition is in terms of a convex set of points in space. Other important ... representation and as the convex hull of a set of points vertex representation . Vertex representation Convex hull In his book Convex polytopes , Gr nbaum defines a convex polytope as a compact space compact convex set with a finite number of extreme points A set K of R sup n sup is convex if, for each ... K . This is equivalent to defining a bounded convex polytope as the convex hull of a finite set ... more details
More footnotes date February 2012 Convex minimization , a subfield of mathematical optimization optimization , studies the problem of minimizing convex function s over convex set s. The convexity property ... math together with a convex function convex , real valued function mathematics function math f mathcal X to mathbb R math defined on a convex set convex subset math mathcal X math of math X math , the problem ... X math . The convexity of math f math makes the powerful tools of convex analysis applicable. In locally convex F finite dimensional normed space s, the Hahn Banach theorem and the existence ... computational methods. Convex minimization has applications in a wide range of disciplines, such as automatic ... improvements in computing and in optimization theory, convex minimization is nearly as straightforward as linear programming . Many optimization problems can be reformulated as convex minimization problems ... as a problem of minimizing the function f , which is convex . Convex optimization problem An optimization ... math is the objective , is called convex if math mathcal X math is a closed convex set and math f x math is convex on math mathbb R n math . ref cite book last1 Hiriart Urruty first1 Jean Baptiste last2 Lemar chal first2 Claude title Convex analysis and minimization algorithms Fundamentals page 291 year 1996 url http books.google.de books?id Gdl4Jc3RVjcC&printsec frontcover&dq lemarechal convex analysis and minimization&hl de&sa X&ei E602T4GXGMzQsgaPtJ2VDA&ved 0CDUQ6AEwAA v onepage&q convex 20minimization ... title Lectures on modern convex optimization analysis, algorithms, and engineering applications ... on Modern Convex Optimization Analysis, Algorithms,&hl de&sa X&ei 26c2T6G7HYrIswac0d2uDA&ved 0CDIQ6AEwAA v onepage&q convex 20programming&f false ref Alternatively, an optimization problem on the form ... i 1, dots,m end align math is called convex if the functions math f, g 1 ldots g m mathbb R n rightarrow mathbb R math are convex. ref Boyd Vandenberghe, p. 7 ref Theory The following statements ... more details
Original research date November 2010 Convex Computer Corporation was a company that developed, manufactured ... File BSC Convex 240.JPG right thumb 250px Convex 240 supercomputer. Convex was formed in 1982 ... better price performance ratio . In order to lower costs, the Convex designs were not as technologically ... programs were ported to their systems. The machines ran a BSD version of Unix known initially as Convex ... precision . It was Convex s most successful product. The C2 was followed by the C3 in 1991, being ... per CPU. However, the C3 and the Convex business model were overtaken by changes in the computer industry ... more than chasing a business in decline. By this time, even though Convex was the first vendor to ship a GaAs based product, they were losing money. In 1994, Convex introduced an entirely new design ... draw in customers. But the type of customers Convex attracted believed in Fortran and brute force rather ... could not easily be fixed. Eventually, Convex established a working partnership with HP s hardware ..., Hewlett Packard bought Convex. HP sold Convex Exemplar machines under the S Class MP and X Class ... Class products. Culture According to most former employees, Convex was a very fun place at which to work. For some time, there were beer parties every Friday, and an annual Convex Beach Party .... Convex had an unusually thorough interview process, which, for technical positions, included a grilling ... spent most of their waking hours ensuring Convex s success. The culture was one of creativity. Especially ... as What have you done for the customer today? Dubious date July 2011 Convex lasted longer ... of the market, Convex had a graveyard of former competitor companies on its property. ref cite web author Stephanie Anderson Forest title CONVEX WANTS TO BE A FULL FLEDGED HEAVYWEIGHT url http www.businessweek.com archives 1991 b3210058.arc.htm accessdate 2009 05 29 ref Ex employees of Convex jokingly refer to themselves as ex cons . There is a http www.ex convex.org mailing list of Convex ... more details
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