Convex function
In calculus, a real-valued function is called convex if its graph is below each link between two of its points. This is equivalent to the fact that the epigraph of the function, ie the set of points above the graph is a convex set.
A real-valued function is called concave if its graph is above everyone and connecting two of its points. This is equivalent to that of Hypo graph of the function, ie the set of points below the graph, is a convex set.
One of the first, which dealt with the properties of convex and concave functions, was the Danish mathematician Johan Ludwig Jensen. The named after him, Jensen's inequality is the basis of important results in the theory of probability, measure theory and analysis.
The special significance of convex or concave functions is that they form a much larger group than the linear functions, but also have many easy to be examined properties that allow statements about nonlinear systems. For example, since minima strictly convex functions are unique, they are in many optimization problems of meaning ( see also: Convex optimization). Even for convex functionals defined on infinite-dimensional spaces can be, under certain conditions, make similar statements. Therefore, convexity also plays an important role in the calculus of variations.
- 5.1 Positive combinations
- 5.2 limit functions
- Supremum and infimum 5.3
- 5.4 composition
- 5.5 inverse functions
- 7.1 A weaker definition of convexity
- 7.2 boundedness and continuity in normed spaces 7.2.1 In finite dimensional spaces
- 7.2.2 In infinite-dimensional spaces
- 8.1 Convexity and first derivative
- 8.2 convexity and second derivative
Definition
A real-valued function defined on a convex subset of a real vector space is called convex if for all and for all, that
However, always applies the reverse inequality, ie
The function is called concave.
A function is called strictly convex or strictly convex if the above inequality holds in the strict sense; from the mean for two elements and all is that
Similarly, a function is called strictly concave or strictly concave if all applies and that
The set of points above ( or below ) the graph of a convex (or concave ) function is a convex set. In order to avoid misunderstandings related to these geometric meaning, you can see the expressions identify by specifying a direction of view. Thus, the properties are convex and concave rarely referred to as convex from below and convex from above.
Examples
- Linear functions are convex and concave at all, but not strictly.
- The quadratic function is convex on quite strict.
- The additive inverse to their function is concave to very strict.
- The absolute value function is in a very convex, but not strictly convex.
- The exponential function is strictly convex at all.
- The natural logarithm is strictly concave on the interval of positive real numbers.
- The cubic function is strictly concave on the interval and strictly convex on the interval.
- The function which on its distance from the origin point of images a Euclidean plane, that is
History
To convex and concave functions essential statements can be found as early as 1889 by Otto Hölder, said he does not yet used the usual names today. The terms convex and concave function were introduced in 1905 by Johan Ludwig Jensen. However, Jensen used a weaker definition that is still occasionally to be found especially in older works. In this definition, only the inequality
Provided. As Jensen but showed follows then for continuous functions already the inequality used in today's conventional definition
For all between 0 and 1 (see also: Section convexity and continuity )
Elementary Properties
Ratio of convex and concave
The function is exactly then ( strictly ) convex if the function ( strictly ) concave. However, a non- convex function need not necessarily be concave. Convexity and concavity are thus not complementary properties.
Linear functions are the only functions that are both concave and convex.
The cubic function is in a very considered neither convex nor concave. In the interval of all positive real numbers is strictly convex. The additive inverse to their function there is therefore strictly concave.
Because an odd function is so true, it follows that it is strictly concave in the field of all negative numbers.
Jensen's inequality
The Jensen's inequality states that the function value of a convex function on a finite convex combination of interpolation points less than or equal to the convex combination of the function values at the nodes is. For a convex function and for non-negative with so applies:
For concave functions, the inequality holds in the reverse direction.
Inequalities for t < 0 and t > 1
For or the inequalities turn around from the definitions of ( strict ) convexity or concavity. Be an example to convex functions. Then for points of
Provided also the point in the domain is located. If a real convex function, it means the inequality clearly that the values of the function outside the interval always lie above the connecting line through function values .
Calculation rules
Positive combinations
The sum of two convex functions is a convex function. Moreover, convexity is preserved when multiplying by a positive real number. In summary, then, that any positive combination of convex functions is again convex. It is even strictly convex if one of the summands occurring is strictly convex. Similarly, any positive combination of concave functions is concave. The product convex functions is not necessarily convex.
The functions
Are convex on all of the standard parabola is even strictly convex. It follows that all the functions of the form
With strictly convex are all up. This is also intuitively clear, it is curved upward parabolas. The product of the functions and the cubic function, which ( over all considered ) is not convex.
Limit functions
The limit function is a pointwise convergent sequence of convex functions is a convex function. Similarly, the limit function of a pointwise convergent series of convex functions is a convex function. The same applies obviously for concave functions. However, strict convexity is not necessarily obtained, as can be seen from the first of the two following example with the limit value education.
- The sequence of functions with is a sequence of strictly convex on all functions. Your pointwise limit function is the constant zero function. This is a linear function, although convex but not strictly convex.
- The hyperbolic cosine can be developed as a power series as follows:
Supremum and infimum
Is a lot of convex functions and exists pointwise supremum
For all, so is a convex function. The transition to the function shows that the infimum of a set of concave functions (if it exists ) is also back is a concave function. Making the infimum is replaced but not necessarily convexity as shows the following example.
The real functions
Are linear and therefore both convex and concave. The supremum of and is the absolute value function. Although this is convex, not concave. The infimum of the negative and the absolute value function. This is concave, but not convex.
Composition
About the composition of two convex functions and can generally make no statement. However, where, in addition, that is monotonically increasing, so the composition is also convex.
Furthermore, the composition of a concave function with a convex, monotonically decreasing real function is again a convex function.
Each composition of a convex function with the exponential function delivers again a convex function. This also works in the general case, is defined in the on a real vector space. For example, for
Again a convex function. In particular, therefore each logarithmically convex function is a convex function.
Inverse functions
If a defined on an interval, invertible and convex function, it follows from the Konvexitätsungleichung
Be a monotonically increasing function. Then above inequality turns around while applying. It is thus:
So the inverse function is a concave ( and monotonically increasing ) function. For an invertible, monotonically increasing, and convex or concave function, therefore, the inverse has the opposite type of convexity.
However, applies to a monotonically decreasing and convex function
Therefore, an invertible monotonically decreasing and convex or concave function, the inverse function of the same type of convexity.
- The standard parabola is monotonically increasing and strictly convex on. Its inverse function, the square root function is strictly concave on their interval of definition
- The function is monotonically decreasing and strictly convex on all. Its inverse function is strictly convex on the interval
Extreme values
Convexity or concavity of functions are often used to ensure the uniqueness of extreme values . If the output space is a topological vector space (which in particular all finite dimensional real vector spaces and thus also on the case), statements can be made about local extreme points.
A local minimum of a convex function is also a global minimum. A strictly convex function has at most one global minimum. The existence of such a minimum is for example ensured if the strictly convex function is continuous and the domain of a compact set. However, a strictly convex function does not necessarily have a global minimum: has no global minimum in.
The same is true for concave functions: A local maximum of a concave function is always also a global maximum. A strictly concave function has at most one global maximum. A continuous strictly concave function on a compact convex set has precisely on this set a global maximum. but for example, has no global maximum for.
Convexity and continuity
Substituting the continuity of a real function advance, sufficient to exhibit their convexity already the condition that the following inequality holds for all of the definition interval:
This corresponds to the Konvexitätsdefinition according to Jensen. Conversely, that each defined on an interval function that satisfies the above inequality, the interior points is continuous. Discontinuities can occur at most boundary points, as the example of the function with
Shows that, although convex but has a discontinuity at the edge point.
Thus, the two options are convexity to define at least for open intervals equivalent. To what extent this result can be transferred to general topological spaces is treated in the following two sections.
A weaker definition of convexity
A continuous function on a convex subset of a real topological vector space is convex if there exists a fixed with, so, from applies to all:
This can be shown by appropriate nested intervals. A proof is completely executed in the proof archive.
The fact that in this weaker definition of convexity continuity is needed, it can be seen by the following counter-example.
Be a Hamel basis of the vector space of real numbers over the field of rational numbers, ie a linearly independent over the rational numbers set of real numbers in which each real number has a representation of the type with only finitely many rational. Then, satisfied for any choice of the function while but is not necessarily convex.
Boundedness and continuity in normed spaces
If, for a function in addition to the condition that for a fixed relationship
, applies to all of a convex subset of a normed vector space, still requires that is bounded above, then it already follows the continuity of the interior points of. This is clearly evident from the fact that you can pull on a point of discontinuity an arbitrarily steep connecting line between two function values, the function between the two values below the connecting line and outside the two values above the connecting line must lie. Can now be arbitrarily steep the connecting line, so you will eventually reach about the upper bound of the function.
Formally, the proof is somewhat more complicated. A full version is in the proof archive.
The statement that a convex bounded function is continuous at the interior points, is also relevant for solving the Cauchy functional equation
From this statement it follows that this functional equation has a unique solution if it is additionally required that is limited.
In finite-dimensional spaces
Convex functions that are defined on a convex subset of a finite dimensional real vector space, are continuous in the interior points. To see this, consider an interior point. For this there is a simplex with vertices containing again as interior point. Each dot is, however, in the form
And displayed for all. After jens between inequality now applies
Is therefore bounded from above and thus, as indicated above, continuous in the inner point.
In infinite-dimensional spaces
In the infinite-dimensional case are convex functions are not necessarily continuous, as it particularly linear (and thus convex ) functionals are not continuous.
Convexity and differentiability
Convexity and first derivative
A defined on an open interval, convex or concave function is locally Lipschitz continuous and thus differentiable with respect to the set of Rademacher almost everywhere. She is left -and right- differentiable at every point.
A continuously differentiable real function is exactly then convex if its derivative is monotonically increasing, and strictly convex if and only if its derivative is strictly increasing. Analog is a continuously differentiable function iff ( strictly ) concave if its derivative ( strictly ) is monotonically decreasing. This result is essentially already in 1889 Otto Hölder.
Differentiable convex functions lie above each of its tangents; it applies. Moreover, the strict inequality holds for strictly convex functions for. Similarly, concave functions are always below its tangents. From these two properties follows, for example, the generalization of Bernoulli's inequality:
Convexity and second derivative
For a real, twice differentiable function, further statements can be made. if and only convex if its second derivative is non-negative. Is consistently positive, so always left curved, then it follows that is strictly convex. Analogously, if and concave if applies. Is consistently negative, always quite curved so is strictly concave.
If the multi-dimensional function is twice continuously differentiable, then applies that is exactly then convex if the Hessian matrix is positive semi-definite of. If the Hessian matrix of positive definite, so is strictly convex. Conversely concave if and only if the Hessian matrix is semidefinite of negative. If the Hessian matrix of negative definite, so is strictly concave.
Convex functions in geometry
A subset is a convex set if and only if the by
Defined distance function is a convex function.
The same property is true not only for subsets of, but also for general subsets of CAT ( 0) - spaces, particularly of non- Riemannian manifold of positive sectional curvature. The convexity of the distance function is an important tool in the investigation of non- positive curved spaces.