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 or on 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 \cup (or a straight line like a linear function), while a concave function's graph is shaped like a cap \cap. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. Well-known examples of convex functions of a single variable include a linear function f(x) = cx (where c is a real number), a quadratic function cx^2 (c as a nonnegative real number) and an exponential function ce^x (c 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.

Definition

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<li>For all and all : The right hand side represents the straight line between and in the graph of f as a function of t; increasing t from 0 to 1 or decreasing t from 1 to 0 sweeps this line. Similarly, the argument of the function f in the left hand side represents the straight line between x_1 and x_2 in X or the x-axis of the graph of f. So, this condition requires that the straight line between any pair of points on the curve of f to be above or just meets the graph. </li> <li>For all 0 < t < 1 and all such that : The difference of this second condition with respect to the first condition above is that this condition does not include the intersection points (for example, and ) between the straight line passing through a pair of points on the curve of f (the straight line is represented by the right hand side of this condition) and the curve of f; the first condition includes the intersection points as it becomes or at t = 0 or 1, or x_1 = x_2. In fact, the intersection points do not need to be considered in a condition of convex using because and are always true (so not useful to be a part of a condition). </li> </ol> ****T****h****e**** ****s****e****c****o****n****d**** ****s****t****a****t****e****m****e****n****t**** ****c****h****a****r****a****c****t****e****r****i****z****i****n****g**** ****c****o****n****v****e****x**** ****f****u****n****c****t****i****o****n****s**** ****t****h****a****t**** ****a****r****e**** ****v****a****l****u****e****d**** ****i****n**** ****t****h****e**** ****r****e****a****l**** ****l****i****n****e**** ****\****R**** ****i****s**** ****a****l****s****o**** ****t****h****e**** ****s****t****a****t****e****m****e****n****t**** ****u****s****e****d**** ****t****o**** ****d****e****f****i****n****e**** **** ****t****h****a****t**** ****a****r****e**** ****v****a****l****u****e****d**** ****i****n**** ****t****h****e**** ****e****x****t****e****n****d****e****d**** ****r****e****a****l**** ****n****u****m****b****e****r**** ****l****i****n****e**** ****w****h****e****r****e**** ****s****u****c****h**** ****a**** ****f****u****n****c****t****i****o****n**** ****f**** ****i****s**** ****a****l****l****o****w****e****d**** ****t****o**** ****t****a****k****e**** ****\****p****m****\****i****n****f****t****y**** ****a****s**** ****a**** ****v****a****l****u****e****.**** The first statement is not used because it permits t to take 0 or 1 as a value, in which case, if or respectively, then would be undefined (because the multiplications and are undefined). The sum is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of -\infty and +\infty as a value. The second statement can also be modified to get the definition of, where the latter is obtained by replacing \,\leq\, with the strict inequality \,<. ****E****x****p****l****i****c****i****t****l****y****,**** ****t****h****e**** ****m****a****p**** ****f**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****f****o****r**** ****a****l****l**** ****r****e****a****l**** ****0**** ****<**** ****t**** ****<**** ****1**** ****a****n****d**** ****a****l****l**** ****s****u****c****h**** ****t****h****a****t**** ****:**** A strictly convex function f is a function that the straight line between any pair of points on the curve f is above the curve f except for the intersection points between the straight line and the curve. An example of a function which is convex but not strictly convex is. This function is not strictly convex because any two points sharing an x coordinate will have a straight line between them, while any two points NOT sharing an x coordinate will have a greater value of the function than the points between them. The function f is said to be **' (resp. **') if -f (f multiplied by −1) is convex (resp. strictly convex).

Alternative naming

The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward. If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph \cup. As an example, Jensen's inequality refers to an inequality involving a convex or convex-(down), function.

Properties

Many properties of convex functions have the same simple formulation for functions of many variables as for functions of one variable. See below the properties for the case of many variables, as some of them are not listed for functions of one variable.

Functions of one variable

Functions of several variables

Operations that preserve convexity

Strongly convex functions

The concept of strong convexity extends and parametrizes the notion of strict convexity. Intuitively, a strongly-convex function is a function that grows as fast as a quadratic function. A strongly convex function is also strictly convex, but not vice versa. If a one-dimensional function f is twice continuously differentiable and the domain is the real line, then we can characterize it as follows: For example, let f be strictly convex, and suppose there is a sequence of points (x_n) such that. Even though, the function is not strongly convex because f''(x) will become arbitrarily small. More generally, a differentiable function f is called strongly convex with parameter m > 0 if the following inequality holds for all points x, y in its domain: or, more generally, where is any inner product, and |\cdot| is the corresponding norm. Some authors, such as refer to functions satisfying this inequality as elliptic functions. An equivalent condition is the following: It is not necessary for a function to be differentiable in order to be strongly convex. A third definition for a strongly convex function, with parameter m, is that, for all x, y in the domain and Notice that this definition approaches the definition for strict convexity as m \to 0, and is identical to the definition of a convex function when m = 0. Despite this, functions exist that are strictly convex but are not strongly convex for any m > 0 (see example below). If the function f is twice continuously differentiable, then it is strongly convex with parameter m if and only if for all x in the domain, where I is the identity and \nabla^2f is the Hessian matrix, and the inequality \succeq means that is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue of be at least m for all x. If the domain is just the real line, then is just the second derivative f''(x), so the condition becomes. If m = 0 then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that ), which implies the function is convex, and perhaps strictly convex, but not strongly convex. Assuming still that the function is twice continuously differentiable, one can show that the lower bound of implies that it is strongly convex. Using Taylor's Theorem there exists such that Then by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above. A function f is strongly convex with parameter m if and only if the function is convex. A twice continuously differentiable function f on a compact domain X that satisfies f''(x)>0 for all x\in X is strongly convex. The proof of this statement follows from the extreme value theorem, which states that a continuous function on a compact set has a maximum and minimum. Strongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class. Like strictly convex functions, strongly convex functions have unique minima on compact sets.

Properties of strongly-convex functions

If f is a strongly-convex function with parameter m, then:

Uniformly convex functions

A uniformly convex function, with modulus \phi, is a function f that, for all x, y in the domain and satisfies where \phi is a function that is non-negative and vanishes only at 0. This is a generalization of the concept of strongly convex function; by taking we recover the definition of strong convexity. It is worth noting that some authors require the modulus \phi to be an increasing function, but this condition is not required by all authors.

Examples

Functions of one variable

Functions of n variables