Convex function: Difference between revisions

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<li>For all <math>0 \leq t \leq 1</math> and all <math>x_1, x_2 \in X</math>:

<math display=block>f\left( t x_1 + (1-t) x_2 \right) \leq t f\left( x_1 \right) + (1-t) f\left( x_2 \right)</math>

The right hand side represents the straight line between <math>\left( x_1, f\left(x_1\right) \right)</math> and <math>\left( x_2, f\left(x_2\right) \right)</math> in the graph of <math>f</math> as a function of <math>t;</math> increasing <math>t</math> from <math>0</math> to <math>1</math> or decreasing <math>t</math> from <math>1</math> to <math>0</math> sweeps this line. Similarly, the argument of the function <math>f</math> in the left hand side represents the straight line between <math>x_1</math> and <math>x_2</math> in <math>X</math> or the <math>x</math>-axis of the graph of <math>f.</math> So, this condition requires that the straight line between any pair of points on the curve of <math>f</math> to be above or just meets the graph.<ref>{{Cite web|last=|first=|date=|title=Concave Upward and Downward|url=https://www.mathsisfun.com/calculus/concave-up-down-convex.html|url-status=live|archive-url=https://web.archive.org/web/20131218034748/http://www.mathsisfun.com:80/calculus/concave-up-down-convex.html |archive-date=2013-12-18 |access-date=|website=}}</ref>

</li>

<li>For all <math>0 < t < 1</math> and all <math>x_1, x_2 \in X</math> such that <math>x_1 \neq x_2</math>:

<math display=block>f\left( t x_1 + (1-t) x_2 \right) \leq t f\left( x_1 \right) + (1-t) f\left( x_2 \right)</math>

The difference of this second condition with respect to the first condition above is that this condition does not include the intersection points (e.g.for example, <math>\left( x_1, f\left(x_1\right) \right)</math> and <math>\left( x_2, f\left(x_2\right) \right)</math>) between the straight line passing through a pair of points on the curve of <math>f</math> (the straight line is represented by the right hand side of this condition) and the curve of <math>f;</math> the first condition includes the intersection points as it becomes <math>f\left(x_1\right) \leq f\left(x_1\right)</math> or <math>f\left(x_2\right) \leq f\left(x_2\right)</math> at <math>t = 0</math> or <math>1,</math> or <math>x_1 = x_2.</math> In fact, the intersection points do not need to be considered in a condition of convex using <math display=block>f\left( t x_1 + (1-t) x_2 \right) \leq t f\left( x_1 \right) + (1-t) f\left( x_2 \right)</math> because <math>f\left(x_1\right) \leq f\left(x_1\right)</math> and <math>f\left(x_2\right) \leq f\left(x_2\right)</math> are always true (so not useful to be a part of a condition).

</li>

</ol>

The second statement characterizing convex functions that are valued in the real line <math>\R</math> is also the statement used to define '''{{em|convex functions}}''' that are valued in the [[extended real number line]] <math>[-\infty, \infty] = \R \cup \{ \pm\infty \},</math> where such a function <math>f</math> is allowed to (but is not required to) take <math>\pm\infty</math> as a value. The first statement is not used because it permits <math>t</math> to take <math>0</math> or <math>1</math> as a value, in which case, if <math>f\left( x_1 \right) = \pm\infty</math> or <math>f\left( x_2 \right) = \pm\infty,</math> respectively, then <math>t f\left( x_1 \right) + (1 - t) f\left( x_2 \right)</math> would be undefined (because the multiplications <math>0 \cdot \infty</math> and <math>0 \cdot (-\infty)</math> are undefined). The sum <math>-\infty + \infty</math> is also undefined so a convex extended real-valued function is typically only allowed to take exactly one of <math>-\infty</math> and <math>+\infty</math> 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 <math>\,\leq\,</math> with the strict inequality <math>\,<.</math>

Explicitly, the map <math>f</math> is called '''{{em|strictly convex}}''' if and only if for all real <math>0 < t < 1</math> and all <math>x_1, x_2 \in X</math> such that <math>x_1 \neq x_2</math>:

<math display=block>f\left( t x_1 + (1-t) x_2 \right) < t f\left( x_1 \right) + (1-t) f\left( x_2 \right)</math>

A strictly convex function <math>f</math> is a function that the straight line between any pair of points on the curve <math>f</math> is above the curve <math>f</math> except for the intersection points between the straight line and the curve.

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* Suppose <math>f</math> is a function of one [[real number|real]] variable defined on an interval, and let <math display=block>R(x_1, x_2) = \frac{f(x_2) - f(x_1)}{x_2 - x_1}</math> (note that <math>R(x_1, x_2)</math> is the slope of the purple line in the above drawing; the function <math>R</math> is [[Symmetric function|symmetric]] in <math>(x_1, x_2),</math> means that <math>R</math> does not change by exchanging <math>x_1</math> and <math>x_2</math>). <math>f</math> is convex if and only if <math>R(x_1, x_2)</math> is [[monotonically non-decreasing]] in <math>x_1,</math> for every fixed <math>x_2</math> (or vice versa). This characterization of convexity is quite useful to prove the following results.

* A convex function <math>f</math> of one real variable defined on some [[open interval]] {{mvar|<math>C}}</math> is [[Continuous function|continuous]] on <math>C.</math> <math>f</math> admits [[Semi-differentiability|left and right derivatives]], and these are [[monotonically non-decreasing]]. As a consequence, <math>f</math> is [[differentiable function|differentiable]] at all but at most [[countable|countably many]] points, the set on which <math>f</math> is not differentiable can however still be dense. If <math>C</math> is closed, then <math>f</math> may fail to be continuous at the endpoints of <math>C</math> (an example is shown in the [[#Examples|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|differentiable]] function of one variable is convex on an interval if and only if its graph lies above all of its [[tangent]]s:<ref name="boyd">{{cite book| title=Convex Optimization| first1=Stephen P.|last1=Boyd |first2=Lieven| last2=Vandenberghe | year = 2004 |publisher=Cambridge University Press| isbn=978-0-521-83378-3| url= https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=83 |format=pdf | access-date=October 15, 2011}}</ref>{{rp|69}} <math display=block>f(x) \geq f(y) + f'(y) (x-y)</math> for all ''<math>x''</math> and ''<math>y''</math> 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 point]]s). If its second derivative is positive at all points then the function is strictly convex, but the [[Theorem#Converse|converse]] does not hold. For example, the second derivative of <math>f(x) = x^4</math> is <math>f''(x) = 12x^{2}</math>, which is zero for <math>x = 0,</math> but <math>x^4</math> is strictly convex.

**This property and the above property in terms of "...its derivative is monotonically non-decreasing..." are not equal since if <math>f''</math> is non-negative on an interval <math>X</math> then <math>f'</math> is monotonically non-decreasing on <math>X</math> while its converse is not true, for example, <math>f'</math> is monotonically non-decreasing on <math>X</math> while its derivative <math>f''</math> is not defined at some points on <math>X</math>.

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}}

* A function is midpoint convex on an interval <math>C</math> if for all <math>x_1, x_2 \in C</math> <math display=block>f\left( \frac{x_1 + x_2}{2} \right) \leq \frac{f(x_1) + f(x_2)}{2}.</math> 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 [[Wacław Sierpiński|Sierpinski]].<ref>{{cite book|last=Donoghue|first=William F.| title= Distributions and Fourier Transforms|year=1969|publisher=Academic Press | isbn=9780122206504 |url= https://books.google.com/books?id=P30Y7daiGvQC&pg=PA12|access-date=August 29, 2012|page=12}}</ref> In particular, a continuous function that is midpoint convex will be convex.

=== Functions of several variables ===

* A function <math>f : X \to [-\infty, \infty]</math> valued in the [[extended real number]]s <math>[-\infty, \infty] = \R \cup \{ \pm\infty \}</math> is convex if and only if its [[Epigraph (mathematics)|epigraph]] <math display=block>\{ (x, r) \in X \times \R ~:~ r \geq f(x) \}</math> is a convex set.

* A differentiable function <math>f</math> defined on a convex ___domain is convex if and only if <math>f(x) \geq f(y) + \nabla f(y) \cdot (x-y)</math> holds for all <math>x, y</math> 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 derivative]]s is [[Positive-definite matrix|positive semidefinite]] on the interior of the convex set.

* For a convex function <math>f,</math> the [[sublevel set]]s <math>\{ x : f(x) < a \}</math> and <math>\{ x : f(x) \leq a \}</math> with <math>a \in \R</math> are convex sets. A function that satisfies this property is called a '''{{em|[[quasiconvex function]]}}''' and may fail to be a convex function.

* Consequently, the set of [[Arg min|global minimisers]] of a convex function <math>f</math> is a convex set: <math>{\operatorname{argmin}}\,f</math> - convex.

* Any [[local minimum]] of a convex function is also a [[global minimum]]. A {{em|strictly}} convex function will have at most one global minimum.<ref>{{cite web | url=https://math.stackexchange.com/q/337090 | title=If f is strictly convex in a convex set, show it has no more than 1 minimum | publisher=Math StackExchange | date=21 Mar 2013 | access-date=14 May 2016}}</ref>

* [[Jensen's inequality]] applies to every convex function <math>f</math>. If <math>X</math> is a random variable taking values in the ___domain of <math>f,</math> then <math>\operatorname{E}(f(X)) \geq f(\operatorname{E}(X)),</math> where <math>\operatorname{{math|E}}</math> denotes the [[Expected value|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 <math>x</math> and <math>y,</math> (that is, a function satisfying <math>f(a x, a y) = a f(x, y)</math> for all positive real <math>a, x, y > 0</math>) that is convex in one variable must be convex in the other variable.<ref>Altenberg, L., 2012. Resolvent positive linear operators exhibit the reduction phenomenon. Proceedings of the National Academy of Sciences, 109(10), pp.3705-3710.</ref>

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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 <math>f</math> is called strongly convex with parameter {{<math|''>m'' > 0}}</math> if the following inequality holds for all points {{<math|''>x'', ''y''}}</math> in its ___domain:<ref name="bertsekas">{{cite book|page=[https://archive.org/details/convexanalysisop00bert_476/page/n87 72]|title=Convex Analysis and Optimization|url=https://archive.org/details/convexanalysisop00bert_476|url-access=limited|author=Dimitri Bertsekas| others= Contributors: Angelia Nedic and Asuman E. Ozdaglar|publisher=Athena Scientific|year=2003|isbn=9781886529458}}</ref>

<math display=block>(\nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2 </math>

or, more generally,

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<math display=block>f(y) \ge f(x) + \nabla f(x)^T (y-x) + \frac{m}{2} \|y-x\|_2^2 </math>

It is not necessary for a function to be differentiable in order to be strongly convex. A third definition<ref name="nesterov"/> for a strongly convex function, with parameter ''<math>m'',</math> is that, for all ''<math>x'', ''y''</math> in the ___domain and <math>t \in [0,1],</math>

<math display=block>f(tx+(1-t)y) \le t f(x)+(1-t)f(y) - \frac{1}{2} m t(1-t) \|x-y\|_2^2</math>

Notice that this definition approaches the definition for strict convexity as ''<math>m'' →\to 0,</math> and is identical to the definition of a convex function when ''<math>m'' = 0.</math> Despite this, functions exist that are strictly convex but are not strongly convex for any ''<math>m'' > 0</math> (see example below).

If the function <math>f</math> is twice continuously differentiable, then it is strongly convex with parameter ''<math>m''</math> if and only if <math>\nabla^2 f(x) \succeq mI</math> for all ''<math>x''</math> in the ___domain, where ''<math>I''</math> is the identity and <math>\nabla^2f</math> is the [[Hessian matrix]], and the inequality <math>\succeq</math> means that <math>\nabla^2 f(x) - mI</math> is [[Positive-definite matrix|positive semi-definite]]. This is equivalent to requiring that the minimum [[eigenvalue]] of <math>\nabla^2 f(x)</math> be at least ''<math>m''</math> for all ''<math>x''.</math> If the ___domain is just the real line, then <math>\nabla^2 f(x)</math> is just the second derivative <math>f''(x),</math> so the condition becomes <math>f''(x) \ge m</math>. If ''<math>m'' = 0,</math> then this means the Hessian is positive semidefinite (or if the ___domain is the real line, it means that <math>f''(x) \ge 0</math>), 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 <math>\nabla^2 f(x)</math> implies that it is strongly convex. Using [[Taylor's theorem|Taylor's Theorem]] there exists

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The distinction between convex, strictly convex, and strongly convex can be subtle at first glance. If <math>f</math> is twice continuously differentiable and the ___domain is the real line, then we can characterize it as follows:

*<math>f</math> convex if and only if <math>f''(x) \ge 0</math> for all {{mvar|<math>x}}.</math>

*<math>f</math> strictly convex if <math>f''(x) > 0</math> for all {{mvar|<math>x}}</math> (note: this is sufficient, but not necessary).

*<math>f</math> strongly convex if and only if <math>f''(x) \ge m > 0</math> for all {{mvar|<math>x}}.</math>

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

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===Uniformly convex functions===

A uniformly convex function,<ref name="Zalinescu">{{cite book|title=Convex Analysis in General Vector Spaces|author=C. Zalinescu|publisher=World Scientific|year=2002|isbn=9812380671}}</ref><ref name="Bauschke">{{cite book|page=[https://archive.org/details/convexanalysismo00hhba/page/n161 144]|title=Convex Analysis and Monotone Operator Theory in Hilbert Spaces |url=https://archive.org/details/convexanalysismo00hhba|url-access=limited|author=H. Bauschke and P. L. Combettes |publisher=Springer |year=2011 |isbn=978-1-4419-9467-7}}</ref> with modulus <math>\phi</math>, is a function <math>f</math> that, for all ''<math>x'', ''y''</math> in the ___domain and {{<math|''>t'' ∈\in [0, 1]}},</math> satisfies

<math display=block>f(tx+(1-t)y) \le t f(x)+(1-t)f(y) - t(1-t) \phi(\|x-y\|) </math>

where <math>\phi</math> is a function that is non-negative and vanishes only at&nbsp;0. This is a generalization of the concept of strongly convex function; by taking <math>\phi(\alpha) = \tfrac{m}{2} \alpha^2</math> we recover the definition of strong convexity.

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* The function <math>f(x)=x^2</math> has <math>f''(x)=2>0</math>, so {{mvar|f}} is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.

* The function <math>f(x)=x^4</math> has <math>f''(x)=12x^2\ge 0</math>, 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 <math>f(x)=|x|</math> is convex (as reflected in the [[triangle inequality]]), even though it does not have a derivative at the point&nbsp;'' <math>x''&nbsp; =&nbsp; 0.</math> It is not strictly convex.

* The function <math>f(x)=|x|^p</math> for <math>p \ge 1</math> is convex.

* The [[exponential function]] <math>f(x)=e^x</math> is convex. It is also strictly convex, since <math>f''(x)=e^x >0 </math>, but it is not strongly convex since the second derivative can be arbitrarily close to zero. More generally, the function <math>g(x) = e^{f(x)}</math> is [[Logarithmically convex function|logarithmically convex]] if {{mvar|<math>f}}</math> is a convex function. The term "superconvex" is sometimes used instead.<ref>{{Cite journal | last1 = Kingman | first1 = J. F. C. | doi = 10.1093/qmath/12.1.283 | title = A Convexity Property of Positive Matrices | journal = The Quarterly Journal of Mathematics | volume = 12 | pages = 283–284 | year = 1961 }}</ref>

* The function <math>f</math> with ___domain [0,1] defined by <math>f(0) = f(1) = 1, f(x) = 0</math> for <math>0 < x < 1</math> is convex; it is continuous on the open interval <math>(0,&nbsp; 1),</math> but not continuous at 0 and&nbsp;1.

* The function ''x''<supmath>x^3</supmath> has second derivative <math>6'' x''</math>; thus it is convex on the set where ''<math>x'' ≥\geq 0</math> and [[concave function|concave]] on the set where&nbsp;'' <math>x''&nbsp;≤&nbsp; \leq 0.</math>

* Examples of functions that are [[Monotonic function|monotonically increasing]] but not convex include <math>f(x)=\sqrt{x}</math> and <math>g(x)=\log x</math>.

* Examples of functions that are convex but not [[Monotonic function|monotonically increasing]] include <math>h(x)= x^2</math> and <math>k(x)=-x</math>.

* The function <math>f(x) = \tfrac{1}{x}</math> has <math>f''(x)=\tfrac{2}{x^3}</math> which is greater than 0 if ''<math>x'' > 0,</math> so <math>f(x)</math> is convex on the interval <math>(0, \infty)</math>. It is concave on the interval <math>(-\infty, 0)</math>.

* The function <math>f(x)=\tfrac{1}{x^2}</math> with <math>f(0)=\infty</math>, is convex on the interval <math>(0, \infty)</math> and convex on the interval <math>(-\infty, 0)</math>, but not convex on the interval <math>(-\infty, \infty)</math>, because of the singularity at&nbsp;'' <math>x''&nbsp; =&nbsp; 0.</math>

===Functions of ''n'' variables===

* [[LogSumExp]] function, also called softmax function, is a convex function.

*The function <math>-\log\det(X)</math> on the ___domain of [[Positive-definite matrix|positive-definite matrices]] is convex.<ref name="boyd" />{{rp|74}}

* Every real-valued [[linear transformation]] is convex but not strictly convex, since if {{mvar|<math>f}}</math> is linear, then <math>f(a + b) = f(a) + f(b)</math>. This statement also holds if we replace "convex" by "concave".

* Every real-valued [[affine function]], i.e.that is, each function of the form <math>f(x) = a^T x + b,</math>, is simultaneously convex and concave.

* Every [[norm (mathematics)|norm]] is a convex function, by the [[triangle inequality]] and [[Homogeneous function#Positive homogeneity|positive homogeneity]].

* The [[spectral radius]] of a [[nonnegative matrix]] is a convex function of its diagonal elements.<ref>Cohen, J.E., 1981. [https://www.ams.org/journals/proc/1981-081-04/S0002-9939-1981-0601750-2/S0002-9939-1981-0601750-2.pdf Convexity of the dominant eigenvalue of an essentially nonnegative matrix]. Proceedings of the American Mathematical Society, 81(4), pp.657-658.</ref>

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* [[Convex optimization]]

* [[Geodesic convexity]]

* [[Hahn–Banach theorem]]

* [[Hermite–Hadamard inequality]]

* [[Invex function]]