GetWiki

(File:Convex vs. Not-convex.jpg|thumb|right|300px|Convex vs. Not convex)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 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.WEB,www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf, Lecture Notes 2, www.stat.cmu.edu, 3 March 2017, 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

(File:Convex 01.ogg|thumb|right|Visualizing a convex function and Jensen’s Inequality)Let X be a convex subset of a real vector space and let f: X to R be a function.Then f is called {{em|convex}} if and only if any of the following equivalent conditions hold:For all 0 leq t leq 1 and all x_1, x_2 in X:fleft(t x_1 + (1-t) x_2right) leq t fleft(x_1right) + (1-t) fleft(x_2right)The right hand side represents the straight line between left(x_1, fleft(x_1right)right) and left(x_2, fleft(x_2right)right) 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.WEB, Concave Upward and Downward,www.mathsisfun.com/calculus/concave-up-down-convex.html, live,www.mathsisfun.com:80/calculus/concave-up-down-convex.html," title="web.archive.org/web/20131218034748www.mathsisfun.com:80/calculus/concave-up-down-convex.html,">web.archive.org/web/20131218034748www.mathsisfun.com:80/calculus/concave-up-down-convex.html, 2013-12-18, For all 0 < t < 1 and all x_1, x_2 in X such that x_1 neq x_2:fleft(t x_1 + (1-t) x_2right) leq t fleft(x_1right) + (1-t) fleft(x_2right)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, left(x_1, fleft(x_1right)right) and left(x_2, fleft(x_2right)right)) 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 fleft(x_1right) leq fleft(x_1right) or fleft(x_2right) leq fleft(x_2right) 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 fleft(t x_1 + (1-t) x_2right) leq t fleft(x_1right) + (1-t) fleft(x_2right) because fleft(x_1right) leq fleft(x_1right) and fleft(x_2right) leq fleft(x_2right) are always true (so not useful to be a part of a condition). The second statement characterizing convex functions that are valued in the real line R is also the statement used to define {{em|convex functions}} that are valued in the extended real number line [-infty, infty] = R cup {pminfty}, where such a function f is allowed to take pminfty as a value. The first statement is not used because it permits t to take 0 or 1 as a value, in which case, if fleft(x_1right) = pminfty or fleft(x_2right) = pminfty, respectively, then t fleft(x_1right) + (1 - t) fleft(x_2right) would be undefined (because the multiplications 0 cdot infty and 0 cdot (-infty) are undefined). The sum -infty + infty 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 {{em|strict convexity}}, where the latter is obtained by replacing ,leq, with the strict inequality , 0) that is convex in one variable must be convex in the other variable.Altenberg, L., 2012. Resolvent positive linear operators exhibit the reduction phenomenon. Proceedings of the National Academy of Sciences, 109(10), pp.3705-3710.

Operations that preserve convexity

• -f is concave if and only if f is convex.
• If r is any real number then r + f is convex if and only if f is convex.
• Nonnegative weighted sums:
  • if w_1, ldots, w_n geq 0 and f_1, ldots, f_n are all convex, then so is w_1 f_1 + cdots + w_n f_n. In particular, the sum of two convex functions is convex.
  • this property extends to infinite sums, integrals and expected values as well (provided that they exist).
• Elementwise maximum: let {f_i}_{i in I} be a collection of convex functions. Then g(x) = supnolimits_{i in I} f_i(x) is convex. The domain of g(x) is the collection of points where the expression is finite. Important special cases:
  • If f_1, ldots, f_n are convex functions then so is g(x) = max left{f_1(x), ldots, f_n(x)right}.
  • Danskin’s theorem: If f(x,y) is convex in x then g(x) = supnolimits_{yin C} f(x,y) is convex in x even if C is not a convex set.
• Composition:
  • If f and g are convex functions and g is non-decreasing over a univariate domain, then h(x) = g(f(x)) is convex. For example, if f is convex, then so is e^{f(x)} because e^x is convex and monotonically increasing.
  • If f is concave and g is convex and non-increasing over a univariate domain, then h(x) = g(f(x)) is convex.
  • Convexity is invariant under affine maps: that is, if f is convex with domain D_f subseteq mathbf{R}^m, then so is g(x) = f(Ax+b), where A in mathbf{R}^{m times n}, b in mathbf{R}^m with domain D_g subseteq mathbf{R}^n.
• Minimization: If f(x,y) is convex in (x,y) then g(x) = infnolimits_{yin C} f(x,y) is convex in x, provided that C is a convex set and that g(x) neq -infty.
• If f is convex, then its perspective g(x, t) = t f left(tfrac{x}{t} right) with domain left{(x,t) : tfrac{x}{t} in operatorname{Dom}(f), t > 0 right} is convex.
• Let X be a vector space. f : X to mathbf{R} is convex and satisfies f(0) leq 0 if and only if f(ax+by) leq a f(x) + bf(y) for any x, y in X and any non-negative real numbers a, b that satisfy a + b leq 1.

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.WEB, Strong convexity · Xingyu Zhou’s blog,xingyuzhou.org/blog/notes/strong-convexity, 2023-09-27, xingyuzhou.org, 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:

• 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, let f be strictly convex, and suppose there is a sequence of points (x_n) such that f(x_n) = tfrac{1}{n}. Even though f(x_n) > 0, 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:BOOK, 72, Convex Analysis and Optimization,archive.org/details/convexanalysisop00bert_476, limited, Dimitri Bertsekas, Contributors: Angelia Nedic and Asuman E. Ozdaglar, Athena Scientific, 2003, 9781886529458, (nabla f(x) - nabla f(y) )^T (x-y) ge m |x-y|_2^2 or, more generally,langle nabla f(x) - nabla f(y), x-y rangle ge m |x-y|^2 where langle cdot, cdotrangle is any inner product, and |cdot| is the corresponding norm. Some authors, such as BOOK, Introduction to numerical linear algebra and optimisation, Philippe G. Ciarlet, Cambridge University Press, 1989, 9780521339841, refer to functions satisfying this inequality as elliptic functions.An equivalent condition is the following:BOOK, 63–64, Introductory Lectures on Convex Optimization: A Basic Course,archive.org/details/introductorylect00nest, limited, Yurii Nesterov, Kluwer Academic Publishers, 2004, 9781402075537, 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 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|_2^2Notice 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 nabla^2 f(x) succeq mI 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 semi-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, one can show that the lower bound of nabla^2 f(x) implies that it is strongly convex. Using Taylor’s Theorem there existsz in { t x + (1-t) y : t in [0,1] }such thatf(y) = f(x) + nabla f(x)^T (y-x) + frac{1}{2} (y-x)^T nabla^2f(z) (y-x)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.A function f is strongly convex with parameter m if and only if the functionxmapsto f(x) -frac m 2 |x|^2is convex.A twice continuously differentiable function f on a compact domain X that satisfies f’’(x)>0 for all xin 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:WEB, Nemirovsky and Ben-Tal, 2023, Optimization III: Convex Optimization,www2.isye.gatech.edu/~nemirovs/OPTIIILN2023Spring.pdf, {{Rp|location=Prop.6.1.4}}

• For every real number r, the level set {x | f(x) ≤ r} is compact.
• The function f has a unique global minimum on Rn.

Uniformly convex functions

A uniformly convex function,BOOK, Convex Analysis in General Vector Spaces, C. Zalinescu, World Scientific, 2002, 9812380671, BOOK, 144, Convex Analysis and Monotone Operator Theory in Hilbert Spaces,archive.org/details/convexanalysismo00hhba, limited, H. Bauschke and P. L. Combettes, Springer, 2011, 978-1-4419-9467-7, with modulus phi, is a function f that, for all x, y in the domain and t in [0, 1], satisfiesf(tx+(1-t)y) le t f(x)+(1-t)f(y) - t(1-t) phi(|x-y|)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 phi(alpha) = tfrac{m}{2} alpha^2 we recover the definition of strong convexity.It is worth noting that some authors require the modulus phi to be an increasing function,BOOK, 144, Convex Analysis and Monotone Operator Theory in Hilbert Spaces,archive.org/details/convexanalysismo00hhba, limited, H. Bauschke and P. L. Combettes, Springer, 2011, 978-1-4419-9467-7, but this condition is not required by all authors.BOOK, Convex Analysis in General Vector Spaces, C. Zalinescu, World Scientific, 2002, 9812380671,

Examples

Functions of one variable

• The function f(x)=x^2 has f’’(x)=2>0, so {{mvar|f}} is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
• The function f(x)=x^4 has f’’(x)=12x^2ge 0, so {{mvar|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 (as reflected in the triangle inequality), 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 p ge 1 is convex.
• The exponential function f(x)=e^x is convex. It is also strictly convex, since f’’(x)=e^x >0 , but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function g(x) = e^{f(x)} is logarithmically convex if f is a convex function. The term “superconvex” is sometimes used instead.JOURNAL, Kingman, J. F. C., 10.1093/qmath/12.1.283, A Convexity Property of Positive Matrices, The Quarterly Journal of Mathematics, 12, 283–284, 1961,
• 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 x^3 has second derivative 6 x; thus it is convex on the set where x geq 0 and concave on the set where x leq 0.
• 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)= x^2 and k(x)=-x.
• The function f(x) = tfrac{1}{x} has f’’(x)=tfrac{2}{x^3} which is greater than 0 if x > 0 so f(x) is convex on the interval (0, infty). It is concave on the interval (-infty, 0).
• The function f(x)=tfrac{1}{x^2} with f(0)=infty, is convex on the interval (0, infty) and convex on the interval (-infty, 0), but not convex on the interval (-infty, infty), because of the singularity at x = 0.

n variables“>

Functions of n variables

• LogSumExp function, also called softmax function, is a convex function.
• The function -logdet(X) on the domain of positive-definite matrices is convex.{{rp|74}}
• Every real-valued linear transformation 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 real-valued affine function, that is, each function of the form f(x) = a^T x + b, is simultaneously convex and concave.
• Every norm is a convex function, by the triangle inequality and positive homogeneity.
• The spectral radius of a nonnegative matrix is a convex function of its diagonal elements.Cohen, J.E., 1981. Convexity of the dominant eigenvalue of an essentially nonnegative matrix. Proceedings of the American Mathematical Society, 81(4), pp.657-658.

See also

{{Div col|colwidth=30em}}

• Concave function
• Convex analysis
• Convex conjugate
• Convex curve
• Convex optimization
• Geodesic convexity
• Hahn–Banach theorem
• Hermite–Hadamard inequality
• Invex function
• Jensen’s inequality
• K-convex function
• Kachurovskii’s theorem, which relates convexity to monotonicity of the derivative
• Karamata’s inequality
• Logarithmically convex function
• Pseudoconvex function
• Quasiconvex function
• Subderivative of a convex function

{{Div col end}}

Notes

{{Reflist}}

References

• BOOK

,

• Borwein, Jonathan, and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
• BOOK

,

• Hiriart-Urruty, Jean-Baptiste, and Lemaréchal, Claude. (2004). Fundamentals of Convex analysis. Berlin: Springer.
• BOOK, Mark Krasnosel’skii, Krasnosel’skii M.A., Rutickii Ya.B., Convex Functions and Orlicz Spaces, P.Noordhoff Ltd, Groningen, 1961,
• BOOK

,

• BOOK

,

• BOOK

,

• BOOK

,

• BOOK

,

• BOOK, Zălinescu, C., Convex analysis in general vector spaces, World Scientific Publishing  Co., Inc, River Edge, NJ, 2002, xx+367, 981-238-067-1, 1921556,

External links

• {{springer|title=Convex function (of a real variable)|id=p/c026240}}
• {{springer|title=Convex function (of a complex variable)|id=p/c026230}}

{{Convex analysis and variational analysis}}{{Authority control}}