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| [[Image:ConvexFunction.svg|thumb|300px|right|Convex function on an interval.]]
| | You understand or the doctor has told we that we have hemorrhoids: today what is the number one hemorrhoid treatment. What is how to do away with hemorrhoids?<br><br>There are two issues associated with using creams. The initially is that some folks experience burning sensations, occasionally thus bad which you need to discontinue the utilization of the cream. The second problem is that lotions do not treat the underlying problems which cause hemorrhoids; therefore creams are a temporary [http://hemorrhoidtreatmentfix.com/prolapsed-hemorrhoid-treatment prolapsed hemorrhoids treatment].<br><br>Because bowel disorders are the most normal cause of hemorrhoids, you need to do something to enhance the health of the digestive system. One great method is to eat healthy. Prepare a thick gravy of rice plus blend this with a full glass of buttermilk and sliced bananas. Have this for breakfast and for an afternoon snack. Not just is this filling and nutritious, it's also fairly yummy. Add glucose to taste. We must see results in a week.<br><br>If you find going to the bathroom for a bowel movement fairly difficult think about a stool softener. These could function well and lower the need for straining. These could be bought conveniently over the counter plus can be highly powerful. If you would like anything a little more natural we might wish to consider eating prunes or drinking prune juice.<br><br>Another tip which would offer certain immediate relief to the hemorrhoid problem is chatting a nice warm shower. The bath will sooth the pain we feel, and in the event you add a little salt into your bath, about a teaspoon or thus, plus massage the hemorrhoid this might even further help with pain relief.<br><br>Or, try to apply phenylephrine or Preparation H to the region where we have hemorrhoid. According to certain experts, use of the ointment is moreover rather powerful. It could actually constrict the blood vessels plus reduce the redness plus all.<br><br>Whenever utilizing a sitz bath tub, we can utilize special soaps plus lotions that is created to be use with all the bath. This could aid treat additional symptoms of the hemorrhoids, also. You are able to choose up any sitz tub at the localized health store or you are able to buy one online. |
| [[Image:Epigraph convex.svg|right|thumb|300px|A function (in black) is convex if and only if the region above its [[Graph of a function|graph]] (in green) is a [[convex set]].]]
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| In [[mathematics]], a [[real-valued function]] <math>f(x)</math> defined on an [[interval (mathematics)|interval]] is called '''convex''' (or '''convex downward''' or '''concave upward''') if the [[line segment]] between any two points on the [[graph of a function|graph of the function]] lies above the graph, in a [[Euclidean space]] (or more generally a [[vector space]]) of at least two dimensions. Equivalently, a function is convex if its [[epigraph (mathematics)|epigraph]] (the set of points on or above the graph of the function) is a [[convex set]]. Well-known examples of convex functions are the [[quadratic function]] <math>f(x)=x^2</math> and the [[exponential function]] <math>f(x)=e^x</math> for any real number ''x''.
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| 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]]).
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| Exponential growth is a special case of convexity. [[Exponential growth]] narrowly means "increasing at a rate ''proportional'' to the current value", while convex growth generally means "increasing at an increasing rate (but not necessarily proportionally to current value)".
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| ==Definition==
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| A [[real number|real]] valued function {{nowrap|''f'' : ''X'' → '''R'''}} defined on a [[convex set]] ''X'' in a [[vector space]] is called ''convex'' if, for any two points <math>x_1</math> and <math> x_2 </math> in ''X'' and any <math>t\in[0,1]</math>,
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| :<math>f(tx_1+(1-t)x_2)\leq t f(x_1)+(1-t)f(x_2).</math>
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| The function is called '''strictly convex''' if | |
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| :<math>f(tx_1+(1-t)x_2) < t f(x_1)+(1-t)f(x_2)\,</math>
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| for every <math>t\,</math>, <math> 0 < t < 1\, </math>, and <math>x_1 \not=x_2</math>.
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| A function ''f'' is said to be (strictly) [[concave function|concave]] if −''f'' is (strictly) convex.
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| == Properties ==
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| Suppose ''f'' is a function of one [[real number|real]] variable defined on an interval, and let
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| :<math> R(x_1,x_2) = \frac{f(x_1) - f(x_2)}{x_1 - x_2}</math>
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| (note that <math>R(x_1,x_2)</math> is the slope of the purple line in the above drawing; note also that the function ''R'' is symmetric in <math>x_1,x_2</math>). ''f'' is convex if and only if <math>R(x_1,x_2)</math> is [[monotonically non-decreasing]] in <math>x_1</math>, for <math>x_2</math> fixed (or viceversa). This characterization of convexity is quite useful to prove the following results.
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| A convex function ''f'' defined on some [[open interval]] ''C'' is [[continuous function|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 function|differentiable]] at all but at most [[countable|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).
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| A function is midpoint convex on an interval ''C'' if
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| :<math>f\left( \frac{x_1+x_2}{2} \right) \le \frac{f(x_1)+f(x_2)}{2}</math>
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| for all <math>x_1</math> and <math>x_2</math> 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 by [[Sierpinski Theorem]].<ref>{{cite book|last=Donoghue|first=William F.|title=Distributions and Fourier Transforms|year=1969|publisher=Academic Press|isbn=9780122206504|url=http://books.google.com/books?id=P30Y7daiGvQC&pg=PA12|accessdate=August 29, 2012|page=12}}</ref> In particular, a continuous function that is midpoint convex will be convex.
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| 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]]. For the basic case of a differentiable function from (a subset of) the real numbers to the real numbers, "convex" is equivalent to "increasing at an increasing rate".
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| A [[continuously differentiable]] function of one variable is convex on an interval if and only if the function lies above all of its [[tangent]]s:
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| :<math>f(x) \geq f(y) + f'(y)(x-y)</math><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=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011}}</ref>{{rp|69}}
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| for all ''x'' and ''y'' in the interval. In particular, if ''f'' '(''c'') = 0, then ''c'' is a [[global minimum]] of ''f''(''x'').
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| 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).
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| 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 ''f''(''x'') = ''x''<sup>4</sup> is ''f'' "(''x'') = 12 ''x''<sup>2</sup>, which is zero for ''x'' = 0, but ''x''<sup>4</sup> is strictly convex.
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| 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-definite matrix|positive semidefinite]] on the interior of the convex set.
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| Any [[local minimum]] of a convex function is also a [[global minimum]]. A ''strictly'' convex function will have at most one global minimum.
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| For a convex function ''f'', the [[sublevel set]]s {''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]]''.
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| [[Jensen's inequality]] applies to every convex function ''f''. If ''X'' is a random variable taking values in the domain of ''f'', then <math>\operatorname{E}(f(X)) \geq f(\operatorname{E}(X)). </math> (Here <math>\operatorname{E}</math> denotes the [[Expected value|mathematical expectation]].)
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| If a function ''f'' is convex, and ''f''(0) ≤ 0, then ''f'' is [[superadditivity|superadditive]] on the positive half-axis. Proof:
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| * since ''f'' is convex, let ''y'' = 0, <math>f(tx) = f(tx+(1-t)\cdot 0) \le t f(x)+(1-t)f(0) \le t f(x)</math> for every <math>t\in\left[0,1\right]</math>
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| * <math>f(a) + f(b) = f \left((a+b) \frac{a}{a+b} \right) + f \left((a+b) \frac{b}{a+b} \right)
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| \le \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) = f(a+b)</math>
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| ==Convex function calculus==
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| *If <math>f</math> and <math>g</math> are convex functions, then so are <math>m(x) = \max\{f(x),g(x)\}</math> and <math>h(x) = f(x) + g(x).</math>
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| *If <math>f</math> and <math>g</math> are convex functions and <math>g</math> is non-decreasing, then <math>h(x) = g(f(x))</math> is convex. As an example, if <math>f(x)</math> is convex, then so is <math>e^{f(x)}</math>, because <math>e^x</math> is convex and monotonically increasing.
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| *If <math>f</math> is concave and <math>g</math> is convex and non-increasing, then <math>h(x) = g(f(x))</math> is convex.
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| *Convexity is invariant under affine maps: that is, if <math>f(x)</math> is convex with <math>x\in\mathbb{R}^n</math>, then so is <math>g(x) = f(Ax+b)</math>, where <math>A\in\mathbb{R}^{n \times n},\; b\in\mathbb{R}^n.</math>
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| *If <math>f(x,y)</math> is convex in <math>x</math> then <math>g(x) = \sup_{y\in C} f(x,y)</math> is convex in <math>x, </math> provided <math>g(x) > -\infty</math> for some <math>x.</math>
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| *If <math>f(x)</math> is convex, then its perspective <math>g(x, t) = t f(x/t)</math> (whose domain is <math>\left\lbrace (x, t) | x/t \in Dom(f), t > 0 \right\rbrace</math>) is convex.
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| *The [[additive inverse]] of a convex function is a [[concave function]].
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| ==Strongly convex functions==
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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.
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| 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:<ref name="bertsekas">{{cite book|page=72|title=Convex Analysis and Optimization|author=Dimitri Bertsekas|others=Contributors: Angelia Nedic and Asuman E. Ozdaglar|publisher=Athena Scientific|year=2003|isbn=9781886529458}}</ref>
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| :<math> ( \nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2 </math>
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| or, more generally,
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| :<math> \langle \nabla f(x) - \nabla f(y), (x-y) \rangle \ge m \|x-y\|^2 </math>
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| where <math>\|\cdot\|</math> is any [[Norm_(mathematics)|norm]].
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| Some authors, such as <ref name="ciarlet">{{cite book|title=Introduction to numerical linear algebra and optimisation|author=Philippe G. Ciarlet|publisher=Cambridge University Press|year=1989|isbn=9780521339841}}</ref>
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| refer to functions satisfying this inequality as [[Elliptic operator|elliptic]] functions.
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| An equivalent condition is the following:<ref name="nesterov">{{cite book|pages=63–64|title=Introductory Lectures on Convex Optimization: A Basic Course|author=Yurii Nesterov|publisher=Kluwer Academic Publishers|year=2004|isbn=9781402075537}}</ref>
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| :<math> f(y) \ge f(x) + \nabla f(x)^T (y-x) + \frac{m}{2} \|y-x\|_2^2 </math>
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| 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 ''m'', is that, for all ''x'',''y'' in the domain and <math>t\in [0,1]</math>,
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| :<math>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>
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| Notice that this definition approaches the definition for strict convexity as <math>m \rightarrow 0</math>, 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).
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| If the function ''f'' is twice continuously differentiable, then ''f'' is strongly convex with parameter ''m'' if and only if <math> \nabla^2 f(x) \succeq m I</math> for all ''x'' in the domain, where ''I'' 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 ''m'' for all ''x''. 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 ''m'' = 0, 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.
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| 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. Start by using [[Taylor's theorem|Taylor's Theorem]]:
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| :<math> f(y) = f(x) + \nabla f(x)^T (y-x) + 1/2 (y-x)^T \nabla^2f(z) (y-x) </math>
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| for some (unknown) <math> z \in [x,y] </math>.
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| Then <math>(y-x)^T \nabla^2f(z) (y-x) \ge m (y-x)^T(y-x) </math> by the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.
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| A function ''f'' is strongly convex with parameter ''m'' if and only if the function <math> x\mapsto f(x) - m/2\|x\|^2</math> is convex.
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| The distinction between convex, strictly convex, and strongly convex can be subtle at first glimpse. If <math>f</math> is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:
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| : <math>f\,\!</math> convex if and only if <math> f''(x) \ge 0 </math> for all <math>x\,\!</math>
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| : <math>f\,\!</math> strictly convex if <math> f''(x) > 0 \,\!</math> for all <math>x\,\!</math> (note: this is sufficient, but not necessary)
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| : <math>f\,\!</math> strongly convex if and only if <math> f''(x) \ge m > 0 </math> for all <math>x\,\!</math>
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| For example, consider a function <math>f</math> that is strictly convex, and suppose there is a sequence of points <math>(x_n)</math> such that <math>f''(x_n) = \frac{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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| 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.
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| ===Uniformly convex functions===
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| 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>
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| <ref name="Bauschke">{{cite book|page=144|title=Convex Analysis and Monotone Operator Theory in Hilbert Spaces|author=H. Bauschke and P. L. Combettes|publisher=Springer|year=2011|isbn=978-1-4419-9467-7}}</ref>
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| with modulus <math>\phi</math>, is a function ''f'' that, for all ''x'',''y'' in the domain and {{nowrap| ''t'' ∈ [0, 1]}}, satisfies
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| :<math>f(tx+(1-t)y) \le t f(x)+(1-t)f(y) - t(1-t) \phi(\|x-y\|), \,</math>
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| where <math>\phi</math> is a function that is increasing and vanishes only at 0. This is a generalization of the concept of strongly convex function; by taking <math>\phi(\alpha) = \frac{m}{2} \alpha^2</math> we recover the definition of strong convexity. | |
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| ==Examples==
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| * The function <math>f(x)=x^2</math> has <math>f''(x)=2>0</math> at all points, so ''f'' is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2.
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| * The function <math>f(x)=x^4</math> has <math>f''(x)=12x^2\ge0</math>, 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.
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| * 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 ''x'' = 0. It is not strictly convex.
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| * The function <math>f(x)=|x|^p</math> for 1 ≤ ''p'' is convex.
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| * 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 ''f'' is a convex function. The term "superconvex" is sometimes used instead.<ref>{{cite doi|10.1093/qmath/12.1.283}}</ref>
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| * 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.
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| * The function ''x''<sup>3</sup> has second derivative 6''x''; thus it is convex on the set where ''x'' ≥ 0 and [[concave function|concave]] on the set where ''x'' ≤ 0.
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| * The function <math>-\log\text{det}(X)</math> on the domain of [[Positive-definite matrix|positive-definite matrices]] is convex.<ref name="boyd" />{{rp|74}}
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| * Every [[linear transformation]] taking values in <math>\mathbb{R}</math> is convex but not strictly convex, since if ''f'' is linear, then <math>f(a + b) = f(a) + f(b).</math> This statement also holds if we replace "convex" by "concave".
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| * Every [[affine function]] taking values in <math>\mathbb{R}</math>, i.e., each function of the form <math>f(x) = a^T x + b </math>, is simultaneously convex and concave.
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| * Every [[norm (mathematics)|norm]] is a convex function, by the [[triangle inequality]] and [[Homogeneous function#Positive homogeneity|positive homogeneity]].
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| * Examples of functions that are [[Monotonic function|monotonically increasing]] but not convex include <math>f(x) = \sqrt x</math> and ''g''(''x'') = log(''x'').
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| * 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>.
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| * The function ''f''(''x'') = 1/''x'' has <math>f\,''(x)=\frac{2}{x^3}</math> which is greater than 0 if ''x'' > 0, so ''f''(''x'') is convex on the interval (0, +∞). It is concave on the interval (-∞,0).
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| * The function ''f''(''x'') = 1/''x''<sup>2</sup>, 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.
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| ==See also==
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| * [[Concave function]]
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| * [[Convex optimization]]
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| * [[Convex conjugate]]
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| * [[Geodesic convexity]]
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| * [[Kachurovskii's theorem]], which relates convexity to [[monotone operator|monotonicity]] of the derivative
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| * [[Logarithmically convex function]]
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| * [[Pseudoconvex function]]
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| * [[Quasiconvex function]]
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| * [[Invex function]]
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| * [[Subderivative]] of a convex function
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| * [[Jensen's inequality]]
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| * [[Karamata's inequality]]
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| * [[Hermite–Hadamard inequality]]
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| ==Notes==
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| {{Reflist}}<!--added under references heading by script-assisted edit-->
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| ==References==
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| * {{cite book
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| | last = Bertsekas
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| | first = Dimitri
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| | authorlink= Dimitri Bertsekas
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| | title = Convex Analysis and Optimization
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| | publisher = Athena Scientific
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| | year = 2003
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| }}
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| * [[Jonathan M. Borwein|Borwein, Jonathan]], and Lewis, Adrian. (2000). Convex Analysis and Nonlinear Optimization. Springer.
| |
| * {{cite book
| |
| | last = Donoghue
| |
| | first = William F.
| |
| | title = Distributions and Fourier Transforms
| |
| | publisher = Academic Press
| |
| | year = 1969
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| }}
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| * Hiriart-Urruty, Jean-Baptiste, and [[Claude Lemaréchal|Lemaréchal, Claude]]. (2004). Fundamentals of Convex analysis. Berlin: Springer.
| |
| *{{cite book | author =[[Mark Krasnosel'skii|Krasnosel'skii M.A.]], Rutickii Ya.B. | title=Convex Functions and Orlicz Spaces | publisher= P.Noordhoff Ltd | location=Groningen | year=1961}}
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| * {{cite book
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| | last = Lauritzen
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| | first = Niels
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| | title = Undergraduate Convexity
| |
| | publisher = World Scientific Publishing
| |
| | year = 2013
| |
| }}
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| * {{cite book
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| | last = Luenberger
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| | first = David
| |
| | authorlink = David Luenberger
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| | title = Linear and Nonlinear Programming
| |
| | publisher = Addison-Wesley
| |
| | year = 1984
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| }}
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| * {{cite book
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| | last = Luenberger
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| | first = David
| |
| | authorlink = David Luenberger
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| | title = Optimization by Vector Space Methods
| |
| | publisher = Wiley & Sons
| |
| | year = 1969
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| }}
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| <!-- * {{cite web
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| | last = Moon
| |
| | first = Todd
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| | title = Tutorial: Convexity and Jensen's inequality
| |
| | url=http://www.neng.usu.edu/classes/ece/7680/lecture2/node5.html
| |
| | accessdate = 2008-09-04
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| }} -->
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| * {{cite book
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| | last = Rockafellar
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| | first = R. T.
| |
| | authorlink= R. Tyrrell Rockafellar
| |
| | title = Convex analysis
| |
| | publisher = Princeton University Press
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| | year = 1970
| |
| | location = Princeton
| |
| }}
| |
| * {{cite book
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| | last = Thomson
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| | first = Brian
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| | title = Symmetric Properties of Real Functions
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| | publisher = CRC Press
| |
| | year = 1994
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| }}
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| * {{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific Publishing Co., Inc|location=River Edge, NJ|year=2002|pages=xx+367|isbn=981-238-067-1|mr=1921556}}
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| ==External links==
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| * Stephen Boyd and Lieven Vandenberghe, [http://www.stanford.edu/~boyd/cvxbook/ Convex Optimization] (PDF)
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| * {{springer|title=Convex function (of a real variable)|id=p/c026240}}
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| * {{springer|title=Convex function (of a complex variable)|id=p/c026230}}
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| [[Category:Types of functions]]
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| [[Category:Convex analysis]]
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| [[Category:Generalized convexity]]
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