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Topic: Proper convex function

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Infimal convolute

List of convexity topics

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Legendre transformation

	Convex conjugate - Wikipedia, the free encyclopedia
		The convex conjugate of the exponential function is
		The convex conjugate of a closed convex function is again a closed convex function.
		The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.
	en.wikipedia.org /wiki/Convex_conjugate (289 words)

	Function - LoveToKnow 1911 (Site not responding. Last check: 2007-10-30)
		The notion of functionality or functional relation of two magnitudes was thus of geometrical origin; but a function soon came to be regarded as an analytical expression, not necessarily an algebraic expression, containing the variable or variables.
		The function may have a limit on the left and a limit on the right at the same point; these may be different from each other, and at least one of them must be different from the value of the function at the point.
		The function may be given by specifying the domain of the argument and the rule of calculation, or else the function may have to be determined in accordance with certain conditions; for example, it may have to satisfy in a prescribed domain an assigned differential equation.
	www.1911encyclopedia.org /Function (13554 words)

	Function (mathematics) - Wikipedia, the free encyclopedia
		Thus, the "is a square root of" relation is a function, and it contains for example, the pairs (3, 9) and (-3, 9); while the converse, "is the square of" relation, is not a function because it contains both the pairs (9, 3) and (9, −3), and 3 is not equal to −3.
		For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input.
		The number of computable functions from integers to integers is countable, because the number of possible algorithms is. The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers.
	en.wikipedia.org /wiki/Function_(mathematics) (3457 words)

	Proper convex function - Wikipedia, the free encyclopedia
		The effective domain of a convex function is always a convex set.
		The sum of two proper convex functions is convex but not necessarily proper convex.
		The infimal convolute of two proper convex functions is convex but not necessarily proper convex.
	en.wikipedia.org /wiki/Proper_convex_function (182 words)

	Convex - Wikipedia, the free encyclopedia
		Convex function, a function with the epigraph (the set of points lying on or above the graph) forming a convex set
		Convex conjugate, is a generalization of the Legendre transformation
		Convex lens, a lens with surfaces that curve outward
	en.wikipedia.org /wiki/Convex (213 words)

	Untitled Document (Site not responding. Last check: 2007-10-30)
		We show that this "quasiconvex" subdifferential is always a cyclically quasimonotone operator that coincides with the Fenchel-Moreau subdifferential whenever the function is convex, and that under mild assumptions, the density of its domain in the domain of the function is equivalent to the quasiconvexity of the function.
		We study some properties of a family of functions defined on a set X which are related to the validity of the necessary and sufficient optimality conditions for differences of convex functions that have been obtained recently by Flores-Bazán for the case when is the family of continuous functions on a metric space.
		Indirect functions are characterized as certain nonincreasing polyhedral convex functions; their relations with the associated charactersitic functions are analyzed by means of Fenchel-Moreau generalized conjugation theory.
	pareto.uab.es /jemartinez/abstracts.html (3580 words)

	Models - Operations Research Models and Methods
		A function of a single variable with a decreasing derivative.
		A concave function in the objective function of a maximization problem can be represented by the sum of several linear expressions with a piecewise linear approximation (3 for the figure).
		A convex function in the objective function of a minimization problem can be represented by the sum of several linear expressions with a piecewise linear approximation (3 for the figure).
	www.me.utexas.edu /~jensen/ORMM/models/unit/integer/subunits/terminology (363 words)

	SEP: Recursive Functions
		The recursive functions are characterized by the process in virtue of which the value of a function for some argument is defined in terms of the value of that function for some other (in some appropriate sense "smaller") arguments, as well as the values of certain other functions.
		The primary example of a doubly-recursive function that is not primitive recursive is due to Wilhelm Ackermann.
		Whereas Ackermann's function cannot be obtained by means of primitive recursion, it can be obtained by further expanding the class of recursive functions by introducing a "minimization" or "least search" operator μ.
	plato.stanford.edu /entries/recursive-functions (6913 words)

	proper / propriety - a function of musical scales, revealing their perceptual interval properties
		A function defined over the interval matrix of a scale, revealing the perceptual properties of that scale.
		Strictly Proper scales are those in which all intervallic size classes are distinct and non-overlapping.
		A scale is proper if all intervals spanning the same number of scale steps, have a range of sizes (in cents) that does not overlap but may meet, the range of sizes for any other number of scale steps.
	tonalsoft.com /enc/p/proper.aspx (587 words)

	Duals
		Let p be a proper convex function on R^n with convex conjugate p, and let q be a proper* concave function on R^n with concave conjugate q* (q could be the indicator function of a convex feasible region).
		The dual decisions are functionals, which could be specialized to classes defined by a finite number of parameters (which then become the dual variables).
		The Lagrangian dual is a strong dual for a convex program that satisfies Slater's interiority condition.
	orion.math.uwaterloo.ca /~hwolkowi/mirror.d/glossary/duals.html (1508 words)

	Convex Function (Site not responding. Last check: 2007-10-30)
		Definition 1 A function such that for all is said to be logarithmically convex if is a convex function.
		The opposite of a convex function is a concave function.
		50) this.border=1; this.alt='Thumbnails by Thumbshots.org';">Definition of Convex Function and Jensen's Inequality
	www.convexfunction.info (320 words)

	Error Bounds for Convex Inequality Systems - Lewis, Pang (ResearchIndex)
		Abstract: Using convex analysis, this paper gives a systematic and unified treatment for the existence of a global error bound for a convex inequality system.
		We establish a necessary and sufficient condition for a closed convex set defined by a closed proper convex function to possess a global error bound in terms of a natural residual.
		The normal property for two convex cones was introduced by Jameson [26] to study the closedness of the sum of their dual cones....
	citeseer.ist.psu.edu /123721.html (853 words)

	Proper Convex Function (Site not responding. Last check: 2007-10-30)
		It is always possible to consider the restriction of a proper convex function f to its...
		We establish a necessary and sufficient condition for a closed convex set defined by a closed proper convex function...
		of a Conjecture on the Integral Means of the Derivative of a Convex Function Roger W. Barnard...
	www.properconvexfunction.info (465 words)

	Common Musical Instruments Used In Jazz (Site not responding. Last check: 2007-10-30)
		Classified technically as a board zither--an instrument in which strings are passed over a board that functions as a resonator--the piano belongs to the DULCIMER (struck zither) family in contradistinction to the HARPSICHORD, played by plucking the strings, and the CLAVICHORD, in which the strings are touched by tangents.
		The early trombone (sackbut) differs from its modern counterpart primarily by its narrow tube of heavier metal, its small expansion at the end, and its funnel-shaped mouthpiece similar to the horn, the result being a mellow tone with command of soft through moderately loud dynamics, eminently suitable for combining with voices, soft woodwinds, or strings.
		Changed in function, but little in design, from their Renaissance origins, trombones together with trumpets provide a magnificent choir of brilliant brass tone for both orchestra and band.
	www.nw-cybermall.com /jazzworld/common_musical_instruments.htm (4104 words)

	GameDev.net - Fast Convex Polygons
		You will better understand the performance implications of the available GUI functions, and you will be able to fill in any gaps in the functions provided.
		Fast Convex Polygon Filling When last we left the topic of filling convex polygons, the implementation we had met all of our functional requirements.
		The memset function linked in from the library is tied to the memory model in use; in small (tiny, small, or medium) data models memset accepts only near pointers, so it can't be used to access screen memory.
	www.gamedev.net /reference/articles/article368.asp (2476 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		A set X of points in the plane is called convex if for every two points a, b in X the set of all points on the line segment, joining a and b, belongs to X.
		X is a z-containing set if z is in the interior of the convex hull of X.
		X is a z-avoiding set if z is not contained in the interior of the convex hull of X.
	dimacs.rutgers.edu /~dkrasner/2001 (695 words)

	A Glossary of Frequently Misused or Misunderstood Physics Terms and Concepts.
		When a function is written in the form of an equation relating values of variables, y = y(x), y must be single-valued, that is each value of x corresponds to only one value of y.
		Thus a surface convex to the incident light is positive, one concave to the incident light is negative.
		A diverging mirror is convex to the incoming light, with negative f.
	www.lhup.edu /~dsimanek/glossary.htm (9901 words)

	Journal of Convex Analysis, Vol. 5, No. 2, pp. 279-301, 1998 (Site not responding. Last check: 2007-10-30)
		A new theorem for $\mathbb{R}^d$ concerns the free separation of two convex sets, where this means that there is a separating hyperplane $H$ such that all sufficiently small perturbations of $H$ still separate the two sets.
		A theorem for general $\mathbb{X}$ implies that if a proper convex subset $C$ of $\mathbb{X}$ is s-closed, then $C$ is an intersection of open halfspaces.
		In conjunction with the relevance of s-closedness to line-searches, and the related fact that the standard theorems on extremal structure of convex sets in $\mathbb{R}^d$ extend to s-closed subsets of $\mathbf{F}^d$, this suggests that results on the behavior of s-closed sets may eventually provide useful tools in the development of genuinely rational optimization algorithms.
	www.emis.de /journals/JCA/vol.5_no.2/6.html (338 words)

	Gottesman, Research Interests (Site not responding. Last check: 2007-10-30)
		Many aspects of bacterial and viral physiology are affected by the heat shock response.
		We have been focussing on the role of the heat shock proteins in the proper folding of newly synthesized and denatured polypeptides.
		Lu, S.-D., Lu, D. and Gottesman, M.E. (1989) Stimulation of IS1 excision by phage P1 ref function.
	convex.hhmi.columbia.edu /faculty/gottesman.html (346 words)

	Math Forum Discussions
		I came across a proof which relies on positive homogeneity of a recession function.
		Let f(x) be a closed proper convex function of x \in R^1.
		The recession function for f, lets call it f", is
	mathforum.org /kb/thread.jspa?threadID=1223676&messageID=3943581 (125 words)

	Hiriart-Urruty: New concepts in nondifferentiable programming
		On the differentiability almost everywhere of functions of many variables.
		A class of almost-differentiable functions and a minimization method for functions of this class.
		Conditions for an extremum of a functional in case of constraints.
	www.numdam.org /numdam-bin/item?id=MSMF_1979__60__57_0 (468 words)

	An Asymptotical Variational Principle Associated with the Steepest Descent Method for a Convex Function - Lemaire ...
		for results along these lines) When f is strongly convex, it is possible to estimate the rate of convergence of the trajectory u(t) see...
		deal with non convex functions that have a priori many local minima.
		2: A dynamical approach to convex minimization coupling approximation with the stee..
	citeseer.ist.psu.edu /9030.html (321 words)

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