Real function with secant line between points above the graph itself
In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set.
In simple terms, a convex function graph is shaped like a cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap .
A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.[1] Well-known examples of convex functions of a single variable include a linear function (where is a real number), a quadratic function ( as a nonnegative real number) and an exponential function ( as a nonnegative real number).
Convex functions play an important role in many areas of mathematics. They are especially important in the study of optimization problems where they are distinguished by a number of convenient properties. For instance, a strictly convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional hypotheses, convex functions continue to satisfy such properties and as a result, they are the most well-understood functionals in the calculus of variations. In probability theory, a convex function applied to the expected value of a random variable is always bounded above by the expected value of the convex function of the random variable. This result, known as Jensen's inequality, can be used to deduce inequalities such as the arithmetic–geometric mean inequality and Hölder's inequality.
^"Lecture Notes 2" (PDF). www.stat.cmu.edu. Retrieved 3 March 2017.
mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between...
In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with...
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...
a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination...
the function f {\displaystyle f} is closed. This definition is valid for any function, but most used for convexfunctions. A proper convexfunction is...
the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convexfunctions over convex sets...
particular the subfields of convex analysis and optimization, a proper convexfunction is an extended real-valued convexfunction with a non-empty domain...
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convexfunctions over convex sets (or, equivalently...
optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convexfunctions. It is also known...
piecewise linear functions and the convex piecewise linear functions. In general, for every n-dimensional continuous piecewise linear function f : R n → R...
Convex analysis is the branch of mathematics devoted to the study of properties of convexfunctions and convex sets, often with applications in convex...
Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convexfunctions. Important subclasses of convex curves...
In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of...
Strictly convex may refer to: Strictly convexfunction, a function having the line between any two points above its graph Strictly convex polygon, a polygon...
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...
that point. Subderivatives arise in convex analysis, the study of convexfunctions, often in connection to convex optimization. Let f : I → R {\displaystyle...
Convexfunction, when the line segment between any two points on the graph of the function lies above or on the graph Convex conjugate, of a function...
related to the problems on convex sets is the following problem on a convexfunction f: Rn → R: Strong unconstrained convexfunction minimization (SUCFM):...
In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it...
functional on X . {\displaystyle X.} A function p : X → R {\displaystyle p:X\to \mathbb {R} } which is subadditive, convex, and satisfies p ( 0 ) ≤ 0 {\displaystyle...
Intuitively, subharmonic functions are related to convexfunctions of one variable as follows. If the graph of a convexfunction and a line intersect at...
In mathematical optimization, the Ackley function is a non-convexfunction used as a performance test problem for optimization algorithms. It was proposed...
function over the positive reals which is logarithmically convex (super-convex), meaning that y = ln f ( x ) {\displaystyle y=\ln f(x)} is convex....
transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent...
Rastrigin function of two variables In mathematical optimization, the Rastrigin function is a non-convexfunction used as a performance test problem for...