In mathematics, a real-valued function f defined on an interval (or on any convex subset of some vector space) is called convex, or concave up, if for any two points x and y in its domain C and any t in [0,1], we have

Convex function on an interval. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, an 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 In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined

In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set. ↔ In Mathematics, the epigraph of a function f    R n→ R is the set of points lying on or above its In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the

A function is called strictly convex if

for any t in (0,1) and

A function f is said to be concave if − f is convex. In Mathematics, a concave function is the negative of a Convex function.

Properties

A function (in blue) is convex if and only if the region above its graph (in green) is a convex set. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. In Euclidean space, an object is convex if for every pair of points within the object every point on the Straight line segment that joins them is also within the

A convex function f defined on some open interval C is continuous on C and differentiable at all but at most countably many points. In Mathematics, an 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 In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change If C is closed, then f may fail to be continuous at the endpoints of C.

A function is midpoint convex on an interval "C" if

for all x and y in C. This condition is only slightly weaker than convexity. For example, a real valued measurable function that is midpoint convex will be convex. In Mathematics, measurable functions are Well-behaved functions between measurable spaces. In particular, a continuous function that is midpoint convex will be convex.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents: f(y) ≥ f(x) + f '(x) (y − x) for all x and y in the interval. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. In particular, if f '(c) = 0, then c is a global minimum of f(x). In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that

A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the converse does not hold, as shown by f(x) = x⁴.

More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set. In Mathematics, the Hessian matrix is the Square matrix of second-order Partial derivatives of a function. In Linear algebra, a positive-definite matrix is a (Hermitian matrix which in many ways is analogous to a Positive Real number.

Any local minimum of a convex function is also a global minimum. In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that A strictly convex function will have at most one global minimum.

For a convex function f, the sublevel sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets. In Mathematics, a level set of a real -valued function f of n variables is a set of the form { ( x 1 However, a function whose sublevel sets are convex sets may fail to be a convex function; such a function is called a 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

Jensen's inequality applies to every convex function f. In Mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a Convex function of an Integral If X is a random variable taking values in the domain of f, then (Here E denotes the mathematical expectation. )

Convex function calculus

• If f and g are convex functions, then so are m(x) = max{f(x),g(x)} and h(x) = f(x) + g(x).
• If f and g are convex functions and if g is increasing, then h(x) = g(f(x)) is convex.
• Convexity is invariant under affine maps: that is, if f(x) is convex with , then so is g(y) = f(Ay + b), where
• If f(x,y) is convex in (x,y) and C is a convex nonempty set, then is convex in x, provided for some x.

Examples

• The function f(x) = x² has f''(x) = 2 > 0 at all points, so f is a (strictly) convex function.
• The absolute value function f(x) = | x | is convex, even though it does not have a derivative at the point x = 0. In Mathematics, the absolute value (or modulus) of a Real number is its numerical value without regard to its sign.
• The function f(x) = | x | ^p for 1 ≤ p is convex.
• The function f with domain [0,1] defined by f(0)=f(1)=1, f(x)=0 for 0<x<1 is convex; it is continuous on the open interval (0,1), but not continuous at 0 and 1.
• The function x³ has second derivative 6x; thus it is convex on the set where x ≥ 0 and concave on the set where x ≤ 0. In Mathematics, a concave function is the negative of a Convex function.
• Every linear transformation taking values in is convex but not strictly convex, since if f is linear, then f(a + b) = f(a) + f(b). In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This statement also holds if we replace "convex" by "concave".
• Every affine function taking values in , i. In Geometry, an affine transformation or affine map or an affinity (from the Latin affinis, "connected with" between two Vector e. , each function of the form f(x) = a^Tx + b, is simultaneously convex and concave.
• Every norm is a convex function, by the triangle inequality. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Mathematics, the triangle inequality states that for any Triangle, the length of a given side must be less than or equal to the sum of the other two sides but greater
• If f is convex, the perspective function g(x,t) = tf(x / t) is convex for t > 0.
• Examples of functions that are monotonically increasing but not convex include and g(x) = log(x).
• Examples of functions that are convex but not monotonically increasing include h(x) = x² and k(x) = − x.
• The function f(x) = 1/x², with f(0)=+∞, is convex on the interval [0,+∞) and convex on the interval (-∞,0], but not convex on the interval (-∞,+∞), because of the singularity at x = 0.

See also

• Convex optimization
• Geodesic convexity
• Kachurovskii's theorem, which relates convexity to monotonicity of the derivative
• Logarithmically convex function
• Quasiconvex function
• Subderivative of a convex function
• Jensen's inequality
• Hermite–Hadamard inequality

References

• Rockafellar, R. Convex optimization is a subfield of mathematical optimization. In Mathematics &mdash specifically in Riemannian geometry &mdash geodesic convexity is a natural generalization of convexity for sets and functions In Mathematics, Kachurovskii's theorem is a theorem relating the convexity of a function on a Banach space to the monotonicity of its Fréchet In Mathematics, a function f defined on an convex subset of a real Vector space and taking positive values is said to In Mathematics, a quasiconvex function is a real -valued function defined on an interval or on a convex subset of a real Vector In Mathematics, the concepts of subderivative, subgradient, and subdifferential arise in Convex analysis, that is in the study of Convex In Mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a Convex function of an Integral In Mathematics, the Hermite–Hadarmard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called T. (1970). Convex analysis. Princeton: Princeton University Press.  

• Luenberger, David (1984). Linear and Nonlinear Programming. Addison-Wesley.  

• Luenberger, David (1969). Optimization by Vector Space Methods. Wiley & Sons.  

• Bertsekas, Dimitri (2003). Convex Analysis and Optimization. Athena Scientific.  

• Thomson, Brian (1994). Symmetric Properties of Real Functions. CRC Press.  

• Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.

• Krasnosel'skii M.A., Rutickii Ya. Mark Alexandrovich Krasnosel'skii (Марк Александрович Красносельский ( April 27, 1920 &ndash February 13 1997 B. (1961). Convex Functions and Orlicz Spaces. Groningen: P. Noordhoff Ltd.  

• Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.

External links

• Stephen Boyd and Lieven Vandenberghe, Convex Optimization (PDF)
• Jon Dattorro, Convex Optimization & Euclidean Distance Geometry