[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/conjugated-function-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/conjugated-function-wikipedia\/","headline":"Conjugated function – Wikipedia","name":"Conjugated function – Wikipedia","description":"before-content-x4 In mathematics, and more precisely in convex analysis, the conjugated function is a function built from a real function","datePublished":"2018-07-01","dateModified":"2018-07-01","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/132e57acb643253e7810ee9702d9581f159a1c61","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/132e57acb643253e7810ee9702d9581f159a1c61","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/conjugated-function-wikipedia\/","wordCount":17854,"articleBody":" (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4In mathematics, and more precisely in convex analysis, the conjugated function is a function built from a real function f {displaystyle f} (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Defined on a vector space AND {displaystyle mathbb {E} } , which is useful: The combined function of (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4f {displaystyle f} is most often noted f \u2217 {displaystyle f^{*}} . It is a convex function, even if f {displaystyle f} is not, defined on slopes , that is to say on the elements of the dual vector space of AND {displaystyle mathbb {E} } . The definition is motivated and specified below. (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4L’application f \u2192 f \u2217 {displaystyle fto f^{*}} is called transformation de Fenchel or Legendre transformation or Transformation de Legendre-Fnechel , the APRIIIIr Adrien-Mary Legendrere it Werner Fenchel. Supposed knowledge : linear algebra, differential calculation, the bases of convex analysis (in particular the main concepts attached to convex sets and functions); The subdifferential of a convex function is only used to motivate the definition of combined function. The apparent complexity of the concept of combined function requires motivating its definition. Table of ContentsFunction convexification [ modifier | Modifier and code ] Access route to the subdifferential [ modifier | Modifier and code ] The combined and its properties [ modifier | Modifier and code ] The biconjugated and its properties [ modifier | Modifier and code ] Conjugated rules for calculating [ modifier | Modifier and code ] Inf-image under a linear application [ modifier | Modifier and code ] Pre-composition with a linear application [ modifier | Modifier and code ] Examples [ modifier | Modifier and code ] Rules [ modifier | Modifier and code ] Distance to a convex [ modifier | Modifier and code ] Maximum proper value [ modifier | Modifier and code ] Elements of history [ modifier | Modifier and code ] Related articles [ modifier | Modifier and code ] Bibliography [ modifier | Modifier and code ] Function convexification [ modifier | Modifier and code ] By definition, a function is convex if its epigraph is convex. Convex a function f : AND \u2192 R\u00af{displaystyle f:mathbb {E} to {bar {mathbb {R} }}} consists in determining the largest convex function, let’s say closed, reduced f {displaystyle f} . In terms of epigraph, this amounts to finding the smallest closed convex containing the epigraph of f {displaystyle f} , that is to say to take the closed convex envelope of the epigraph and f \u2282 AND \u00d7 R . {displaystyle operatorname {epi} ,fsubset mathbb {E} times mathbb {R} .} Like any closed convex envelope, that of and f {displaystyle operatorname {epi} ,f} is the intersection of every closed half-space containing and f {displaystyle operatorname {epi} ,f} , that is, sets of the form H \u2212( X , T , t ) : = { ( x , a ) \u2208 AND \u00d7 R : \u27e8 X , x \u27e9 + T a \u2a7d t } , {displaystyle H^{-}(xi ,tau ,t):={(x,alpha )in mathbb {E} times mathbb {R} :langle xi ,xrangle +tau alpha leqslant t},} Or ( X , T , t ) \u2208 AND \u00d7 R \u00d7 R {displaystyle (xi ,tau ,t)in mathbb {E} times mathbb {R} times mathbb {R} } . It is easy to see that when H – ( X , T , t ) {Displaystyle h^{-} (xi, tau, t)} contains an epigraph, we must have T \u2a7d 0 {displaystyle tau leqslant 0} . It is more thin to show that, in the expression of the closed convex envelope of and f {displaystyle operatorname {epi} ,f} as an intersection of H – ( X , T , t ) {Displaystyle h^{-} (xi, tau, t)} , we can only keep half-spaces with T < 0 {displaystyle tau \u03be\u03c4,x\u27e9 + t\u03c4. {displaystyle xmapsto leftlangle {frac {-xi }{tau }},xrightrangle +{frac {t}{tau }}.} Summons. As you would expect, the closed convenience of f {displaystyle f} is the upper envelope of all the Minurant Affines of f {displaystyle f} . Parmi les diminishing affines x \u21a6 \u27e8 s , x \u27e9 + a {displaystyle xmapsto langle s,xrangle +alpha } having a slope s \u2208 AND {displaystyle sin mathbb {E} } fixed, one can also keep only the one that is the highest, that is to say the one that has the greatest a {displaystyle alpha } . We must therefore determine the greatest a {displaystyle alpha } such as \u2200 x \u2208 AND : \u27e8 s , x \u27e9 + a \u2a7d f ( x ) ou\u27e8 s , x \u27e9 – f ( x ) \u2a7d – a . {displaystyle forall ,xin mathbb {E} ,:quad langle s,xrangle +alpha leqslant f(x)quad {mbox{ou}}quad langle s,xrangle -f(x)leqslant -alpha .} We clearly see that the smallest value of – a {displaystyle -alpha } is given by f \u2217( s ) : = sup x\u2208E(\u27e8 s , x \u27e9 – f ( x ) ). {displaystyle f^{*}(s):=sup _{xin mathbb {E} }{Bigl (}langle s,xrangle -f(x){Bigr )}.} It is the value of the combination of f {displaystyle f} in s {displaystyle s} . We are interested in – a {displaystyle -alpha } rather than a {displaystyle alpha } to define f \u2217 {displaystyle f^{*}} in order to obtain a conjugated conjugated function. Access route to the subdifferential [ modifier | Modifier and code ] In this section, we show the interest of the concept of combined function as a tool to calculate the subdifferential of a convex function. The subdifferential \u2202 f ( x ) {displaystyle partial f(x)} of a convex function f {displaystyle f} in x {displaystyle x} is the set of slopes of minorating the affines of f {displaystyle f} (i.e., des fonctions affines that less f {displaystyle f} ), which are accurate in x {displaystyle x} (i.e., who have the same value as f {displaystyle f} in x {displaystyle x} ). For x {displaystyle x} given, it is not always easy to specify all the minor affins exact in x {displaystyle x} . It is sometimes easier to give yourself a slope s {displaystyle s} of Minurant Affine x \u21a6 \u27e8 s , x \u27e9 + a {displaystyle xmapsto langle s,xrangle +alpha } and seek the points it may be exact by modifying a {displaystyle alpha } . From this point of view, we are looking for the largest a {displaystyle alpha } such as \u2200 x \u2208 AND : \u27e8 s , x \u27e9 + a \u2a7d f ( x ) ou\u27e8 s , x \u27e9 – f ( x ) \u2a7d – a . {displaystyle forall ,xin mathbb {E} ,:quad langle s,xrangle +alpha leqslant f(x)quad {mbox{ou}}quad langle s,xrangle -f(x)leqslant -alpha .} We clearly see that the smallest value of – a {displaystyle -alpha } is given by f \u2217( s ) : = sup x\u2208E(\u27e8 s , x \u27e9 – f ( x ) ). {displaystyle f^{*}(s):=sup _{xin mathbb {E} }{Bigl (}langle s,xrangle -f(x){Bigr )}.} It is the value of the combination of f {displaystyle f} in s {displaystyle s} . We are interested in – a {displaystyle -alpha } rather than a {displaystyle alpha } to define f \u2217 {displaystyle f^{*}} in order to obtain a conjugated conjugated function. Let’s get back to the problem we were asking at the start of this paragraph: if x {displaystyle x} is solution of the maximization problem above, then \u27e8 s , x \u27e9 – f ( x ) \u2a7e \u27e8 s , and \u27e9 – f ( and ) {DisplayStyle Langle S, Xrangle -F (x) Geqslant Langle S, Yrangle -F (Y)} for everything and \u2208 AND {displaystyle yin mathbb {E} } or \u2200 and \u2208 AND : f ( and ) \u2a7e f ( x ) + \u27e8 s , and – x \u27e9 . {displaystyle forall ,yin mathbb {E} ,:quad f(y)geqslant f(x)+langle s,y-xrangle .} These inequalities show that and \u21a6 f ( x ) + \u27e8 s , and – x \u27e9 {displaystyle ymapsto f(x)+langle s,y-xrangle } is a minor refine f {displaystyle f} , Exact and x {displaystyle x} . It is assumed in this section that the functions are defined in a euclidean vector space AND {displaystyle mathbb {E} } (of finished dimension therefore), whose scalar product is noted \u27e8 \u22c5 , \u22c5 \u27e9 {displaystyle langle cdot ,cdot rangle } and the associated standard \u2016 \u22c5 \u2016 {displaystyle |cdot |} . On note The combined and its properties [ modifier | Modifier and code ] We chose to designate by s {displaystyle s} l’Argment of f \u2217 {displaystyle f^{*}} to remember that this is a slope ( slope in English), that is to say an element of the dual space of AND {displaystyle mathbb {E} } , here identified with AND {displaystyle mathbb {E} } via the scalar product. We are interested below in the convexity and the clean and closed character of the conjugated f \u2217 {displaystyle f^{*}} . The convexity of f \u2217 {displaystyle f^{*}} is a remarkable property of the combined, since it is remembered that f {displaystyle f} is not necessarily convex. It will be remembered that a convex and clean function necessarily has an affine minor; It therefore checks the properties of point 1 above, so that f \u2208 Conv \u2061 ( AND ) \u27f9 f \u2217\u2208 C onv\u00af\u2061 ( AND ) . {displaystyle fin operatorname {Conv} (mathbb {E} )quad Longrightarrow quad f^{*}in operatorname {C{overline {onv}}} (mathbb {E} ).} The biconjugated and its properties [ modifier | Modifier and code ] We can of course apply the transformation of Legendre-Fenchel to the combined function f \u2217 {displaystyle f^{*}} ; We thus obtain the biconjugated of f {displaystyle f} , noted f \u2217 \u2217 {displaystyle f^{**}} . The convex, closed and clean character of the biconjugated is examined in the following result. Closed convex biconjugate – Regardless f : AND \u2192 R\u00af{displaystyle f:mathbb {E} to {bar {mathbb {R} }}} , Biconjuga\u00e9e f \u2217\u2217{displaystyle f^{**}} is convex and closed. In addition, the following properties are equivalent: f {displaystyle f} is clean and has a Minurant Affine, f\u2217\u2217\u2208 Conv\u00af\u2061 ( E) {displaystyle f^{**}in operatorname {C{overline {onv}}} (mathbb {E} )} . If the argument s {displaystyle s} of f \u2217 {displaystyle f^{*}} is a slope (identified with an element of AND {displaystyle mathbb {E} } ), l’argument x {displaystyle x} of f \u2217 \u2217 {displaystyle f^{**}} is in the starting space AND {displaystyle mathbb {E} } . We can then wonder if there is a link between f {displaystyle f} And f \u2217 \u2217 {displaystyle f^{**}} . The following proposal examines this question. We noted there f\u00af{displaystyle {bar {f}}} , the closure of f {displaystyle f} . This result makes it possible to compare the values \u200b\u200bof f \u2217 \u2217 {displaystyle f^{**}} And f {displaystyle f} . If we take the combined of the biconjugated, we find the conjugated. There is therefore no concept of triconjugu\u00e9e . No triciejugated – Either f : AND \u2192 R\u00af{displaystyle f:mathbb {E} to {bar {mathbb {R} }}} A clean function with a minor refine. SO ( f \u2217\u2217) \u2217= f \u2217{displaystyle (f^{**})^{*}=f^{*}} . Conjugated rules for calculating [ modifier | Modifier and code ] Inf-image under a linear application [ modifier | Modifier and code ] Let us recall the definition of Inf-image of a function under a linear application . We give ourselves two Euclidian spaces AND {displaystyle mathbb {E} } And F {displaystyle mathbb {F} } (we will need a scalar product here on AND {displaystyle mathbb {E} } And F {displaystyle mathbb {F} } , while this structure is not necessary in the definition of f \u22bb A {displaystyle fveebar A} ), a function f : AND \u2192 R\u00af{displaystyle f:mathbb {E} to {bar {mathbb {R} }}} and a linear application A : AND \u2192 F {displaystyle A:mathbb {E} to mathbb {F} } . Then the inf-image of f {displaystyle f} below A {displaystyle A} is the noted application f \u22bb A : F \u2192 R\u00af{displaystyle fveebar A:mathbb {F} to {bar {mathbb {R} }}} and defined in and \u2208 F {displaystyle yin mathbb {F} } about ( f \u22bb A ) ( and ) = inf x\u2208EAx=yf ( x ) . {displaystyle (fveebar A)(y)=inf _{{xin mathbb {E} } atop {Ax=y}};f(x).} Pre-composition with a linear application [ modifier | Modifier and code ] Examples [ modifier | Modifier and code ] Rules [ modifier | Modifier and code ] Either f : AND \u2192 R : x \u21a6 \u2016 x \u2016 {displaystyle f:mathbb {E} to mathbb {R} :xmapsto |x|} A standard on a Euclidean space AND {displaystyle mathbb {E} } , not necessarily derived from the scalar product \u27e8 \u22c5 , \u22c5 \u27e9 {displaystyle langle cdot ,cdot rangle } of AND {displaystyle mathbb {E} } . We introduce the dual standard \u2016 s \u2016 D: = sup \u2016x\u2016\u2a7d1\u27e8 s , x \u27e9 {displaystyle |s|_{_{D}}:=sup _{|x|leqslant 1};langle s,xrangle } and the closed duality ball B\u00afD: = { s \u2208 AND : \u2016 s \u2016 D\u2a7d first } . {Displaystyle {bar {b}} _ {_ {d}}: = {sin mathbb {e}: | S | _ {_ {d}} ledlant 1}.} The combined function f \u2217 : AND \u2192 R {displaystyle f^{*}:mathbb {E} to mathbb {R} } of f {displaystyle f} is the indicator of the dual unit ball; So she takes in s \u2208 AND {displaystyle sin mathbb {E} } the following value f \u2217 ( s ) = IB\u00afD( s ) : = { 0si\u2016s\u2016D\u22641+\u221esinon.{displaystyle f^{*}(s)={mathcal {I}}_{{bar {B}}_{_{D}}}(s):=left{{begin{array}{lll}0&{mbox{si}}&|s|_{_{D}}leq 1\\+infty &{mbox{sinon}}.end{array}}right.} Now consider the power 1″>standard f : AND \u2192 R : x \u21a6 1p\u2016 x \u2016 p. {displaystyle f:mathbb {E} to mathbb {R} :xmapsto {frac {1}{p}},|x|^{p}.} Its conjugated function f \u2217 : AND \u2192 R {displaystyle f^{*}:mathbb {E} to mathbb {R} } Take in s \u2208 AND {displaystyle sin mathbb {E} } the following value f \u2217 ( s ) = first p\u2032\u2016 s \u2016 Dp\u2032, {displaystyle f^{*}(s)={frac {1}{p’}},|s|_{_{D}}^{p’},} Or "},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/all2en\/wiki32\/conjugated-function-wikipedia\/#breadcrumbitem","name":"Conjugated function – Wikipedia"}}]}]