Fenchel-Moreau conjugate function

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)