Concave function

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Concave_function

In mathematics, 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 of the values at the endpoints. Equivalently, a concave function is any function for which the hypograph is convex. The class of concave functions is in a sense the opposite of the class of convex functions. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

Definition

A real-valued function ${\displaystyle f}$ on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any ${\displaystyle x}$ and ${\displaystyle y}$ in the interval and for any ${\displaystyle \alpha \in [0,1]}$,

${\displaystyle f((1-\alpha )x+\alpha y)\ge (1-\alpha )f(x)+\alpha f(y)}$

A function is called strictly concave if

${\displaystyle f((1-\alpha )x+\alpha y)>(1-\alpha )f(x)+\alpha f(y)}$

for any ${\displaystyle \alpha \in (0,1)}$ and ${\displaystyle x\ne y}$.

For a function ${\displaystyle f:\mathbb{R}\to \mathbb{R}}$, this second definition merely states that for every ${\displaystyle z}$ strictly between ${\displaystyle x}$ and ${\displaystyle y}$, the point ${\displaystyle (z,f(z))}$ on the graph of ${\displaystyle f}$ is above the straight line joining the points ${\displaystyle (x,f(x))}$ and ${\displaystyle (y,f(y))}$.

A function ${\displaystyle f}$ is quasiconcave if the upper contour sets of the function ${\displaystyle S(a)=\{x:f(x)\ge a\}}$ are convex sets.

Properties

A cubic function is concave (left half) when its first derivative (red) is monotonically decreasing i.e. its second derivative (orange) is negative, and convex (right half) when its first derivative is monotonically increasing i.e. its second derivative is positive

Functions of a single variable

1. A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.
2. Points where concavity changes (between concave and convex) are inflection points.
3. If f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, informally, if the "acceleration" is non-positive). If f ′′ is negative then f is strictly concave, but the converse is not true, as shown by f(x) = −x⁴.
4. If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation: $${\displaystyle f(y)\le f(x)+f^{\prime }(x)[y-x]}$$
5. A Lebesgue measurable function on an interval C is concave if and only if it is midpoint concave, that is, for any x and y in C $${\displaystyle f\left(\frac{x+y}{2}\right)\ge \frac{f(x)+f(y)}{2}}$$
6. If a function f is concave, and f(0) ≥ 0, then f is subadditive on ${\displaystyle [0,\mathrm{\infty })}$. Proof:
   • Since f is concave and 1 ≥ t ≥ 0, letting y = 0 we have $${\displaystyle f(tx)=f(tx+(1-t)\cdot 0)\ge tf(x)+(1-t)f(0)\ge tf(x).}$$
   • For ${\displaystyle a,b\in [0,\mathrm{\infty })}$: $${\displaystyle f(a)+f(b)=f\left((a+b)\frac{a}{a+b}\right)+f\left((a+b)\frac{b}{a+b}\right)\ge \frac{a}{a+b}f(a+b)+\frac{b}{a+b}f(a+b)=f(a+b)}$$

Functions of n variables

1. A function f is concave over a convex set if and only if the function −f is a convex function over the set.
2. The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
3. Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.
4. Any local maximum of a concave function is also a global maximum. A strictly concave function will have at most one global maximum.

Examples

• The functions ${\displaystyle f(x)=-x^2}$ and ${\displaystyle g(x)=\sqrt{x}}$ are concave on their domains, as their second derivatives ${\displaystyle f^{″}(x)=-2}$ and $g^{″}(x)=-\frac{1}{4x^{3/2}}$ are always negative.
• The logarithm function ${\displaystyle f(x)=\log x}$ is concave on its domain ${\displaystyle (0,\mathrm{\infty })}$, as its derivative ${\displaystyle \frac{1}{x}}$ is a strictly decreasing function.
• Any affine function ${\displaystyle f(x)=ax+b}$ is both concave and convex, but neither strictly-concave nor strictly-convex.
• The sine function is concave on the interval ${\displaystyle [0,\pi ]}$.
• The function ${\displaystyle f(B)=\log |B|}$, where ${\displaystyle |B|}$ is the determinant of a nonnegative-definite matrix B, is concave.

Applications

Rays bending in the computation of radiowave attenuation in the atmosphere involve concave functions.

In expected utility theory for choice under uncertainty, cardinal utility functions of risk averse decision makers are concave.

In microeconomic theory, production functions are usually assumed to be concave over some or all of their domains, resulting in diminishing returns to input factors.

In Thermodynamics and Information Theory, Entropy is a concave function. In the case of thermodynamic entropy, without phase transition, entropy as a function of extensive variables is strictly concave. If the system can undergo phase transition, if it is allowed to split into two subsystems of different phase (phase separation, e.g. boiling), the entropy-maximal parameters of the subsystems will result in a combined entropy precisely on the straight line between the two phases. This means that the "Effective Entropy" of a system with phase transition is the convex envelope of entropy without phase separation; therefore, the entropy of a system including phase separation will be non-strictly concave.

Bounded operator

From Wikipedia, the free encyclopedia

Not to be confused with bounded function (set theory).

In normed vector spaces

Every bounded operator is Lipschitz continuous at ${\displaystyle 0.}$

Equivalence of boundedness and continuity

A linear operator between normed spaces is bounded if and only if it is continuous.

Proof

Suppose that ${\displaystyle L}$ is bounded. Then, for all vectors ${\displaystyle x,h\in X}$ with ${\displaystyle h}$ nonzero we have $${\displaystyle \Vert L(x+h)-L(x)\Vert =\Vert L(h)\Vert \le M\Vert h\Vert .}$$ Letting ${\displaystyle h}$ go to zero shows that ${\displaystyle L}$ is continuous at ${\displaystyle x.}$ Moreover, since the constant ${\displaystyle M}$ does not depend on ${\displaystyle x,}$ this shows that in fact ${\displaystyle L}$ is uniformly continuous, and even Lipschitz continuous.

Conversely, it follows from the continuity at the zero vector that there exists a ${\displaystyle \epsilon >0}$ such that ${\displaystyle \Vert L(h)\Vert =\Vert L(h)-L(0)\Vert \le 1}$ for all vectors ${\displaystyle h\in X}$ with ${\displaystyle \Vert h\Vert \le \epsilon .}$ Thus, for all non-zero ${\displaystyle x\in X,}$ one has $${\displaystyle \Vert Lx\Vert =\Vert \frac{\Vert x\Vert }{\epsilon }L\left(\epsilon \frac{x}{\Vert x\Vert }\right)\Vert =\frac{\Vert x\Vert }{\epsilon }\Vert L\left(\epsilon \frac{x}{\Vert x\Vert }\right)\Vert \le \frac{\Vert x\Vert }{ϵ}\cdot 1=\frac{1}{\epsilon }\Vert x\Vert .}$$ This proves that ${\displaystyle L}$ is bounded. Q.E.D.

In topological vector spaces

A linear operator ${\displaystyle F:X\to Y}$ between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever ${\displaystyle B\subseteq X}$ is bounded in ${\displaystyle X}$ then ${\displaystyle F(B)}$ is bounded in ${\displaystyle Y.}$ A subset of a TVS is called bounded (or more precisely, von Neumann bounded) if every neighborhood of the origin absorbs it. In a normed space (and even in a seminormed space), a subset is von Neumann bounded if and only if it is norm bounded. Hence, for normed spaces, the notion of a von Neumann bounded set is identical to the usual notion of a norm-bounded subset.

Continuity and boundedness

Every sequentially continuous linear operator between TVS is a bounded operator. This implies that every continuous linear operator between metrizable TVS is bounded. However, in general, a bounded linear operator between two TVSs need not be continuous.

This formulation allows one to define bounded operators between general topological vector spaces as an operator which takes bounded sets to bounded sets. In this context, it is still true that every continuous map is bounded, however the converse fails; a bounded operator need not be continuous. This also means that boundedness is no longer equivalent to Lipschitz continuity in this context.

If the domain is a bornological space (for example, a pseudometrizable TVS, a Fréchet space, a normed space) then a linear operators into any other locally convex spaces is bounded if and only if it is continuous. For LF spaces, a weaker converse holds; any bounded linear map from an LF space is sequentially continuous.

If ${\displaystyle F:X\to Y}$ is a linear operator between two topological vector spaces and if there exists a neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X}$ such that ${\displaystyle F(U)}$ is a bounded subset of ${\displaystyle Y,}$ then ${\displaystyle F}$ is continuous. This fact is often summarized by saying that a linear operator that is bounded on some neighborhood of the origin is necessarily continuous. In particular, any linear functional that is bounded on some neighborhood of the origin is continuous (even if its domain is not a normed space).

Bornological spaces

Main article: Bornological space

Bornological spaces are exactly those locally convex spaces for which every bounded linear operator into another locally convex space is necessarily continuous. That is, a locally convex TVS ${\displaystyle X}$ is a bornological space if and only if for every locally convex TVS ${\displaystyle Y,}$ a linear operator ${\displaystyle F:X\to Y}$ is continuous if and only if it is bounded.

Every normed space is bornological.

Characterizations of bounded linear operators

Let ${\displaystyle F:X\to Y}$ be a linear operator between topological vector spaces (not necessarily Hausdorff). The following are equivalent:

1. ${\displaystyle F}$ is (locally) bounded;
2. (Definition): ${\displaystyle F}$ maps bounded subsets of its domain to bounded subsets of its codomain;
3. ${\displaystyle F}$ maps bounded subsets of its domain to bounded subsets of its image ${\displaystyle \mathrm{Im}F:=F(X)}$;
4. ${\displaystyle F}$ maps every null sequence to a bounded sequence;
   • A null sequence is by definition a sequence that converges to the origin.
   • Thus any linear map that is sequentially continuous at the origin is necessarily a bounded linear map.
5. ${\displaystyle F}$ maps every Mackey convergent null sequence to a bounded subset of ${\displaystyle Y.}$
   • A sequence ${\displaystyle x_{\bullet }={\left(x_i\right)}_{i=1}^{\mathrm{\infty }}}$ is said to be Mackey convergent to the origin in ${\displaystyle X}$ if there exists a divergent sequence ${\displaystyle r_{\bullet }={\left(r_i\right)}_{i=1}^{\mathrm{\infty }}\to \mathrm{\infty }}$ of positive real number such that ${\displaystyle r_{\bullet }={\left(r_i{x}_i\right)}_{i=1}^{\mathrm{\infty }}}$ is a bounded subset of ${\displaystyle X.}$

if ${\displaystyle X}$ and ${\displaystyle Y}$ are locally convex then the following may be add to this list:

6. ${\displaystyle F}$ maps bounded disks into bounded disks.
7. ${\displaystyle F^{-1}}$ maps bornivorous disks in ${\displaystyle Y}$ into bornivorous disks in ${\displaystyle X.}$

if ${\displaystyle X}$ is a bornological space and ${\displaystyle Y}$ is locally convex then the following may be added to this list:

8. ${\displaystyle F}$ is sequentially continuous at some (or equivalently, at every) point of its domain.
   • A sequentially continuous linear map between two TVSs is always bounded, but the converse requires additional assumptions to hold (such as the domain being bornological and the codomain being locally convex).
   • If the domain ${\displaystyle X}$ is also a sequential space, then ${\displaystyle F}$ is sequentially continuous if and only if it is continuous.
9. ${\displaystyle F}$ is sequentially continuous at the origin.

Examples

• Any linear operator between two finite-dimensional normed spaces is bounded, and such an operator may be viewed as multiplication by some fixed matrix.
• Any linear operator defined on a finite-dimensional normed space is bounded.
• On the sequence space ${\displaystyle c_{00}}$ of eventually zero sequences of real numbers, considered with the ${\displaystyle {\ell }^1}$ norm, the linear operator to the real numbers which returns the sum of a sequence is bounded, with operator norm 1. If the same space is considered with the ${\displaystyle {\ell }^{\mathrm{\infty }}}$ norm, the same operator is not bounded.
• Many integral transforms are bounded linear operators. For instance, if $${\displaystyle K:[a,b]\times [c,d]\to \mathbb{R}}$$ is a continuous function, then the operator ${\displaystyle L}$ defined on the space ${\displaystyle C[a,b]}$ of continuous functions on ${\displaystyle [a,b]}$ endowed with the uniform norm and with values in the space ${\displaystyle C[c,d]}$ with ${\displaystyle L}$ given by the formula $${\displaystyle (Lf)(y)={\int }_a^{b}K(x,y)f(x)dx,}$$ is bounded. This operator is in fact a compact operator. The compact operators form an important class of bounded operators.
• The Laplace operator $${\displaystyle \mathrm{\Delta }:H^2({\mathbb{R}}^n)\to L^2({\mathbb{R}}^n)}$$ (its domain is a Sobolev space and it takes values in a space of square-integrable functions) is bounded.
• The shift operator on the Lp space ${\displaystyle {\ell }^2}$ of all sequences ${\displaystyle \left(x_0,x_1,x_2,\dots \right)}$ of real numbers with ${\displaystyle x_0^2+x_1^2+x_2^2+\cdots <\mathrm{\infty },}$ $${\displaystyle L(x_0,x_1,x_2,\dots )=\left(0,x_0,x_1,x_2,\dots \right)}$$ is bounded. Its operator norm is easily seen to be ${\displaystyle 1.}$

Unbounded linear operators

Let ${\displaystyle X}$ be the space of all trigonometric polynomials on ${\displaystyle [-\pi ,\pi ],}$ with the norm

$${\displaystyle \Vert P\Vert ={\int }_{-\pi }^{\pi }|P(x)|dx.}$$

The operator ${\displaystyle L:X\to X}$ that maps a polynomial to its derivative is not bounded. Indeed, for ${\displaystyle v_n=e^{inx}}$ with ${\displaystyle n=1,2,\dots ,}$ we have ${\displaystyle \Vert v_n\Vert =2\pi ,}$ while ${\displaystyle \Vert L(v_n)\Vert =2\pi n\to \mathrm{\infty }}$ as ${\displaystyle n\to \mathrm{\infty },}$ so ${\displaystyle L}$ is not bounded.

Properties of the space of bounded linear operators

The space of all bounded linear operators from ${\displaystyle X}$ to ${\displaystyle Y}$ is denoted by ${\displaystyle B(X,Y)}$.

• ${\displaystyle B(X,Y)}$ is a normed vector space.
• If ${\displaystyle Y}$ is Banach, then so is ${\displaystyle B(X,Y)}$; in particular, dual spaces are Banach.
• For any ${\displaystyle A\in B(X,Y)}$ the kernel of ${\displaystyle A}$ is a closed linear subspace of ${\displaystyle X}$.
• If ${\displaystyle B(X,Y)}$ is Banach and ${\displaystyle X}$ is nontrivial, then ${\displaystyle Y}$ is Banach.