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A generalized dual maximizer for the Monge–Kantorovich transport problem

Mathias Beiglböck, Christian Léonard, Walter Schachermayer (2012)

ESAIM: Probability and Statistics

The dual attainment of 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 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.

A generalized dual maximizer for the Monge–Kantorovich transport problem∗

Mathias Beiglböck, Christian Léonard, Walter Schachermayer (2012)

ESAIM: Probability and Statistics

The dual attainment of 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 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...

A "hidden" characterization of approximatively polyhedral convex sets in Banach spaces

Taras Banakh, Ivan Hetman (2012)

Studia Mathematica

A closed convex subset C of a Banach space X is called approximatively polyhedral if for each ε > 0 there is a polyhedral (= intersection of finitely many closed half-spaces) convex set P ⊂ X at Hausdorff distance < ε from C. We characterize approximatively polyhedral convex sets in Banach spaces and apply the characterization to show that a connected component of the space C o n v ( X ) of closed convex subsets of X endowed with the Hausdorff metric is separable if and only if contains a polyhedral convex...

A new algorithm for approximating the least concave majorant

Martin Franců, Ron Kerman, Gord Sinnamon (2017)

Czechoslovak Mathematical Journal

The least concave majorant, F ^ , of a continuous function F on a closed interval, I , is defined by F ^ ( x ) = inf { G ( x ) : G ≥ F , G concave } , x ∈ I . We present an algorithm, in the spirit of the Jarvis March, to approximate the least concave majorant of a differentiable piecewise polynomial function of degree at most three on I . Given any function F ∈ 𝒞 4 ( I ) , it can be well-approximated on I by a clamped cubic spline S . We show that S ^ is then a good approximation to F ^ . We give two examples, one to illustrate, the other to apply our algorithm.

A saddle-point approach to the Monge-Kantorovich optimal transport problem

Christian Léonard (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-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.

A saddle-point approach to the Monge-Kantorovich optimal transport problem

Christian Léonard (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The Monge-Kantorovich problem is revisited by means of a variant of the saddle-point method without appealing to c-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.

A two-stage stochastic optimization model for a gas sale retailer

F. Maggioni, Maria Teresa Vespucci, E. Allevi, Marida Bertocchi, M. Innorta (2008)

Kybernetika

The paper deals with a new stochastic optimization model, named OMoGaS–SV (Optimization Modelling for Gas Seller–Stochastic Version), to assist companies dealing with gas retail commercialization. Stochasticity is due to the dependence of consumptions on temperature uncertainty. Due to nonlinearities present in the objective function, the model can be classified as an NLP mixed integer model, with the profit function depending on the number of contracts with the final consumers, the typology of...

A viscosity-proximal gradient method with inertial extrapolation for solving certain minimization problems in Hilbert space

L.O. Jolaoso, H.A. Abass, O.T. Mewomo (2019)

Archivum Mathematicum

In this paper, we study the strong convergence of the proximal gradient algorithm with inertial extrapolation term for solving classical minimization problem and finding the fixed points of δ -demimetric mapping in a real Hilbert space. Our algorithm is inspired by the inertial proximal point algorithm and the viscosity approximation method of Moudafi. A strong convergence result is achieved in our result without necessarily imposing the summation condition ∑ n = 1 ∞ β n ∥ x n - 1 - x n ∥ < + ∞ on the inertial term. Finally, we provide...

Applications of the Fréchet subdifferential

Durea, M. (2003)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 46A30, 54C60, 90C26.In this paper we prove two results of nonsmooth analysis involving the Fréchet subdifferential. One of these results provides a necessary optimality condition for an optimization problem which arise naturally from a class of wide studied problems. In the second result we establish a sufficient condition for the metric regularity of a set-valued map without continuity assumptions.

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