{"history":[],"article":{"id":"b09d6b94-21de-4834-94e0-0b8f55744597","title":"Convex function","slug":"convex-function","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a [real-valued function](https://bliptext.com/articles/real-valued-function) is called **convex** if the [line segment](https://bliptext.com/articles/line-segment) between any two distinct points on the [graph of the function](https://bliptext.com/articles/graph-of-a-function) lies above or on the graph between the two points.\nEquivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a [convex set](https://bliptext.com/articles/convex-set).\nIn simple terms, a convex function graph is shaped like a cup \\cup (or a straight line like a linear function), while a [concave function](https://bliptext.com/articles/concave-function)'s graph is shaped like a cap \\cap.\nA twice-[differentiable](https://bliptext.com/articles/differentiable-function) function of a single variable is convex [if and only if](https://bliptext.com/articles/if-and-only-if) its [second derivative](https://bliptext.com/articles/second-derivative) is nonnegative on its entire [domain](https://bliptext.com/articles/domain-of-a-function).\nWell-known examples of convex functions of a single variable include a [linear function](https://bliptext.com/articles/linear-function) f(x) = cx (where c is a [real number](https://bliptext.com/articles/real-number)), a [quadratic function](https://bliptext.com/articles/quadratic-function) cx^2 (c as a nonnegative [real number](https://bliptext.com/articles/real-number)) and an [exponential function](https://bliptext.com/articles/exponential-function) ce^x (c as a nonnegative [real number](https://bliptext.com/articles/real-number)).\nConvex functions play an important role in many areas of mathematics.\nThey are especially important in the study of [optimization](https://bliptext.com/articles/optimization) problems where they are distinguished by a number of convenient properties.\nFor instance, a strictly convex function on an [open set](https://bliptext.com/articles/open-set) has no more than one [minimum](https://bliptext.com/articles/maximum-and-minimum).\nEven 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](https://bliptext.com/articles/calculus-of-variations).\nIn [probability theory](https://bliptext.com/articles/probability-theory), a convex function applied to the [expected value](https://bliptext.com/articles/expected-value) of a [random variable](https://bliptext.com/articles/random-variable) is always bounded above by the [expected value](https://bliptext.com/articles/expected-value) of the convex function of the [random variable](https://bliptext.com/articles/random-variable).\nThis result, known as [Jensen's inequality](https://bliptext.com/articles/jensen-s-inequality), can be used to deduce [inequalities](https://bliptext.com/articles/inequality-mathematics) such as the [arithmetic–geometric mean inequality](https://bliptext.com/articles/inequality-of-arithmetic-and-geometric-means) and [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality).\n# [Convex function on an interval. | upload.wikimedia.org/wikipedia/commons/c/c7/ConvexFunction.svg]\n# [A function (in black) is convex if and only if the region above its graph (in green) is a convex set. | upload.wikimedia.org/wikipedia/commons/3/31/Epigraph///convex.svg]\n# [A graph of the bivariate convex function x2 + xy + y2. | upload.wikimedia.org/wikipedia/commons/6/6e/Grafico///3d///x2%2Bxy%2By2.png]\n# [Convex vs. Not convex | upload.wikimedia.org/wikipedia/commons/e/ec/Convex///vs.///Not-convex.jpg]\n\n# Definition\nLet X be a [convex subset](https://bliptext.com/articles/convex-set) of a real [vector space](https://bliptext.com/articles/vector-space) and let f: X \\to \\R be a function.\n****T****h****e****n**** ****f**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****a****n****y**** ****o****f**** ****t****h****e**** ****f****o****l****l****o****w****i****n****g**** ****e****q****u****i****v****a****l****e****n****t**** ****c****o****n****d****i****t****i****o****n****s**** ****h****o****l****d****:****\n

For all and all :\nThe right hand side represents the straight line between and 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.\nSimilarly, 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.\n

For all 0 < t < 1 and all such that :\nThe 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 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 or at t = 0 or 1, or x_1 = x_2.\nIn 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).\n

\n\n****T****h****e**** ****s****e****c****o****n****d**** ****s****t****a****t****e****m****e****n****t**** ****c****h****a****r****a****c****t****e****r****i****z****i****n****g**** ****c****o****n****v****e****x**** ****f****u****n****c****t****i****o****n****s**** ****t****h****a****t**** ****a****r****e**** ****v****a****l****u****e****d**** ****i****n**** ****t****h****e**** ****r****e****a****l**** ****l****i****n****e**** ****\\****R**** ****i****s**** ****a****l****s****o**** ****t****h****e**** ****s****t****a****t****e****m****e****n****t**** ****u****s****e****d**** ****t****o**** ****d****e****f****i****n****e**** **** ****t****h****a****t**** ****a****r****e**** ****v****a****l****u****e****d**** ****i****n**** ****t****h****e**** ****e****x****t****e****n****d****e****d**** ****r****e****a****l**** ****n****u****m****b****e****r**** ****l****i****n****e**** ****w****h****e****r****e**** ****s****u****c****h**** ****a**** ****f****u****n****c****t****i****o****n**** ****f**** ****i****s**** ****a****l****l****o****w****e****d**** ****t****o**** ****t****a****k****e**** ****\\****p****m****\\****i****n****f****t****y**** ****a****s**** ****a**** ****v****a****l****u****e****.****\nThe first statement is not used because it permits t to take 0 or 1 as a value, in which case, if or respectively, then would be undefined (because the multiplications and are undefined).\nThe sum 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.\nThe second statement can also be modified to get the definition of, where the latter is obtained by replacing \\,\\leq\\, with the strict inequality \\,<.\n****E****x****p****l****i****c****i****t****l****y****,**** ****t****h****e**** ****m****a****p**** ****f**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****f****o****r**** ****a****l****l**** ****r****e****a****l**** ****0**** ****<**** ****t**** ****<**** ****1**** ****a****n****d**** ****a****l****l**** ****s****u****c****h**** ****t****h****a****t**** ****:****\nA strictly convex function f is a function that the straight line between any pair of points on the curve f is above the curve f except for the intersection points between the straight line and the curve.\nAn example of a function which is convex but not strictly convex is.\nThis 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.\nThe function f is said to be **' (resp. **') if -f (f multiplied by −1) is convex (resp. strictly convex).\n\n# Alternative naming\nThe term convex is often referred to as convex down or [concave](https://bliptext.com/articles/concave-function) upward, and the term [concave](https://bliptext.com/articles/concave-function) is often referred as [concave](https://bliptext.com/articles/concave-function) down or convex upward.\nIf the term \"convex\" is used without an \"up\" or \"down\" keyword, then it refers strictly to a cup shaped graph \\cup.\nAs an example, [Jensen's inequality](https://bliptext.com/articles/jensen-s-inequality) refers to an inequality involving a convex or convex-(down), function.\n\n# Properties\nMany properties of convex functions have the same simple formulation for functions of many variables as for functions of one variable.\nSee below the properties for the case of many variables, as some of them are not listed for functions of one variable.\n\n## Functions of one variable\n\n## Functions of several variables\n\n# Operations that preserve convexity\n\n# Strongly convex functions\nThe concept of strong convexity extends and parametrizes the notion of strict convexity.\nIntuitively, a strongly-convex function is a function that grows as fast as a quadratic function.\nA strongly convex function is also strictly convex, but not vice versa.\nIf a one-dimensional function f is twice continuously differentiable and the domain is the real line, then we can characterize it as follows:\nFor example, let f be strictly convex, and suppose there is a sequence of points (x_n) such that.\nEven though, the function is not strongly convex because f''(x) will become arbitrarily small.\nMore 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:\nor, more generally,\nwhere is any [inner product](https://bliptext.com/articles/inner-product), and \\|\\cdot\\| is the corresponding [norm](https://bliptext.com/articles/norm-mathematics).\nSome authors, such as refer to functions satisfying this inequality as [elliptic](https://bliptext.com/articles/elliptic-operator) functions.\nAn equivalent condition is the following:\nIt is not necessary for a function to be differentiable in order to be strongly convex.\nA third definition for a strongly convex function, with parameter m, is that, for all x, y in the domain and\nNotice 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.\nDespite this, functions exist that are strictly convex but are not strongly convex for any m > 0 (see example below).\nIf the function f is twice continuously differentiable, then it is strongly convex with parameter m if and only if for all x in the domain, where I is the identity and \\nabla^2f is the [Hessian matrix](https://bliptext.com/articles/hessian-matrix), and the inequality \\succeq means that is [positive semi-definite](https://bliptext.com/articles/positive-definite-matrix).\nThis is equivalent to requiring that the minimum [eigenvalue](https://bliptext.com/articles/eigenvalue) of be at least m for all x. If the domain is just the real line, then is just the second derivative f''(x), so the condition becomes.\nIf m = 0 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.\nAssuming still that the function is twice continuously differentiable, one can show that the lower bound of implies that it is strongly convex.\nUsing [Taylor's Theorem](https://bliptext.com/articles/taylor-s-theorem) there exists\nsuch that\nThen\nby the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.\nA function f is strongly convex with parameter m if and only if the function\nis convex.\nA twice continuously differentiable function f on a compact domain X that satisfies f''(x)>0 for all x\\in X is strongly convex.\nThe proof of this statement follows from the [extreme value theorem](https://bliptext.com/articles/extreme-value-theorem), which states that a continuous function on a compact set has a maximum and minimum.\nStrongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class.\nLike strictly convex functions, strongly convex functions have unique minima on compact sets.\n\n## Properties of strongly-convex functions\nIf f is a strongly-convex function with parameter m, then:\n\n# Uniformly convex functions\nA uniformly convex function, with modulus \\phi, is a function f that, for all x, y in the domain and satisfies\nwhere \\phi is a function that is non-negative and vanishes only at 0.\nThis is a generalization of the concept of strongly convex function; by taking we recover the definition of strong convexity.\nIt is worth noting that some authors require the modulus \\phi to be an increasing function, but this condition is not required by all authors.\n\n# Examples\n\n## Functions of one variable\n\n## Functions of n variables","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T18:51:44.373Z","updated_at":"2025-01-25T18:51:44.373Z","search_vector":"'convex':1 'function':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), a [real-valued function](https://bliptext.com/articles/real-valued-function) is called **convex** if the [line segment](https://bliptext.com/articles/line-segment) between any two distinct points on the [graph of the function](https://bliptext.com/articles/graph-of-a-function) lies above or on the graph between the two points.\nEquivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a [convex set](https://bliptext.com/articles/convex-set).\nIn simple terms, a convex function graph is shaped like a cup \\cup (or a straight line like a linear function), while a [concave function](https://bliptext.com/articles/concave-function)'s graph is shaped like a cap \\cap.\nA twice-[differentiable](https://bliptext.com/articles/differentiable-function) function of a single variable is convex [if and only if](https://bliptext.com/articles/if-and-only-if) its [second derivative](https://bliptext.com/articles/second-derivative) is nonnegative on its entire [domain](https://bliptext.com/articles/domain-of-a-function).\nWell-known examples of convex functions of a single variable include a [linear function](https://bliptext.com/articles/linear-function) f(x) = cx (where c is a [real number](https://bliptext.com/articles/real-number)), a [quadratic function](https://bliptext.com/articles/quadratic-function) cx^2 (c as a nonnegative [real number](https://bliptext.com/articles/real-number)) and an [exponential function](https://bliptext.com/articles/exponential-function) ce^x (c as a nonnegative [real number](https://bliptext.com/articles/real-number)).\nConvex functions play an important role in many areas of mathematics.\nThey are especially important in the study of [optimization](https://bliptext.com/articles/optimization) problems where they are distinguished by a number of convenient properties.\nFor instance, a strictly convex function on an [open set](https://bliptext.com/articles/open-set) has no more than one [minimum](https://bliptext.com/articles/maximum-and-minimum).\nEven 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](https://bliptext.com/articles/calculus-of-variations).\nIn [probability theory](https://bliptext.com/articles/probability-theory), a convex function applied to the [expected value](https://bliptext.com/articles/expected-value) of a [random variable](https://bliptext.com/articles/random-variable) is always bounded above by the [expected value](https://bliptext.com/articles/expected-value) of the convex function of the [random variable](https://bliptext.com/articles/random-variable).\nThis result, known as [Jensen's inequality](https://bliptext.com/articles/jensen-s-inequality), can be used to deduce [inequalities](https://bliptext.com/articles/inequality-mathematics) such as the [arithmetic–geometric mean inequality](https://bliptext.com/articles/inequality-of-arithmetic-and-geometric-means) and [Hölder's inequality](https://bliptext.com/articles/h-lder-s-inequality).\n# [Convex function on an interval. | upload.wikimedia.org/wikipedia/commons/c/c7/ConvexFunction.svg]\n# [A function (in black) is convex if and only if the region above its graph (in green) is a convex set. | upload.wikimedia.org/wikipedia/commons/3/31/Epigraph///convex.svg]\n# [A graph of the bivariate convex function x2 + xy + y2. | upload.wikimedia.org/wikipedia/commons/6/6e/Grafico///3d///x2%2Bxy%2By2.png]\n# [Convex vs. Not convex | upload.wikimedia.org/wikipedia/commons/e/ec/Convex///vs.///Not-convex.jpg]\n\n# Definition\nLet X be a [convex subset](https://bliptext.com/articles/convex-set) of a real [vector space](https://bliptext.com/articles/vector-space) and let f: X \\to \\R be a function.\n****T****h****e****n**** ****f**** ****i****s**** ****c****a****l****l****e****d**** **** ****i****f**** ****a****n****d**** ****o****n****l****y**** ****i****f**** ****a****n****y**** ****o****f**** ****t****h****e**** ****f****o****l****l****o****w****i****n****g**** ****e****q****u****i****v****a****l****e****n****t**** ****c****o****n****d****i****t****i****o****n****s**** ****h****o****l****d****:****\n For all and all :\nThe right hand side represents the straight line between and 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.\nSimilarly, 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.\n \n For all 0 \n \n****T****h****e**** ****s****e****c****o****n****d**** ****s****t****a****t****e****m****e****n****t**** ****c****h****a****r****a****c****t****e****r****i****z****i****n****g**** ****c****o****n****v****e****x**** ****f****u****n****c****t****i****o****n****s**** ****t****h****a****t**** ****a****r****e**** ****v****a****l****u****e****d**** ****i****n**** ****t****h****e**** ****r****e****a****l**** ****l****i****n****e**** ****\\****R**** ****i****s**** ****a****l****s****o**** ****t****h****e**** ****s****t****a****t****e****m****e****n****t**** ****u****s****e****d**** ****t****o**** ****d****e****f****i****n****e**** **** ****t****h****a****t**** ****a****r****e**** ****v****a****l****u****e****d**** ****i****n**** ****t****h****e**** ****e****x****t****e****n****d****e****d**** ****r****e****a****l**** ****n****u****m****b****e****r**** ****l****i****n****e**** ****w****h****e****r****e**** ****s****u****c****h**** ****a**** ****f****u****n****c****t****i****o****n**** ****f**** ****i****s**** ****a****l****l****o****w****e****d**** ****t****o**** ****t****a****k****e**** ****\\****p****m****\\****i****n****f****t****y**** ****a****s**** ****a**** ****v****a****l****u****e****.****\nThe first statement is not used because it permits t to take 0 or 1 as a value, in which case, if or respectively, then would be undefined (because the multiplications and are undefined).\nThe sum 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.\nThe second statement can also be modified to get the definition of, where the latter is obtained by replacing \\,\\leq\\, with the strict inequality \\, 0 if the following inequality holds for all points x, y in its domain:\nor, more generally,\nwhere is any [inner product](https://bliptext.com/articles/inner-product), and \\|\\cdot\\| is the corresponding [norm](https://bliptext.com/articles/norm-mathematics).\nSome authors, such as refer to functions satisfying this inequality as [elliptic](https://bliptext.com/articles/elliptic-operator) functions.\nAn equivalent condition is the following:\nIt is not necessary for a function to be differentiable in order to be strongly convex.\nA third definition for a strongly convex function, with parameter m, is that, for all x, y in the domain and\nNotice 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.\nDespite this, functions exist that are strictly convex but are not strongly convex for any m > 0 (see example below).\nIf the function f is twice continuously differentiable, then it is strongly convex with parameter m if and only if for all x in the domain, where I is the identity and \\nabla^2f is the [Hessian matrix](https://bliptext.com/articles/hessian-matrix), and the inequality \\succeq means that is [positive semi-definite](https://bliptext.com/articles/positive-definite-matrix).\nThis is equivalent to requiring that the minimum [eigenvalue](https://bliptext.com/articles/eigenvalue) of be at least m for all x. If the domain is just the real line, then is just the second derivative f''(x), so the condition becomes.\nIf m = 0 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.\nAssuming still that the function is twice continuously differentiable, one can show that the lower bound of implies that it is strongly convex.\nUsing [Taylor's Theorem](https://bliptext.com/articles/taylor-s-theorem) there exists\nsuch that\nThen\nby the assumption about the eigenvalues, and hence we recover the second strong convexity equation above.\nA function f is strongly convex with parameter m if and only if the function\nis convex.\nA twice continuously differentiable function f on a compact domain X that satisfies f''(x)>0 for all x\\in X is strongly convex.\nThe proof of this statement follows from the [extreme value theorem](https://bliptext.com/articles/extreme-value-theorem), which states that a continuous function on a compact set has a maximum and minimum.\nStrongly convex functions are in general easier to work with than convex or strictly convex functions, since they are a smaller class.\nLike strictly convex functions, strongly convex functions have unique minima on compact sets.\n\n## Properties of strongly-convex functions\nIf f is a strongly-convex function with parameter m, then:\n\n# Uniformly convex functions\nA uniformly convex function, with modulus \\phi, is a function f that, for all x, y in the domain and satisfies\nwhere \\phi is a function that is non-negative and vanishes only at 0.\nThis is a generalization of the concept of strongly convex function; by taking we recover the definition of strong convexity.\nIt is worth noting that some authors require the modulus \\phi to be an increasing function, but this condition is not required by all authors.\n\n# Examples\n\n## Functions of one variable\n\n## Functions of n variables","revisionCount":0},"hasMore":false,"nextCursor":null}

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