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 (or a straight line like a linear function), while a concave function's graph is shaped like a cap .

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.
A graph of the bivariate convex function x2 + xy + y2.
Convex vs. Not convex

A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.[1] Well-known examples of convex functions of a single variable include a linear function (where is a real number), a quadratic function ( as a nonnegative real number) and an exponential function ( 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 edit

Visualizing a convex function and Jensen's Inequality

Let   be a convex subset of a real vector space and let   be a function.

Then   is called convex if and only if any of the following equivalent conditions hold:

  1. For all   and all  :
    The right hand side represents the straight line between   and   in the graph of   as a function of   increasing   from   to   or decreasing   from   to   sweeps this line. Similarly, the argument of the function   in the left hand side represents the straight line between   and   in   or the  -axis of the graph of   So, this condition requires that the straight line between any pair of points on the curve of   to be above or just meets the graph.[2]
  2. For all   and all   such that  :
    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,   and  ) between the straight line passing through a pair of points on the curve of   (the straight line is represented by the right hand side of this condition) and the curve of   the first condition includes the intersection points as it becomes   or   at   or   or   In fact, the intersection points do not need to be considered in a condition of convex using
    because   and   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   is also the statement used to define convex functions that are valued in the extended real number line   where such a function   is allowed to take   as a value. The first statement is not used because it permits   to take   or   as a value, in which case, if   or   respectively, then   would be undefined (because the multiplications   and   are undefined). The sum   is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of   and   as a value.

The second statement can also be modified to get the definition of strict convexity, where the latter is obtained by replacing   with the strict inequality   Explicitly, the map   is called strictly convex if and only if for all real   and all   such that  :


A strictly convex function   is a function that the straight line between any pair of points on the curve   is above the curve   except for the intersection points between the straight line and the curve. An example of a function which is convex but not strictly convex is  . This function is not strictly convex because any two points sharing an x coordinate will have a straight line between them, while any two points NOT sharing an x coordinate will have a greater value of the function than the points between them.

The function   is said to be concave (resp. strictly concave) if   (  multiplied by −1) is convex (resp. strictly convex).

Alternative naming edit

The term convex is often referred to as convex down or concave upward, and the term concave is often referred as concave down or convex upward.[3][4][5] If the term "convex" is used without an "up" or "down" keyword, then it refers strictly to a cup shaped graph  . As an example, Jensen's inequality refers to an inequality involving a convex or convex-(down), function.[6]

Properties edit

Many properties of convex functions have the same simple formulation for functions of many variables as for functions of one variable. See below the properties for the case of many variables, as some of them are not listed for functions of one variable.

Functions of one variable edit

  • Suppose   is a function of one real variable defined on an interval, and let
    (note that   is the slope of the purple line in the above drawing; the function   is symmetric in   means that   does not change by exchanging   and  ).   is convex if and only if   is monotonically non-decreasing in   for every fixed   (or vice versa). This characterization of convexity is quite useful to prove the following results.
  • A convex function   of one real variable defined on some open interval   is continuous on     admits left and right derivatives, and these are monotonically non-decreasing. As a consequence,   is differentiable at all but at most countably many points, the set on which   is not differentiable can however still be dense. If   is closed, then   may fail to be continuous at the endpoints of   (an example is shown in the examples section).
  • 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 differentiable function of one variable is convex on an interval if and only if its graph lies above all of its tangents:[7]: 69 
    for all   and   in the interval.
  • 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. Visually, a twice differentiable convex function "curves up", without any bends the other way (inflection points). If its second derivative is positive at all points then the function is strictly convex, but the converse does not hold. For example, the second derivative of   is  , which is zero for   but   is strictly convex.
    • This property and the above property in terms of "...its derivative is monotonically non-decreasing..." are not equal since if   is non-negative on an interval   then   is monotonically non-decreasing on   while its converse is not true, for example,   is monotonically non-decreasing on   while its derivative   is not defined at some points on  .
  • If   is a convex function of one real variable, and  , then   is superadditive on the positive reals, that is   for positive real numbers   and  .

Since   is convex, by using one of the convex function definitions above and letting   it follows that for all real  

From this it follows that
  • A function   is midpoint convex on an interval   if for all  
    This condition is only slightly weaker than convexity. For example, a real-valued Lebesgue measurable function that is midpoint-convex is convex: this is a theorem of Sierpiński.[8] In particular, a continuous function that is midpoint convex will be convex.

Functions of several variables edit

  • A function   valued in the extended real numbers   is convex if and only if its epigraph
    is a convex set.
  • A differentiable function   defined on a convex domain is convex if and only if   holds for all   in the domain.
  • A twice differentiable function of several variables is convex on a convex set if and only if its Hessian matrix of second partial derivatives is positive semidefinite on the interior of the convex set.
  • For a convex function   the sublevel sets   and   with   are convex sets. A function that satisfies this property is called a quasiconvex function and may fail to be a convex function.
  • Consequently, the set of global minimisers of a convex function   is a convex set:   - convex.
  • Any local minimum of a convex function is also a global minimum. A strictly convex function will have at most one global minimum.[9]
  • Jensen's inequality applies to every convex function  . If   is a random variable taking values in the domain of   then   where   denotes the mathematical expectation. Indeed, convex functions are exactly those that satisfies the hypothesis of Jensen's inequality.
  • A first-order homogeneous function of two positive variables   and   (that is, a function satisfying   for all positive real  ) that is convex in one variable must be convex in the other variable.[10]

Operations that preserve convexity edit

  •   is concave if and only if   is convex.
  • If   is any real number then   is convex if and only if   is convex.
  • Nonnegative weighted sums:
    • if   and   are all convex, then so is   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   be a collection of convex functions. Then   is convex. The domain of   is the collection of points where the expression is finite. Important special cases:
    • If   are convex functions then so is  
    • Danskin's theorem: If   is convex in   then   is convex in   even if   is not a convex set.
  • Composition:
    • If   and   are convex functions and   is non-decreasing over a univariate domain, then   is convex. For example, if   is convex, then so is   because   is convex and monotonically increasing.
    • If   is concave and   is convex and non-increasing over a univariate domain, then   is convex.
    • Convexity is invariant under affine maps: that is, if   is convex with domain  , then so is  , where   with domain  
  • Minimization: If   is convex in   then   is convex in   provided that   is a convex set and that  
  • If   is convex, then its perspective   with domain   is convex.
  • Let   be a vector space.   is convex and satisfies   if and only if   for any   and any non-negative real numbers   that satisfy  

Strongly convex functions edit

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.[11] A strongly convex function is also strictly convex, but not vice versa. If a one-dimensional function   is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:

  •   convex if and only if   for all  
  •   strictly convex if   for all   (note: this is sufficient, but not necessary).
  •   strongly convex if and only if   for all  

For example, let   be strictly convex, and suppose there is a sequence of points   such that  . Even though  , the function is not strongly convex because   will become arbitrarily small.

More generally, a differentiable function   is called strongly convex with parameter   if the following inequality holds for all points   in its domain:[12]

or, more generally,
where   is any inner product, and   is the corresponding norm. Some authors, such as [13] refer to functions satisfying this inequality as elliptic functions.

An equivalent condition is the following:[14]


It is not necessary for a function to be differentiable in order to be strongly convex. A third definition[14] for a strongly convex function, with parameter   is that, for all   in the domain and  


Notice that this definition approaches the definition for strict convexity as   and is identical to the definition of a convex function when   Despite this, functions exist that are strictly convex but are not strongly convex for any   (see example below).

If the function   is twice continuously differentiable, then it is strongly convex with parameter   if and only if   for all   in the domain, where   is the identity and   is the Hessian matrix, and the inequality   means that   is positive semi-definite. This is equivalent to requiring that the minimum eigenvalue of   be at least   for all   If the domain is just the real line, then   is just the second derivative   so the condition becomes  . If   then this means the Hessian is positive semidefinite (or if the domain is the real line, it means that  ), 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   implies that it is strongly convex. Using Taylor's Theorem there exists

such that
by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.

A function   is strongly convex with parameter m if and only if the function

is convex.

A twice continuously differentiable function   on a compact domain   that satisfies   for all   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 edit

If f is a strongly-convex function with parameter m, then:[15]: Prop.6.1.4 

Uniformly convex functions edit

A uniformly convex function,[16][17] with modulus  , is a function   that, for all   in the domain and   satisfies

where   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   we recover the definition of strong convexity.

It is worth noting that some authors require the modulus   to be an increasing function,[17] but this condition is not required by all authors.[16]

Examples edit

Functions of one variable edit

  • The function   has  , so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
  • The function   has  , 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   is convex (as reflected in the triangle inequality), even though it does not have a derivative at the point   It is not strictly convex.
  • The function   for   is convex.
  • The exponential function   is convex. It is also strictly convex, since  , but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function   is logarithmically convex if   is a convex function. The term "superconvex" is sometimes used instead.[18]
  • The function   with domain [0,1] defined by   for   is convex; it is continuous on the open interval   but not continuous at 0 and 1.
  • The function   has second derivative  ; thus it is convex on the set where   and concave on the set where  
  • Examples of functions that are monotonically increasing but not convex include   and  .
  • Examples of functions that are convex but not monotonically increasing include   and  .
  • The function   has   which is greater than 0 if   so   is convex on the interval  . It is concave on the interval  .
  • The function   with  , is convex on the interval   and convex on the interval  , but not convex on the interval  , because of the singularity at  

Functions of n variables edit

  • LogSumExp function, also called softmax function, is a convex function.
  • The function   on the domain of positive-definite matrices is convex.[7]: 74 
  • Every real-valued linear transformation is convex but not strictly convex, since if   is linear, then  . This statement also holds if we replace "convex" by "concave".
  • Every real-valued affine function, that is, each function of the form   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.[19]

See also edit

Notes edit

  1. ^ "Lecture Notes 2" (PDF). Retrieved 3 March 2017.
  2. ^ "Concave Upward and Downward". Archived from the original on 2013-12-18.
  3. ^ Stewart, James (2015). Calculus (8th ed.). Cengage Learning. pp. 223–224. ISBN 978-1305266643.
  4. ^ W. Hamming, Richard (2012). Methods of Mathematics Applied to Calculus, Probability, and Statistics (illustrated ed.). Courier Corporation. p. 227. ISBN 978-0-486-13887-9. Extract of page 227
  5. ^ Uvarov, Vasiliĭ Borisovich (1988). Mathematical Analysis. Mir Publishers. p. 126-127. ISBN 978-5-03-000500-3.
  6. ^ Prügel-Bennett, Adam (2020). The Probability Companion for Engineering and Computer Science (illustrated ed.). Cambridge University Press. p. 160. ISBN 978-1-108-48053-6. Extract of page 160
  7. ^ a b Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
  8. ^ Donoghue, William F. (1969). Distributions and Fourier Transforms. Academic Press. p. 12. ISBN 9780122206504. Retrieved August 29, 2012.
  9. ^ "If f is strictly convex in a convex set, show it has no more than 1 minimum". Math StackExchange. 21 Mar 2013. Retrieved 14 May 2016.
  10. ^ Altenberg, L., 2012. Resolvent positive linear operators exhibit the reduction phenomenon. Proceedings of the National Academy of Sciences, 109(10), pp.3705-3710.
  11. ^ "Strong convexity · Xingyu Zhou's blog". Retrieved 2023-09-27.
  12. ^ Dimitri Bertsekas (2003). Convex Analysis and Optimization. Contributors: Angelia Nedic and Asuman E. Ozdaglar. Athena Scientific. p. 72. ISBN 9781886529458.
  13. ^ Philippe G. Ciarlet (1989). Introduction to numerical linear algebra and optimisation. Cambridge University Press. ISBN 9780521339841.
  14. ^ a b Yurii Nesterov (2004). Introductory Lectures on Convex Optimization: A Basic Course. Kluwer Academic Publishers. pp. 63–64. ISBN 9781402075537.
  15. ^ Nemirovsky and Ben-Tal (2023). "Optimization III: Convex Optimization" (PDF).
  16. ^ a b C. Zalinescu (2002). Convex Analysis in General Vector Spaces. World Scientific. ISBN 9812380671.
  17. ^ a b H. Bauschke and P. L. Combettes (2011). Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer. p. 144. ISBN 978-1-4419-9467-7.
  18. ^ Kingman, J. F. C. (1961). "A Convexity Property of Positive Matrices". The Quarterly Journal of Mathematics. 12: 283–284. doi:10.1093/qmath/12.1.283.
  19. ^ 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.

References edit

  • 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.
  • Lauritzen, Niels (2013). Undergraduate Convexity. World Scientific Publishing.
  • 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. (2002). Convex analysis in general vector spaces. River Edge, NJ: World Scientific Publishing  Co., Inc. pp. xx+367. ISBN 981-238-067-1. MR 1921556.

External links edit