The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to -conjugates. A new abstract characterization of the optimal plans is obtained in the case where the cost function takes infinite values. It leads us to new explicit sufficient and necessary optimality conditions. As by-products, we obtain a new proof of the well-known Kantorovich dual equality and an improvement of the convergence of the minimizing sequences.
The Monge-Kantorovich problem is revisited by means of a variant
of the saddle-point method without appealing to -conjugates. A
new abstract characterization of the optimal plans is obtained in
the case where the cost function takes infinite values. It leads
us to new explicit sufficient and necessary optimality conditions.
As by-products, we obtain a new proof of the well-known
Kantorovich dual equality and an improvement of the convergence of
the minimizing sequences.
Entropic projections and dominating points are solutions to convex
minimization problems related to conditional laws of large
numbers. They appear in many areas of applied mathematics such as
statistical physics, information theory, mathematical statistics,
ill-posed inverse problems or large deviation theory. By means of convex conjugate
duality and functional analysis, criteria are derived for the
existence of entropic projections, generalized entropic
projections and dominating points. Representations...
The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general setting. The spaces are assumed to be polish and equipped with Borel probability measures and . The transport cost function : × → [0,∞] is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel’s perturbation technique.
The duality theory for the Monge-Kantorovich transport problem is analyzed in a general setting. The spaces X,Y are assumed to be Polish and equipped with Borel probability measures μ and ν. The transport cost function c: X × Y → [0,∞] is assumed to be Borel. Our main result states that in this setting there is no duality gap provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport...
The dual attainment of the Monge–Kantorovich transport problem is analyzed in a general
setting. The spaces are assumed to be polish and equipped with Borel
probability measures and . The transport cost
function : × → [0,∞] is assumed
to be Borel measurable. We show that a dual optimizer always exists, provided we interpret
it as a projective limit of certain finitely additive measures. Our methods are functional
analytic and rely...
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