Namespaces
Variants
Actions

Difference between revisions of "Fenchel-Moreau conjugate function"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (Automatically changed introduction)
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
Given two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200401.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200402.png" /> and a  "coupling"  function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200403.png" />, the Fenchel–Moreau conjugate to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200404.png" /> with respect to the coupling function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200405.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200406.png" /> defined by
+
<!--This article has been texified automatically. Since there was no Nroff source code for this article,  
 +
the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
 +
was used.
 +
If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200407.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
Out of 52 formulas, 49 were replaced by TEX code.-->
  
with the convention <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200408.png" /> [[#References|[a1]]]. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f1200409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004010.png" /> are linear spaces in duality, via a bilinear coupling function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004011.png" /> (cf. also [[Linear space|Linear space]]; [[Duality|Duality]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004012.png" /> is just the usual Fenchel conjugate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004013.png" /> (called also the Young–Fenchel conjugate, or Legendre–Fenchel conjugate; cf. also [[Legendre transform|Legendre transform]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004014.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004015.png" /> is a [[Locally convex space|locally convex space]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004016.png" /> the conjugate space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004017.png" />, with the coupling function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004018.png" />, then the second Fenchel conjugate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004019.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004020.png" /> coincides with the greatest lower semi-continuous minorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004021.png" /> (Moreau's theorem); this result admits a natural extension to Fenchel–Moreau conjugates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004022.png" />.
+
{{TEX|semi-auto}}{{TEX|part}}
 +
Given two sets $X$, $W$ and "coupling" function $\varphi : X \times W \rightarrow \overline { \mathbf{R} }$, the Fenchel–Moreau conjugate to a function $f : X \rightarrow \overline { \mathbf{R} }$ with respect to the coupling function $\varphi$ is the function $f ^ { c \langle \varphi \rangle } : W \rightarrow \overline { \mathbf{R} }$ defined by
  
Another important particular class of Fenchel–Moreau conjugates is obtained for coupling functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004023.png" /> that take only the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004025.png" /> or, equivalently, the conjugates for which there exists a (unique) subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004026.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004027.png" /> such that
+
\begin{equation} \tag{a1} f ^ { c ( \varphi ) } ( w ) = \operatorname { sup } _ { x \in X } \{ \varphi ( x , w ) - f ( x ) \} ( w \in W ), \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
with the convention $( + \infty ) - ( + \infty ) = - \infty - ( - \infty ) = - \infty$ [[#References|[a1]]]. When $X$ and $W$ are linear spaces in duality, via a bilinear coupling function $\varphi$ (cf. also [[Linear space|Linear space]]; [[Duality|Duality]]), $f ^ { c  ( \varphi )}$ is just the usual Fenchel conjugate $f ^ { * }$ (called also the Young–Fenchel conjugate, or Legendre–Fenchel conjugate; cf. also [[Legendre transform|Legendre transform]]) of $f$. If $X$ is a [[Locally convex space|locally convex space]] and $W = X ^ { * },$ the conjugate space of $X$, with the coupling function $\varphi ( x , w ) = w ( x )$, then the second Fenchel conjugate $f ^ { * * } = ( f ^ { * } ) ^ { * }$ of $f$ coincides with the greatest lower semi-continuous minorant of $f$ (Moreau's theorem); this result admits a natural extension to Fenchel–Moreau conjugates $f ^ { c  ( \varphi )}$.
 +
 
 +
Another important particular class of Fenchel–Moreau conjugates is obtained for coupling functions $\varphi : X \times W \rightarrow \overline { \mathbf{R} }$ that take only the values $0$ and $- \infty,$ or, equivalently, the conjugates for which there exists a (unique) subset $\Omega$ of $X \times W$ such that
 +
 
 +
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004028.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a2)</td></tr></table>
  
 
these are called conjugates of type Lau or level-set conjugates. While Fenchel conjugates have many applications in [[Convex analysis|convex analysis]], conjugates of type Lau are useful for the study of quasi-convex functions (i.e., of functions all of whose level sets are convex) and for duality theory in micro-economics (duality between direct and indirect utility functions).
 
these are called conjugates of type Lau or level-set conjugates. While Fenchel conjugates have many applications in [[Convex analysis|convex analysis]], conjugates of type Lau are useful for the study of quasi-convex functions (i.e., of functions all of whose level sets are convex) and for duality theory in micro-economics (duality between direct and indirect utility functions).
  
A useful related concept is the Flachs–Pollatschek conjugate function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004029.png" />, defined by
+
A useful related concept is the Flachs–Pollatschek conjugate function $f ^ { \Delta ( \varphi ) } : W \rightarrow \overline {\bf R }$, defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
\begin{equation} \tag{a3} f ^ { \Delta ( \varphi ) } ( w ) = \operatorname { sup } _ { x \in X } \operatorname { min } \{ \varphi ( x , w ) , - f ( x ) \} ( w \in W ), \end{equation}
  
 
which has applications in, e.g., optimization theory.
 
which has applications in, e.g., optimization theory.
  
A unified approach is the conjugate function with respect to a binary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004032.png" />, assumed completely distributive (cf. also [[Completely distributive lattice|Completely distributive lattice]]) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004033.png" /> in the [[Lattice|lattice]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004034.png" />, defined by
+
A unified approach is the conjugate function with respect to a binary operation $\odot$ on $\overline{\mathbf{R}}$, assumed completely distributive (cf. also [[Completely distributive lattice|Completely distributive lattice]]) with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004033.png"/> in the [[Lattice|lattice]] $( \overline { \mathbf{R} } , \leq )$, defined by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} f ^ { b ( \varphi ) } ( w ) = \operatorname { sup } _ { x \in X } \{ - [ - \varphi ( x , w ) \odot f ( x ) ] \} ( w \in W ); \end{equation}
  
in particular, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004036.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004037.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004038.png" /> is the Fenchel–Moreau (respectively, the Flachs–Pollatschek) conjugate function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004039.png" />.
+
in particular, when $\odot = +$ (respectively, $\odot=\max$), $f ^ { b ( \varphi ) }$ is the Fenchel–Moreau (respectively, the Flachs–Pollatschek) conjugate function of $f$.
  
In another direction, the Fenchel–Moreau conjugate has been generalized to functions with values in extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004040.png" /> of ordered groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004041.png" />, with applications to functions in the extension (by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004043.png" />) of the additive group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004044.png" /> and to functions in the extension (by adjoining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004046.png" />) of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004047.png" />. More generally, one has also defined the conjugate function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004048.png" /> with respect to a binary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004050.png" />, encompassing the preceding conjugates as particular cases.
+
In another direction, the Fenchel–Moreau conjugate has been generalized to functions with values in extensions $\overline { G }$ of ordered groups $G$, with applications to functions in the extension (by adjoining $- \infty$ and $+ \infty$) of the additive group $( \mathbf{R} , + , \leq )$ and to functions in the extension (by adjoining $0$ and $+ \infty$) of the multiplicative group $( \mathbf{R} _ { + } \backslash \{ 0 \} , \times , \leq )$. More generally, one has also defined the conjugate function of $f : X \rightarrow \overline { G }$ with respect to a binary operation $\odot$ on $\overline { G }$, encompassing the preceding conjugates as particular cases.
  
One of the main fields of applications of these concepts is optimization theory: When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120040/f12004051.png" /> is the [[Objective function|objective function]] of an optimization problem, a conjugate function is used to define (the objective function of) a  "dual"  optimization problem.
+
One of the main fields of applications of these concepts is optimization theory: When $f$ is the [[Objective function|objective function]] of an optimization problem, a conjugate function is used to define (the objective function of) a  "dual"  optimization problem.
  
 
For more details, see [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]].
 
For more details, see [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]].
Line 32: Line 40:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.-J. Moreau,  "Fonctions convexes en dualité" , Univ. Montpellier  (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Flachs,  M.A. Pollatschek,  "Duality theorems for certain programs involving minimum or maximum operations"  ''Math. Progr.'' , '''16'''  (1979)  pp. 348–370</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.E. Diewert,  "Duality approaches to microeconomic theory"  K.J. Arrow (ed.)  M.D. Intrilligator (ed.) , ''Handbook of Mathematical Economics'' , '''2''' , North-Holland  (1982)  pp. 535–599</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I. Singer,  "Abstract convex analysis" , Wiley–Interscience  (1997)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  J.-J. Moreau,  "Fonctions convexes en dualité" , Univ. Montpellier  (1962)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Flachs,  M.A. Pollatschek,  "Duality theorems for certain programs involving minimum or maximum operations"  ''Math. Progr.'' , '''16'''  (1979)  pp. 348–370</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  W.E. Diewert,  "Duality approaches to microeconomic theory"  K.J. Arrow (ed.)  M.D. Intrilligator (ed.) , ''Handbook of Mathematical Economics'' , '''2''' , North-Holland  (1982)  pp. 535–599</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  I. Singer,  "Abstract convex analysis" , Wiley–Interscience  (1997)</td></tr></table>

Latest revision as of 17:44, 1 July 2020

Given two sets $X$, $W$ and a "coupling" function $\varphi : X \times W \rightarrow \overline { \mathbf{R} }$, the Fenchel–Moreau conjugate to a function $f : X \rightarrow \overline { \mathbf{R} }$ with respect to the coupling function $\varphi$ is the function $f ^ { c \langle \varphi \rangle } : W \rightarrow \overline { \mathbf{R} }$ defined by

\begin{equation} \tag{a1} f ^ { c ( \varphi ) } ( w ) = \operatorname { sup } _ { x \in X } \{ \varphi ( x , w ) - f ( x ) \} ( w \in W ), \end{equation}

with the convention $( + \infty ) - ( + \infty ) = - \infty - ( - \infty ) = - \infty$ [a1]. When $X$ and $W$ are linear spaces in duality, via a bilinear coupling function $\varphi$ (cf. also Linear space; Duality), $f ^ { c ( \varphi )}$ is just the usual Fenchel conjugate $f ^ { * }$ (called also the Young–Fenchel conjugate, or Legendre–Fenchel conjugate; cf. also Legendre transform) of $f$. If $X$ is a locally convex space and $W = X ^ { * },$ the conjugate space of $X$, with the coupling function $\varphi ( x , w ) = w ( x )$, then the second Fenchel conjugate $f ^ { * * } = ( f ^ { * } ) ^ { * }$ of $f$ coincides with the greatest lower semi-continuous minorant of $f$ (Moreau's theorem); this result admits a natural extension to Fenchel–Moreau conjugates $f ^ { c ( \varphi )}$.

Another important particular class of Fenchel–Moreau conjugates is obtained for coupling functions $\varphi : X \times W \rightarrow \overline { \mathbf{R} }$ that take only the values $0$ and $- \infty,$ or, equivalently, the conjugates for which there exists a (unique) subset $\Omega$ of $X \times W$ such that

(a2)

these are called conjugates of type Lau or level-set conjugates. While Fenchel conjugates have many applications in convex analysis, conjugates of type Lau are useful for the study of quasi-convex functions (i.e., of functions all of whose level sets are convex) and for duality theory in micro-economics (duality between direct and indirect utility functions).

A useful related concept is the Flachs–Pollatschek conjugate function $f ^ { \Delta ( \varphi ) } : W \rightarrow \overline {\bf R }$, defined by

\begin{equation} \tag{a3} f ^ { \Delta ( \varphi ) } ( w ) = \operatorname { sup } _ { x \in X } \operatorname { min } \{ \varphi ( x , w ) , - f ( x ) \} ( w \in W ), \end{equation}

which has applications in, e.g., optimization theory.

A unified approach is the conjugate function with respect to a binary operation $\odot$ on $\overline{\mathbf{R}}$, assumed completely distributive (cf. also Completely distributive lattice) with respect to in the lattice $( \overline { \mathbf{R} } , \leq )$, defined by

\begin{equation} \tag{a4} f ^ { b ( \varphi ) } ( w ) = \operatorname { sup } _ { x \in X } \{ - [ - \varphi ( x , w ) \odot f ( x ) ] \} ( w \in W ); \end{equation}

in particular, when $\odot = +$ (respectively, $\odot=\max$), $f ^ { b ( \varphi ) }$ is the Fenchel–Moreau (respectively, the Flachs–Pollatschek) conjugate function of $f$.

In another direction, the Fenchel–Moreau conjugate has been generalized to functions with values in extensions $\overline { G }$ of ordered groups $G$, with applications to functions in the extension (by adjoining $- \infty$ and $+ \infty$) of the additive group $( \mathbf{R} , + , \leq )$ and to functions in the extension (by adjoining $0$ and $+ \infty$) of the multiplicative group $( \mathbf{R} _ { + } \backslash \{ 0 \} , \times , \leq )$. More generally, one has also defined the conjugate function of $f : X \rightarrow \overline { G }$ with respect to a binary operation $\odot$ on $\overline { G }$, encompassing the preceding conjugates as particular cases.

One of the main fields of applications of these concepts is optimization theory: When $f$ is the objective function of an optimization problem, a conjugate function is used to define (the objective function of) a "dual" optimization problem.

For more details, see [a2], [a3], [a4].

See also Conjugate function; Dual functions.

References

[a1] J.-J. Moreau, "Fonctions convexes en dualité" , Univ. Montpellier (1962)
[a2] J. Flachs, M.A. Pollatschek, "Duality theorems for certain programs involving minimum or maximum operations" Math. Progr. , 16 (1979) pp. 348–370
[a3] W.E. Diewert, "Duality approaches to microeconomic theory" K.J. Arrow (ed.) M.D. Intrilligator (ed.) , Handbook of Mathematical Economics , 2 , North-Holland (1982) pp. 535–599
[a4] I. Singer, "Abstract convex analysis" , Wiley–Interscience (1997)
How to Cite This Entry:
Fenchel-Moreau conjugate function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fenchel-Moreau_conjugate_function&oldid=22415
This article was adapted from an original article by Ivan Singer (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article