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# Convex function

In mathematics, a real-valued function defined on an ''n''-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions. [1]

66 relations: Absolute value, Affine transformation, Calculus of variations, Claude Lemaréchal, Concave function, Continuous function, Convex conjugate, Convex optimization, Convex set, Countable set, Derivative, Differentiable function, Eigenvalues and eigenvectors, Elliptic operator, Epigraph (mathematics), Euclidean space, Expected value, Exponential function, Extreme value theorem, Geodesic convexity, Graph of a function, Hölder's inequality, Hermite–Hadamard inequality, Hessian matrix, Homogeneous function, Inequality of arithmetic and geometric means, Inflection point, Interval (mathematics), Invex function, Jensen's inequality, Jonathan Borwein, K-convex function, Kachurovskii's theorem, Karamata's inequality, Level set, Line segment, Linear map, Lipschitz continuity, Logarithmically convex function, Mark Krasnosel'skii, Mathematical optimization, Mathematics, Maxima and minima, Measurable function, Monotonic function, Nonnegative matrix, Norm (mathematics), Partial derivative, Positive real numbers, Positive-definite matrix, ... Expand index (16 more) »

## Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

## Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

## Calculus of variations

Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers.

## Claude Lemaréchal

Claude Lemaréchal is a French applied mathematician, and former senior researcher (directeur de recherche) at INRIA near Grenoble, France.

## Concave function

In mathematics, a concave function is the negative of a convex function.

## Continuous function

In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.

## Convex conjugate

In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions.

## Convex optimization

Convex optimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.

## Convex set

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.

## Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

## Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

## Differentiable function

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.

## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

## Elliptic operator

In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator.

## Epigraph (mathematics)

In mathematics, the epigraph or supergraph of a function f: Rn→R is the set of points lying on or above its graph: The strict epigraph is the epigraph with the graph itself removed: The same definitions are valid for a function that takes values in R &cup; &infin;.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Expected value

In probability theory, the expected value of a random variable, intuitively, is the long-run average value of repetitions of the experiment it represents.

## Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

## Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval, then f must attain a maximum and a minimum, each at least once.

## Geodesic convexity

In mathematics &mdash; specifically, in Riemannian geometry &mdash; geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Hölder's inequality

In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality between integrals and an indispensable tool for the study of ''Lp'' spaces.

In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ: → R is convex, then the following chain of inequalities hold.

## Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

## Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

## Inequality of arithmetic and geometric means

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.

## Inflection point

In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuously differentiable plane curve at which the curve crosses its tangent, that is, the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.

## Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

## Invex function

In vector calculus, an invex function is a differentiable function &fnof; from Rn to R for which there exists a vector valued function g such that for all x and u. Invex functions were introduced by Hanson as a generalization of convex functions.

## Jensen's inequality

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function.

## Jonathan Borwein

Jonathan Michael Borwein (20 May 1951 &ndash; 2 August 2016) was a Scottish mathematician who held an appointment as Laureate Professor of mathematics at the University of Newcastle, Australia.

## K-convex function

K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality of the (s,S) policy in inventory control theory.

## Kachurovskii's theorem

In mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet derivative.

## Karamata's inequality

In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line.

## Level set

In mathematics, a level set of a real-valued function ''f'' of ''n'' real variables is a set of the form that is, a set where the function takes on a given constant value c. When the number of variables is two, a level set is generically a curve, called a level curve, contour line, or isoline.

## Line segment

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

## Logarithmically convex function

In mathematics, a function f defined on a convex subset of a real vector space and taking positive values is said to be logarithmically convex or superconvex if \circ f, the composition of the logarithmic function with f, is a convex function.

## Mark Krasnosel'skii

Mark Alexandrovich Krasnosel'skii (Ма́рк Алекса́ндрович Красносе́льский) (April 27, 1920, Starokostiantyniv &ndash; February 13, 1997, Moscow) was a Soviet, Russian and Ukrainian mathematician renowned for his work on nonlinear functional analysis and its applications.

## Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Maxima and minima

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).

## Measurable function

In mathematics and in particular measure theory, a measurable function is a function between two measurable spaces such that the preimage of any measurable set is measurable, analogously to the definition that a function between topological spaces is continuous if the preimage of each open set is open.

## Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

## Nonnegative matrix

In mathematics, a nonnegative matrix, written is a matrix in which all the elements are equal to or greater than zero, that is, A positive matrix is a matrix in which all the elements are greater than zero.

## Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—save for the zero vector, which is assigned a length of zero.

## Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

## Positive real numbers

In mathematics, the set of positive real numbers, \mathbb_.

## Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

## Probability theory

Probability theory is the branch of mathematics concerned with probability.

## Pseudoconvex function

In convex analysis and the calculus of variations, branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex.

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function in one or more variables in which the highest-degree term is of the second degree.

## Quasiconvex function

In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set.

## Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Real-valued function

In mathematics, a real-valued function is a function whose values are real numbers.

## Second derivative

In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of.

In mathematics, the spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its eigenvalues (i.e. supremum among the absolute values of the elements in its spectrum).

## Subderivative

In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to functions which are not differentiable.

In mathematics, a sequence, n ≥ 1, is called superadditive if it satisfies the inequality for all m and n. The major reason for the use of superadditive sequences is the following lemma due to Michael Fekete.

## Symmetric function

In mathematics, a symmetric function of n variables is one whose value given n arguments is the same no matter the order of the arguments.

## Tangent

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.

## Taylor's theorem

In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial.

## Triangle inequality

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## References

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