Convex function is a kind of characteristic of mathematical function.A convex function is a real valued function defined on the convex subset C (interval) of a vector space.
Chinese name
convex function
Foreign name
convex function
Category
mathematics
Nature
The local minimum is the global minimum
Define Fields
Real linear space
Attention
Different definitions of concavity and convexity at home and abroad
Convex functions are defined in real linear space.[1]
be careful:The definition of concavity and convexity of functions by some institutions in the mathematical circles of mainland China is contrary to that of foreign countries.Convex Function refers to concave function in some mathematical books in mainland China.Concave Function refers to a convex function.However, in many books related to economics in mainland China, the concave convex formulation is consistent with that in other countries, that is, it is opposite to that in mathematics textbooks.For example, the definition of concavity and convexity of functions in the advanced mathematics textbook of Tongji University is opposite to this entry. The concavity and convexity of this entry means that the graph above it is concave set or convex set, while that in the advanced mathematics textbook of Tongji University means that the graph below it is concave set or convex set. The definitions of the two are just opposite.
In addition, some textbooks will define convex as upper convex and concave as lower convex.The definitions in the textbook should prevail when encountered.
A convex function is a real valued function f defined on a convex subset C of a vector space, and for any two vectors in a convex subset C
、
yes
Establishment.
So it is easy to get the rational number in any (0,1)
, Yes
If f is continuous, then
It can be changed to any real number in the interval (0,1).
If here convex set C is an interval I, that is: let f be a function defined on interval I, if any two points on I
And any real number
, always
Then f is called convex function on I. When "≤" in the definition is replaced by "<", it is also true that the corresponding measurable function f is strictly convex function on the corresponding subset or interval.[2]
The determination method can use the definition method, known conclusion method and function'sSecond derivativeFor a convex function on the set of real numbers, the general method is to find its second derivative. If its second derivative is greater than or equal to zero in the interval, it is called a convex function.If its second derivative is always greater than 0 in the interval, it is called strictly convex function.[3]
attribute
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nature
The convex function f defined in an open interval C is continuous in C and differentiable at all points except countable points.If C isClosed interval, then f may not be continuous at the end of C.
Univariate differentiable functionIt is convex on an interval if and only if its derivative is monotone on the interval.
A continuous differentiable function of one variable is convex on the interval if and only if the function is above all its tangents: for all x and y in the interval, there is f (y)>f (x)+f '(x) (y − x).In particular, if f '(c)=0, then c is the minimum value of f (x).
A second order differentiable function of one variable is convex in the interval if and only if its second derivative is nonnegative;This can be used to determine whether a function is convex or not.If itsSecond derivativeIf it is a positive number, then the function is strictly convex, but the reverse is not true.For example, the second derivative of f (x)=x4 is f "(x)=12 x2, which is zero when x=0, but x4 is strictly convex.
More generally, a multivariate quadratic differentiable continuous function is convex on a convex set if and only if itsHesse matrixIt is positive definite inside a convex set.
Any minimum of a convex function is also a minimum.Strictly convex functions can have at most one minimum value.
For convex function f, the horizontal subsets {x | f (x)<a} and {x | f (x) ≤ a} (a ∈ R) are convex sets.However, the horizontal subset isconvex setThe function of is not necessarily convex;Such a function is calledQuasiconvex function。
Convex functions have another important property: for convex functions, the local minimum is the global minimum.
To sum up, the main properties of convex functions are:
1. If f is a convex function defined on convex set S, then for any real number β ≥ 0, the function β f is also a convex function defined on S;
2. If foneAnd ftwoIs two convex functions defined on convex set S, then their sum f=fone+ftwoIs still a convex function defined on S;
3. If fi(i=1, 2,..., m) is a convex function defined on convex set S, then for any real number βi≥ 0, function βifiIt is also a convex function defined on S;
4. If f is a convex function defined on the convex set S, then for each real number c, the level set Sc={x | x ∈ S, f (x) ≤ c} is a convex set[4]
definition
set upf（x）It is defined in interval I,f（x）The interval I is called convex function if and only if:
，
yes
Where "≤" is changed to "<", it is the definition of strictly convex function[5]
Calculus
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If f and g are convex functions, then m (x)=max {f (x), g (x)} and h (x)=f (x)+g (x) are also convex functions.
If f and g are convex functions and g increases, then h (x)=g (f (x)) is convex.
Convexity in affinemappingLower constant: that is, if f (x) is a convex function, then g (y)=f (Ay+b) is also a convex function.
Elementary operation
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1. IffandgIs a convex function, thenm(x)=max{f(x)，g(x)}andh(x)=f(x)+g(x)It is also a convex function.
2. IffandgIs a convex function, andgIncremental, thenh(x)=f(g(x))Is a convex function.
3. Convexity is invariant under affine mapping: that is, iff（x）Is a convex function, theng(y)=f(Ay+b)Also convex function
For example
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The function f (x)=x ² is everywhere, so f is a (strictly) convex function.
The function f whose domain is [0,1] is defined as f (0)=f (1)=1. When the second derivative of the 0 function x3 is 6x, it is a convex function on the set of x ≥ 0 and a convex function on the set of x ≤ 0Concave function。
Each linear transformation taking value inside is convex function, but not strictly convex function, because if f is a linear function, then f (a+b)=f (a)+f (b).If we replace "convex" with "concave", then this proposition is also true.
Every affine transformation taking value inside, that is, every function whose shape is f (x)=aTx+b is both convex and concave.
If f is a convex function, g (x, t)=tf (x/t) is a convex function when t>0.
Monotonic increaseBut non convex functions include and g (x)=log (x).
Non monotonically increasing convex functions include h (x)=x2 and k (x)=− x.
The function f (x)=1/x2, f (0)=+∞, is convex in the interval (0,+∞), is also convex in the interval (- ∞, 0), but is not convex in the interval (- ∞,+∞), because of the singular point at x=0.
notes
The definition of convex function in some textbooks is opposite to this definition, that is, convex function is opposite to concave function.Such as the Peking University version and the mathematics textbook of Sun Yat sen University.
