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Given two sets ,
and a "coupling" function
, the Fenchel–Moreau conjugate to a function
with respect to the coupling function
is the function
defined by
![]() | (a1) |
with the convention [a1]. When
and
are linear spaces in duality, via a bilinear coupling function
(cf. also Linear space; Duality),
is just the usual Fenchel conjugate
(called also the Young–Fenchel conjugate, or Legendre–Fenchel conjugate; cf. also Legendre transform) of
. If
is a locally convex space and
the conjugate space of
, with the coupling function
, then the second Fenchel conjugate
of
coincides with the greatest lower semi-continuous minorant of
(Moreau's theorem); this result admits a natural extension to Fenchel–Moreau conjugates
.
Another important particular class of Fenchel–Moreau conjugates is obtained for coupling functions that take only the values
and
or, equivalently, the conjugates for which there exists a (unique) subset
of
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 , defined by
![]() | (a3) |
which has applications in, e.g., optimization theory.
A unified approach is the conjugate function with respect to a binary operation on
, assumed completely distributive (cf. also Completely distributive lattice) with respect to
in the lattice
, defined by
![]() | (a4) |
in particular, when (respectively,
),
is the Fenchel–Moreau (respectively, the Flachs–Pollatschek) conjugate function of
.
In another direction, the Fenchel–Moreau conjugate has been generalized to functions with values in extensions of ordered groups
, with applications to functions in the extension (by adjoining
and
) of the additive group
and to functions in the extension (by adjoining
and
) of the multiplicative group
. More generally, one has also defined the conjugate function of
with respect to a binary operation
on
, encompassing the preceding conjugates as particular cases.
One of the main fields of applications of these concepts is optimization theory: When 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) |
Fenchel-Moreau conjugate function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fenchel-Moreau_conjugate_function&oldid=15994