Appl Math Optim 44:273–297 (2001)
2001 Springer-Verlag New York Inc.
Minimization of Energy Functionals Applied to
Some Inverse Problems
Equipe Modal-X, Universit´e de Paris X,
Bˆatiment G. 200, Avenue de la R´epublique, 92001 Nanterre Cedex, France
Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique,
91128 Palaiseau Cedex, France
Abstract. We consider a general class of problems of the minimization of convex
integral functionals subject to linear constraints. Using Fenchel duality, we prove the
equality of the values of the minimization problem and its associated dual problem.
This equality is a variational criterion for the existence of a solution to a large class
of inverse problems entering the class of generalized Fredholm integral equations.
In particular, our abstract results are applied to marginal problems for stochastic pro-
cesses. Such problems naturally arise from the probabilistic approaches to quantum
Key Words. Fenchel duality, Maximum entropy method, Convex integral func-
tionals, Fredholm integral equations, Marginal problems.
AMS Classiﬁcation. 52A41, 45B05, 45N05.
We consider the energy functionals deﬁned on the space M() of the signed measures
on the measure space (, A) which are of the following form:
I (Q) =
dR ∈ [0, +∞], Q ∈ M(),
if Q is absolutely continuous with respect to a given nonnegative reference measure R,
and I (Q) =+∞otherwise. The function γ
: R → [0, ∞] is the convex conjugate of