Convex function


Convex function
Convex function on an interval.
A function (in black) is convex if and only if the region above its graph (in green) is a convex set.

In mathematics, a real-valued function f(x) defined on an interval is called convex (or convex downward or concave upward) if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. More generally, this definition of convex functions makes sense for functions defined on a convex subset of any vector space.

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 less or equal to the expected value of the convex function of the random variable. This result, known as Jensen's inequality underlies many important inequalities (including, for instance, the arithmetic-geometric mean inequality and Hölder's inequality).

Contents

Definition

A real valued function f : XR defined on a convex set X in a vector space is called convex if, for any two points x1 and x2 in X and any t\in[0,1],

f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2).

The function is called strictly convex if

f(tx_1+(1-t)x_2) < t f(x_1)+(1-t)f(x_2)\,

for every t\,,  0 < t < 1\, , and x_1 \not=x_2.

A function f is said to be (strictly) concave if −f is (strictly) convex.

Properties

Suppose f is a function of one real variable defined on an interval, and let

 R(x_1,x_2) = \frac{f(x_1) - f(x_2)}{x_1 - x_2}

(note that R(x1,x2) is the slope of the purple line in the above drawing; note also that the function R is symmetric in x1,x2). f is convex if and only if R(x1,x2) is monotonically non-decreasing in x1, for x2 fixed (or viceversa). This characterization of convexity is quite useful to prove the following results.

A convex function f defined on some open interval C is continuous on C and Lipschitz continuous on any closed subinterval. f admits left and right derivatives, and these are monotonically non-decreasing. As a consequence, f is differentiable at all but at most countably many points. If C is closed, then f may fail to be continuous at the endpoints of C (an example is shown in the examples' section).

A function is midpoint convex on an interval C if

f\left( \frac{x_1+x_2}{2} \right) \le  \frac{f(x_1)+f(x_2)}{2}

for all x1 and x2 in C. This condition is only slightly weaker than convexity. For example, a real valued Lebesgue measurable function that is midpoint convex will be convex.[1] In particular, a continuous function that is midpoint convex will be convex.

A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable.

A continuously differentiable function of one variable is convex on an interval if and only if the function lies above all of its tangents:

f(x) \geq f(y) + f'(y)[x-y][2]

for all x and y in the interval. In particular, if f '(c) = 0, then c is a global minimum of f(x).

A twice differentiable function of one variable is convex on an interval if and only if its second derivative is non-negative there; this gives a practical test for convexity. If its second derivative is positive then it is strictly convex, but the converse does not hold. For example, the second derivative of f(x) = x4 is f "(x) = 12 x2, which is zero for x = 0, but x4 is strictly convex.

More generally, a continuous, twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix is positive semidefinite on the interior of the convex set.

Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.

For a convex function f, the sublevel sets {x | f(x) < a} and {x | f(x) ≤ a} with a ∈ R are convex sets. However, a function whose sublevel sets are convex sets may fail to be a convex function. A function whose sublevel sets are convex is called a quasiconvex function.

Jensen's inequality applies to every convex function f. If X is a random variable taking values in the domain of f, then \operatorname{E}(f(X)) \geq f(\operatorname{E}(X)). (Here \operatorname{E} denotes the mathematical expectation.)

If a function f is convex, and f(0) ≤ 0, then f is superadditive on the positive half-axis. Proof:

  • since f is convex, let y = 0, f(tx) = f(tx+(1-t)\cdot 0) \le t f(x)+(1-t)f(0) \le t f(x) for every t\in\left[0,1\right]
  • f(a) + f(b) = f \left((a+b) \frac{a}{a+b} \right) + f \left((a+b) \frac{b}{a+b} \right)
\le \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) = f(a+b)

Convex function calculus

  • If f and g are convex functions, then so are m(x) = max{f(x),g(x)} and h(x) = f(x) + g(x).
  • If f and g are convex functions and g is non-decreasing, then h(x) = g(f(x)) is convex.
  • If f is concave and g is convex and non-increasing, then h(x) = g(f(x)) is convex.
  • Convexity is invariant under affine maps: that is, if f(x) is convex with x\in\mathbb{R}^n, then so is g(y) = f(Ay + b) with y\in\mathbb{R}^m, where A\in\mathbb{R}^{n \times m},\; b\in\mathbb{R}^n.
  • If f(x,y) is convex in x then g(x) = \sup_{y\in C} f(x,y) is convex in x, provided g(x) > -\infty for some x.
  • If f(x) is convex, then its perspective g(x,t) = tf(x / t) (whose domain is \left\lbrace (x, t) | x/t \in Dom(f), t > 0 \right\rbrace) is convex.
  • The additive inverse of a convex function is a concave function.

Strongly convex functions

The concept of strong convexity extends and parametrizes the notion of strict convexity. A strongly convex function is also strictly convex, but not vice-versa.

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[3]:

 ( \nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2.

Some authors, such as [4] refer to functions satisfying this inequality as elliptic functions.

An equivalent condition is the following[5]:

 f(y) \ge f(x) + \nabla f(x)^T (y-x) + \frac{m}{2} \|y-x\|_2^2

It is not necessary for a function to be differentiable in order to be strongly convex. A third definition[5] for a strongly convex function, with parameter m, is that, for all x,y in the domain and t\in [0,1],

f(tx+(1-t)y) \le t f(x)+(1-t)f(y) - \frac{1}{2} m t(1-t) \|x-y\|_2^2 \,

Notice that this definition approaches the definition for strict convexity as m \rightarrow 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 f is strongly convex with parameter m if and only if  \nabla^2 f(x) \succeq m I for all x in the domain, where I is the identity and \nabla^2f is the Hessian matrix, and the inequality \succeq means that  \nabla^2 f(x) - mI is positive definite. This is equivalent to requiring that the minimum eigenvalue of  \nabla^2 f(x) be at least m for all x. If the domain is just the real line, then \nabla^2 f(x) is just the second derivative f''(x)\,\!, so the condition becomes f''(x) \ge m . If m = 0, then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that f''(x) \ge 0), which implies the function is convex, and perhaps strictly convex, but not strongly convex.

Assuming still that the function is twice continuously differentiable, we show that the lower bound of \nabla^2 f(x) implies that it is strongly convex. Start by using Taylor's Theorem:

 f(y) = f(x) + \nabla f(x)^T (y-x) + 1/2 (y-x)^T \nabla^2f(z) (y-x)

for some (unknown)  z \in [x,y] . Then (y-x)^T \nabla^2f(z) (y-x) \ge m (y-x)^T(y-x) by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.

The distinction between convex, strictly convex, and strongly convex can be subtle at first glimpse. If f is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:

f\,\! convex if and only if  f''(x) \ge 0 for all x\,\!
f\,\! strictly convex if  f''(x) > 0 \,\! for all x\,\! (note: this is sufficient, but not necessary)
f\,\! strongly convex if and only if  f''(x) \ge m > 0 for all x\,\!

For example, consider a function f that is strictly convex, and suppose there is a sequence of points (xn) such that f'(x_n) = \frac{1}{n}. Even though  f'(x_n) > 0 \,\!, the function is not strongly convex because f'(x)\,\! will become arbitrarily small.

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.

Examples

  • The function f(x) = x2 has f''(x) = 2 > 0 at all points, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
  • The function f(x) = x4 has f''(x)=12x^2\ge0, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points. It is not strongly convex.
  • The absolute value function f(x) = | x | is convex, even though it does not have a derivative at the point x = 0. It is not strictly convex.
  • The function f(x) = | x | p for 1 ≤ p is convex.
  • The exponential function f(x) = ex is convex. It is also strictly convex, since f''(x) = ex > 0, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function g(x) = ef(x) is logarithmically convex if f is a convex function. The term "superconvex" is sometimes used instead.[6]
  • The function f with domain [0,1] defined by f(0) = f(1) = 1, f(x) = 0 for 0 < x < 1 is convex; it is continuous on the open interval (0, 1), but not continuous at 0 and 1.
  • The function x3 has second derivative 6x; thus it is convex on the set where x ≥ 0 and concave on the set where x ≤ 0.
  • Every linear transformation taking values in \mathbb{R} is convex but not strictly convex, since if f is linear, then f(a + b) = f(a) + f(b). This statement also holds if we replace "convex" by "concave".
  • Every affine function taking values in \mathbb{R}, i.e., each function of the form f(x) = aTx + b, is simultaneously convex and concave.
  • Every norm is a convex function, by the triangle inequality and positive homogeneity.
  • Examples of functions that are monotonically increasing but not convex include f(x) = \sqrt x and g(x) = log(x).
  • Examples of functions that are convex but not monotonically increasing include h(x) = x2 and k(x) = − x.
  • The function f(x) = 1/x has f\,''(x)=\frac{2}{x^3} which is greater than 0 if x > 0, so f(x) is convex on the interval (0, +∞). It is concave on the interval (-∞,0).
  • The function f(x) = 1/x2, with f(0) = +∞, is convex on the interval (0, +∞) and convex on the interval (-∞,0), but not convex on the interval (-∞, +∞), because of the singularity at x = 0.

See also

References

  1. ^ Sierpinski Theorem, Donoghue (1969), p. 12
  2. ^ Boyd, Stephen P.; Vandenberghe, Lieven (2004) (pdf). Convex Optimization. Cambridge University Press. p. 69. ISBN 9780521833783. http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf. Retrieved October 15, 2011. 
  3. ^ p. 72, Convex Analysis and Optimization", by Dimitri Bertsekas, Athena Scientific, 2003
  4. ^ Introduction to numerical linear algebra and optimisation, Philippe G. Ciarlet, Cambridge University Press, 1989
  5. ^ a b p. 63-64, Introductory Lectures on Convex Optimization, Yurii Nesterov, Kluwer Academic Publishers, 2004
  6. ^ Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford (2) 12,283-284.
  • Bertsekas, Dimitri (2003). Convex Analysis and Optimization. Athena Scientific. 
  • Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
  • Donoghue, William F. (1969). Distributions and Fourier Transforms. Academic Press. 
  • Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.
  • Krasnosel'skii M.A., Rutickii Ya.B. (1961). Convex Functions and Orlicz Spaces. Groningen: P.Noordhoff Ltd. 
  • Luenberger, David (1984). Linear and Nonlinear Programming. Addison-Wesley. 
  • Luenberger, David (1969). Optimization by Vector Space Methods. Wiley & Sons. 
  • Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press. 
  • Thomson, Brian (1994). Symmetric Properties of Real Functions. CRC Press. 
  • Zălinescu, C.. Convex analysis in general vector spaces. World Scientific Publishing  Co., Inc. pp. xx+367. ISBN 981-238-067-1. MR1921556. 

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