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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 $( + \infty ) - ( + \infty ) = - \infty - ( - \infty ) = - \infty$ [[#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).

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|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

\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|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]]].

See also [[Conjugate function|Conjugate function]]; [[Dual functions|Dual functions]].

====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>